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Baudhayana

Baudhayana, the traditional author of the Sutra, originally belonged to the Kanva school of the White Yajurveda.

Born in 8th to 7th centuries BCE.

Baudhāyana Sulbasūtra

Pythagorean theorem

It is also referred to as Baudhayana theorem. The most notable of the rules (the Sulbasūtra-s do not contain any proofs for the rules which they describe, since they are sūtra-s, formulae, concise) in the Baudhāyana Sulba Sūtra says:

दीर्घचतुरश्रस्याक्ष्णया रज्जु: पार्श्र्वमानी तिर्यग् मानी च यत् पृथग् भूते कुरूतस्तदुभयं करोति ॥

dīrghachatursrasyākṣaṇayā rajjuḥ pārśvamānī, tiryagmānī,
cha yatpṛthagbhūte kurutastadubhayāṅ karoti.
A rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together.

The lines are to be referring to a rectangle, although some interpretations consider this to refer to a square. In either case, it states that the square of the hypotenuse equals the sum of the squares of the sides. If restricted to right-angled isosceles triangles, however, it would constitute a less general claim, but the text seems to be quite open to unequal sides.

If this refers to a rectangle, it is the earliest recorded statement of the Pythagorean theorem.

Baudhāyana also provides a non-axiomatic demonstration using a rope measure of the reduced form of the Pythagorean theorem for an isosceles right triangle:

The cord which is stretched across a square produces an area double the size of the original square.

Sequences of Pythagorean triples used in cryptography as random sequences and for the generation of keys have been dubbed "Baudhayana sequences" in a 2014 paper.

Circling the square

Another problem tackled by Baudhāyana is that of finding a circle whose area is the same as that of a square (the reverse of squaring the circle). His sūtra i.58 gives this construction:

Draw half its diagonal about the centre towards the East-West line; then describe a circle together with a third part of that which lies outside the square.

Explanation:

Draw the half-diagonal of the square, which is larger than the half-side by $x={a \over 2}{\sqrt {2}}-{a \over 2}$ .
Then draw a circle with radius ${a \over 2}+{x \over 3}$ , or ${a \over 2}+{a \over 6}({\sqrt {2}}-1)$ , which equals ${a \over 6}(2+{\sqrt {2}})$ .
Now $(2+{\sqrt {2}})^{2}\approx 11.66\approx {36.6 \over \pi }$ , so the area ${\pi }r^{2}\approx \pi \times {a^{2} \over 6^{2}}\times {36.6 \over \pi }\approx a^{2}$ .

Square root of 2

Baudhāyana i.61-2 (elaborated in Āpastamba Sulbasūtra i.6) gives the length of the diagonal of a square in terms of its sides, which is equivalent to a formula for the square root of 2:

samasya dvikaraṇī. pramāṇaṃ tṛtīyena vardhayet
tac caturthenātmacatustriṃśonena saviśeṣaḥThe diagonal [lit. "doubler"] of a square. The measure is to be increased by a third and by a fourth decreased by the 34th. That is its diagonal approximately.

That is,

${\sqrt {2}}\approx 1+{\frac {1}{3}}+{\frac {1}{3\cdot 4}}-{\frac {1}{3\cdot 4\cdot 34}}={\frac {577}{408}}\approx 1.414216,$

which is correct to five decimals.[8]

Other theorems include: diagonals of rectangle bisect each other, diagonals of rhombus bisect at right angles, area of a square formed by joining the middle points of a square is half of original, the midpoints of a rectangle joined forms a rhombus whose area is half the rectangle, etc.

Note the emphasis on rectangles and squares; this arises from the need to specify yajña bhūmikās—i.e. the altar on which a rituals were conducted, including fire offerings (yajña). This is an aspect of Vaastu Shastras and Shilpa Shastras. These theroms are derived from those texts.

Kātyāyana(c. 3rd century BC)

was a Sanskrit grammarian, mathematician and Vedic priest who lived in ancient India.

Works

He is known for two works:

The Vārttikakāra, an elaboration on Pāṇini grammar. Along with the Mahābhāṣya of Patañjali, this text became a core part of the Vyākaraṇa (grammar) canon. This was one of the six Vedangas, and constituted compulsory education for students in the following twelve centuries.
He also composed one of the later Śulbasūtras, a series of nine texts on the geometry of altar constructions, dealing with rectangles, right-sided triangles, rhombuses, etc.[1]

Views

Kātyāyana's views on the sentence-meaning connection tended towards naturalism. Kātyāyana believed, that the word-meaning relationship was not a result of human convention. For Kātyāyana, word-meaning relations were siddha, given to us, eternal. Though the object a word is referring to is non-eternal, the substance of its meaning, like a lump of gold used to make different ornaments, remains undistorted, and is therefore permanent.

Realizing that each word represented a categorization, he came up with the following conundrum (following Bimal Krishna Matilal, wikipedia):

"If the 'basis' for the use of the word 'cow' is cowhood (a universal) what would be the 'basis' for the use of the word 'cowhood'?

Clearly, this leads to infinite regress. Kātyāyana's solution to this was to restrict the universal category to that of the word itself — the basis for the use of any word is to be the very same word-universal itself."

This view may have been the nucleus of the Sphoṭa doctrine enunciated by Bhartṛhari in the 5th century, in which he elaborates the word-universal as the superposition of two structures — the meaning-universal or the semantic structure (artha-jāti) is superposed on the sound-universal or the phonological structure (śabda-jāti).

In the tradition of scholars like Pingala, Kātyāyana was also interested in mathematics. Here his text on the sulvasutras dealt with geometry, and extended the treatment of the Pythagorean theorem as first presented in 800 BCE by Baudhayana.[2]

Kātyāyana belonged to the Aindra School of Grammar and may have lived towards the Punjab region of the Indian Subcontinent.

Pingala

Born: unclear, 3rd / 2nd century BCE[1]

Residence: Indian subcontinent

Academic work: Era Maurya or post-Maurya

Main interests: Indian mathematics, Sanskrit grammar

Notable works: Author of the Chandaḥśāstra (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody

Notable ideas: mātrāmeru, binary numeral system, arithmetical triangle

Pingala (Devanagari: पिङ्गल piṅgala) (c. 3rd/2nd century BC[1]), is the influential ancient scholar and the author of the Chandaḥśāstra(also called Pingala-sutras), the earliest known treatise on Sanskrit prosody.[2]

The Chandaḥśāstra is a work of eight chapters in the late Sūtra style, not fully comprehensible without a commentary. It has been dated to the last few centuries BCE.[3][4] The 10th century mathematician Halayudha wrote a commentary on the Chandaḥśāstra and expanded it.

Combinatorics

The Chandaḥśāstra presents the first known description of a binary numeral system in connection with the systematic enumeration of meters with fixed patterns of short and long syllables.[5] The discussion of the combinatorics of meter corresponds to the binomial theorem. Halāyudha's commentary includes a presentation of the Pascal's triangle (called meruprastāra). Pingala's work also contains the Fibonacci numbers, called mātrāmeru.[6]

Use of zero is sometimes ascribed to Pingala due to his discussion of binary numbers, usually represented using 0 and 1 in modern discussion, but Pingala used light (laghu) and heavy (guru) rather than 0 and 1 to describe syllables. As Pingala's system ranks binary patterns starting at one (four short syllables—binary "0000"—is the first pattern), the nth pattern corresponds to the binary representation of n-1 (with increasing positional values).

Pingala is credited with using binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), a notation similar to Morse code.[7] Pingala used the Sanskrit word śūnya explicitly to refer to zero.

Panini:

Father of linguistics.
The history of linguistics begins not with Plato or Aristotle, but with the Indian grammarian Panini.

— Rens Bod, University of Amsterdam

Panini was an ancient Sanskrit scholar from Indian subcontinent.

Panini (IAST Pāṇini, fl. 4th century BCE is an ancient Sanskrit philologist, grammarian, and a revered scholar in Hinduism.Considered the father of Indian linguistics,Pāṇini likely lived in the northwest Indian subcontinent during the Mahajanapada era.

Pāṇini is known for his text Ashtadhyayi, a sutra-style treatise on Sanskrit grammar, 3,959 "verses" or rules on linguistics, syntaxand semantics in "eight chapters" which is the foundational text of the Vyākaraṇa branch of the Vedanga, the auxiliary scholarly disciplines of the Vedic period.His aphoristic text attracted numerous bhashya (commentaries), of which Patanjali's Mahābhāṣyais the most famous in Hindu traditions.His ideas influenced and attracted commentaries from scholars of other Indian religions such as Buddhism.

Pāṇini's analysis of noun compounds still forms the basis of modern linguistic theories of compounding in Indian languages. Pāṇini's comprehensive and scientific theory of grammar is conventionally taken to mark the start of Classical Sanskrit. His systematic treatise inspired and made Sanskrit the preeminent Indian language of learning and literature for two millennia.

Pāṇini's theory of morphological analysis was more advanced than any equivalent Western theory before the 20th century. His treatise is generative and descriptive, and has been compared to the Turing machine wherein the logical structure of any computing device has been reduced to its essentials using an idealized mathematical model.

The name Pāṇini is a patronymic meaning descendant of Paṇina.[16] His full name was "Dakṣiputra Pāṇini" according to verses 1.75.13 and 3.251.12 of Patanjali's Mahābhāṣya, with the first part suggesting his mother's name was Dakṣi.

Date and context

Father of linguistics
The history of linguistics begins not with Plato or Aristotle, but with the Indian grammarian Panini.

— Rens Bod, University of Amsterdam[18]

Nothing definite is known about when Pāṇini lived, not even in which century he lived. Most scholarship suggests he lived in or before mid-fourth century BCE (floruit),[2][3] possibly in the sixth or fifth century BCE.[1][17] Pāṇini's grammar defines Classical Sanskrit, so Pāṇini is chronologically placed in the later part of the Vedic period. According to Rens Bod, a professor of Humanities specializing in comparative history, Pāṇini must have lived sometime between seventh and fifth centuries BCE.[18]

Some proposals have attempted to date Pāṇini from references within the text. The first proposal is based on sutra 2.1.70 of Pāṇini, which mentions kumāraśramaṇa, with the word śramaṇa interpreted to imply that he may have had "Buddhist nuns" in mind, and therefore he should be placed after Gautama Buddha. Other scholars question this theory because nuns in the Indian traditions existed outside of and before Buddhism, such as in Jainism.[19] The second proposal is based on the occurrence of the word yavanānī (in 4.1.49, either "Greek woman", or "Greek alphabet").[19] This occurrence of yavanānī, some suggest a terminus post quem as 519 BCE, i.e. the time of Darius I's Behistun Inscription that included the province of Gandhara (IAST: Gandhāra). However, in 1862 Max Müller objected to this interpretation with the statement that there is no reason to assume that yavana meant "Greek" before and in the century Pāṇini lived, and it could as well might have been a reference in a Semitic or a South Indian context.[20] More recently, Patrick Olivelle – a professor of Sanskrit and Indian religions, concurs and states that the term yona or yavana in Pāṇini is "merely linguistic and does not necessarily indicate that he knew or was in contact with Greek settlers", adding that while Pāṇini is generally estimated around the 5th century BCE, placing Pāṇini in a century "is an educated guess".[21]

It is not certain whether Pāṇini used writing for the composition of his work, though it is generally agreed that he knew of a form of writing, based on references to words such as lipi("script") and lipikara ("scribe") in section 3.2 of the Aṣṭādhyāyī.[22][23][24][note 2] Pāṇini cites ten grammarians and linguists before him, none of whom can be chronologically placed with any certainty. The ten Vedic scholar names he quotes are of Apisali, Kashyapa, Gargya, Galava, Cakravarmana, Bharadvaja, Sakatayana, Sakalya, Senaka and Sphotayana.[31]

Both Brihatkatha and Mañjuśrī-mūla-kalpa mention Paninit to have been a contemporary with the Nanda king.[32]

While Pāṇini is considered a Hindu scholar of grammar and linguistics,[4][33][12] his text is also an important historical source of cultural and geographical information. His work is significant such as in including the word Vasudeva (4.3.98), which scholars disagree whether it refers to a deity or a person.[34] The concept of dharma is attested in his sutra 4.4.41 as, dharmam carati or "he observes dharma (duty, righteousness)" (cf. Taittiriya Upanishad 1.11).[35][36]

Biography and location

Also spelled without diacritics as Panini in scholarly literature,[1][11] nothing certain is known about Pāṇini's personal life. According to the Mahābhāṣya of Patanjali, his mother's name was Dākṣī.[37] Patañjali calls Pāṇini Dākṣīputra (meaning son of Dākṣī) at several places in the Mahābhāṣya.[37] Rambhadracharya gives the name of his father as Paṇina, from which the name Pāṇini derives.[37]

In an inscription of Siladitya VII of Valabhi, he is called Śalāturiya, which means "man from Salatura". This means Panini lived in Salatura of ancient Gandhara, which likely was near Lahur, a town at the junction of Indus and Kabul rivers.[38] According to the memoirs of 7th-century Chinese scholar Xuanzang, there was a town called Suoluoduluo on the Indus where Pāṇini was born, and he composed the Qingming-lun (Sanskrit: Vyākaraṇa).[38][39][37]

According to Hartmut Scharfe, Pāṇini lived in Gandhara close to the borders of the Achaemenid Empire, and Gandhara was then an Achaemenian satrap. He must therefore have been technically a Persian subject, but states Scharfe, his work shows no trace of Persian.[17] Inferences, however, vary between scholars. According to Patrick Olivelle, Pāṇini's text and references to him elsewhere suggest that "he was clearly a northerner, probably from the northwestern region".[40]

Legends and later reception

Panini is mentioned in Indian fables and ancient texts. The Panchatantra, for example, mentions that Pāṇini was killed by a lion.[41][42][43]

Pāṇini was depicted on a five-rupee Indian postage stamp in 2004.[44]

Aṣṭādhyāyī

The Aṣṭādhyāyī is the central part of Pāṇini's grammar, and by far the most complex. The Ashtadhyayi is the oldest surviving complete linguistic and grammar text of Sanskrit, and Pāṇini refers to previous texts and authors such as the Unadisutra, Dhatupatha, and Ganapatha some of which have not survived. It complements the Vedic ancillary sciences such as the Niruktas, Nighantus, and Shiksha.[45] Regarded as extremely compact without sacrificing completeness, it would become the model for later specialist technical texts or sutras.[46][47]

The text takes material from lexical lists (Dhatupatha, Ganapatha) as input and describes algorithms to be applied to them for the generation of well-formed words. It is highly systematised and technical. Inherent in its approach are the concepts of the phoneme, the morpheme and the root. His rules have a reputation for perfection[48] – that is, they tersely describe Sanskrit morphology unambiguously and completely. A consequence of his grammar's focus on brevity is its highly unintuitive structure, reminiscent of modern notations such as the "Backus–Naur form". His sophisticated logical rules and technique have been widely influential in ancient and modern linguistics.

The Aṣṭādhyāyī was not the first description of Sanskrit grammar, but it is the earliest that has survived in full. The Aṣṭādhyāyī became the foundation of Vyākaraṇa, a Vedanga.[49]

In the Aṣṭādhyāyī, language is observed in a manner that has no parallel among Greek or Latin grammarians. Pāṇini's grammar, according to Renou and Filliozat, defines the linguistic expression and a classic that set the standard for Sanskrit language.[50] Pāṇini made use of a technical metalanguage consisting of a syntax, morphology and lexicon. This metalanguage is organised according to a series of meta-rules, some of which are explicitly stated while others can be deduced.[51]

The Aṣṭādhyāyī consists of 3,959 sutras or "aphoristic threads" in eight chapters, which are each subdivided into four sections or padas (pādāḥ). This text attracted a famous and one of the most ancient Bhasya (commentary) called the Mahabhasya.[52] The author of Mahabhasya is named Patanjali, who may or may not be the same person as the one who authored Yogasutras.[53] The Mahabhasya, literally "great commentary", is more than a commentary on Ashtadhyayi. It is the earliest known philosophical text of the Hindu Grammarians.[53][note 3] Non-Hindu texts and traditions on grammar emerged after Patanjali, some of which include the Sanskrit grammar text of Jainendra of Jainism and the Chandra school of Buddhism.[55]

Rules

The first two sutras are as follows:

1.1.1 vṛddhir ādaiC (वृद्धिरादैच् । १।१।१)1.1.2 adeṄ guṇaḥ (अदेङ्गुणः । १।१।२)

In these sutras, the capital letters are special meta-linguistic symbols; they are called IT (इत्) markers or, in later writers such as Katyayana and Patanjali, anubandhas (see below). The C and Ṅ refer to Shiva Sutras 4 ("ai, au, C") and 3 ("e, o, Ṅ"), respectively, forming what are known as the pratyāhāras "comprehensive designations" aiC, eṄ. They denote the list of phonemes {ai, au} and {e, o} respectively. The त् (T) appearing (in its variant form /d/) in both sutras is also an IT marker: Sutra 1.1.70 defines it as indicating that the preceding phoneme does not represent a list, but a single phoneme, encompassing all supra-segmental features such as accent and nasality. For further example, आत् (āT) and अत् (aT) represent आ {ā} and अ {a} respectively.

When a sutra defines a technical term, the term defined comes at the end, so the first sutra should have properly been ādaiJ vṛddhir instead of vṛddhir ādaiC. However the order is reversed to have a good-luck word at the very beginning of the work; vṛddhir happens to mean 'prosperity' in its non-technical use.

Thus the two sūtras consist of a list of phonemes, followed by a technical term; the final interpretation of the two sūtras above is thus:

1.1.1: {ā, ai, au} are called vṛ́ddhi.1.1.2: {a, e, o} are called guṇa.

At this point, one can see they are definitions of terminology: guṇa and vṛ́ddhi are the terms for the full and the lengthened Indo-European ablaut grades, respectively.

List of IT markers

its or anubandhas are defined in P. 1.3.2 through P. 1.3.8. These definitions refer only to items taught in the grammar or its ancillary texts such at the dhātupāţha; this fact is made clear in P. 1.3.2 by the word upadeśe, which is then continued in the following six rules by anuvṛtti, Ellipsis. As these anubandhas are metalinguistic markers and not pronounced in the final derived form, pada (word), they are elided by P. 1.3.9 tasya lopaḥ – 'There is elision of that (i.e. any of the preceding items which have been defined as an it).' Accordingly, Pāṇini defines the anubandhas as follows:

Nasalized vowels, e.g. bhañjO. Cf. P. 1.3.2.
A final consonant (haL). Cf. P. 1.3.3.
2. (a) except a dental, m and s in verbal or nominal endings. Cf. P. 1.3.4.
Initial ñi ṭu ḍu. Cf. P 1.3.5
Initial ṣ of a suffix (pratyaya). Cf. P. 1.3.6.
Initial palatals and cerebrals of a suffix. Cf. P. 1.3.7
Initial l, ś, and k but not in a taddhita 'secondary' suffix. Cf. P. 1.3.8.

A few examples of elements that contain its are as follows:

suP nominal suffix
Ś-IT
- Śi strong case endings
- Ślu elision
- ŚaP active marker
P-IT
- luP elision
- āP ā-stems
  - CāP
  - ṬāP
  - ḌāP
- LyaP (7.1.37)
L-IT
K-IT
- Ktvā
- luK elision
saN Desiderative
C-IT
M-IT
Ṅ-IT
- Ṅí Causative
- Ṅii ī-stems
  - ṄīP
  - ṄīN
  - Ṅī'Ṣ
- tiṄ verbal suffix
- lUṄ Aorist
- lIṄ Precative
S-IT
GHU class of verbal stems (1.1.20)
GHI (1.4.7)

Auxiliary texts

Pāṇini's Ashtadhyayi has three associated texts.

The Shiva Sutras are a brief but highly organised list of phonemes.
The Dhatupatha is a lexical list of verbal roots sorted by present class.
The Ganapatha is a lexical list of nominal stems grouped by common properties.

Shiva Sutras

The Shiva Sutras describe a phonemic notational system in the fourteen initial lines preceding the Ashtadhyayi. The notational system introduces different clusters of phonemes that serve special roles in the morphology of Sanskrit, and are referred to throughout the text. Each cluster, called a pratyāhara ends with a dummy sound called an anubandha (the so-called IT index), which acts as a symbolic referent for the list. Within the main text, these clusters, referred through the anubandhas, are related to various grammatical functions.

Dhatupatha

The Dhatupatha is a lexicon of Sanskrit verbal roots subservient to the Ashtadhyayi. It is organised by the ten present classes of Sanskrit, i.e. the roots are grouped by the form of their stem in the present tense.

The ten present classes of Sanskrit are:

bhū-ādayaḥ (root-full grade thematic presents)
ad-ādayaḥ (root presents)
juhoti-ādayaḥ (reduplicated presents)
div-ādayaḥ (ya thematic presents)
su-ādayaḥ (nu presents)
tud-ādayaḥ (root-zero grade thematic presents)
rudh-ādayaḥ (n-infix presents)
tan-ādayaḥ (no presents)
krī-ādayaḥ (ni presents)
cur-ādayaḥ (aya presents, causatives)

The small number of class 8 verbs are a secondary group derived from class 5 roots, and class 10 is a special case, in that any verb can form class 10 presents, then assuming causative meaning. The roots specifically listed as belonging to class 10 are those for which any other form has fallen out of use (causative deponents, so to speak).

Ganapatha

The Ganapatha (gaṇapāṭha) is a list of groups of primitive nominal stems used by the Ashtadhyayi.

Commentary

After Pāṇini, the Mahābhāṣya ("great commentary") of Patañjali on the Ashtadhyayi is one of the three most famous works in Sanskrit grammar. It was with Patañjali that Indian linguistic science reached its definite form. The system thus established is extremely detailed as to shiksha (phonology, including accent) and vyakarana (morphology). Syntax is scarcely touched, but nirukta (etymology) is discussed, and these etymologies naturally lead to semantic explanations. People interpret his work to be a defence of Pāṇini, whose Sūtras are elaborated meaningfully. He also attacks Katyayana rather severely. But the main contributions of Patañjali lies in the treatment of the principles of grammar enunciated by him.

Editions

Rama Nath Sharma, The Aṣṭādhyāyī of Pāṇini (6 Vols.), 2001, ISBN 8121500516 [56]
Otto Böhtlingk, Panini's Grammatik 1887, reprint 1998 ISBN 3-87548-198-4
Katre, Sumitra M., Astadhyayi of Panini, Austin: University of Texas Press, 1987. Reprint Delhi: Motilal Banarsidass, 1989. ISBN 0-292-70394-5
Misra, Vidya Niwas, The Descriptive Technique of Panini, Mouton and Co., 1966.
The Ashtadhyayi.(of Pāṇini) Translated into English by Srisa Chandra Vasu (1897), from this archive.

Bhaṭṭikāvya

The learning of Indian curriculum in late classical times had at its heart a system of grammatical study and linguistic analysis.[57] The core text for this study was the Aṣṭādhyāyī of Pāṇini, the sine qua non of learning.[58] This grammar of Pāṇini had been the object of intense study for the ten centuries prior to the composition of the Bhaṭṭikāvya. It was plainly Bhaṭṭi's purpose to provide a study aid to Pāṇini's text by using the examples already provided in the existing grammatical commentaries in the context of the gripping and morally improving story of the Rāmāyaṇa. To the dry bones of this grammar Bhaṭṭi has given juicy flesh in his poem. The intention of the author was to teach this advanced science through a relatively easy and pleasant medium. In his own words:

This composition is like a lamp to those who perceive the meaning of words and like a hand mirror for a blind man to those without grammar. This poem, which is to be understood by means of a commentary, is a joy to those sufficiently learned: through my fondness for the scholar I have here slighted the dullard.
Bhaṭṭikāvya 22.33–34.

Modern linguistics

Pāṇini's work became known in 19th-century Europe, where it influenced modern linguistics initially through Franz Bopp, who mainly looked at Pāṇini. Subsequently, a wider body of work influenced Sanskrit scholars such as Ferdinand de Saussure, Leonard Bloomfield, and Roman Jakobson. Frits Staal (1930–2012) discussed the impact of Indian ideas on language in Europe. After outlining the various aspects of the contact, Staal notes that the idea of formal rules in language – proposed by Ferdinand de Saussure in 1894 and developed by Noam Chomsky in 1957 – has origins in the European exposure to the formal rules of Pāṇinian grammar.[citation needed] In particular, de Saussure, who lectured on Sanskrit for three decades, may have been influenced by Pāṇini and Bhartrihari; his idea of the unity of signifier-signified in the sign somewhat resembles the notion of Sphoṭa. More importantly, the very idea that formal rules can be applied to areas outside of logic or mathematics may itself have been catalysed by Europe's contact with the work of Sanskrit grammarians.[59]

De Saussure

Pāṇini, and the later Indian linguist Bhartrihari, had a significant influence on many of the foundational ideas proposed by Ferdinand de Saussure, professor of Sanskrit, who is widely considered the father of modern structural linguistics and with Charles S. Peirce on the other side, to semiotics, although the concept Saussure used was semiology. Saussure himself cited Indian grammar as an influence on some of his ideas. In his Mémoire sur le système primitif des voyelles dans les langues indo-européennes (Memoir on the Original System of Vowels in the Indo-European Languages) published in 1879, he mentions Indian grammar as an influence on his idea that "reduplicated aorists represent imperfects of a verbal class." In his De l'emploi du génitif absolu en sanscrit (On the Use of the Genitive Absolute in Sanskrit) published in 1881, he specifically mentions Pāṇini as an influence on the work.[60]

Prem Singh, in his foreword to the reprint edition of the German translation of Pāṇini's Grammar in 1998, concluded that the "effect Panini's work had on Indo-European linguistics shows itself in various studies" and that a "number of seminal works come to mind," including Saussure's works and the analysis that "gave rise to the laryngeal theory," further stating: "This type of structural analysis suggests influence from Panini's analytical teaching." George Cardona, however, warns against overestimating the influence of Pāṇini on modern linguistics: "Although Saussure also refers to predecessors who had taken this Paninian rule into account, it is reasonable to conclude that he had a direct acquaintance with Panini's work. As far as I am able to discern upon rereading Saussure's Mémoire, however, it shows no direct influence of Paninian grammar. Indeed, on occasion, Saussure follows a path that is contrary to Paninian procedure."[60][61]

Leonard Bloomfield

The founding father of American structuralism, Leonard Bloomfield, wrote a 1927 paper titled "On some rules of Pāṇini".[62]

Comparison with modern formal systems

Pāṇini's grammar is the world's first formal system, developed well before the 19th century innovations of Gottlob Frege and the subsequent development of mathematical logic. In designing his grammar, Pāṇini used the method of "auxiliary symbols", in which new affixes are designated to mark syntactic categories and the control of grammatical derivations. This technique, rediscovered by the logician Emil Post, became a standard method in the design of computer programming languages.[63] Sanskritists now accept that Pāṇini's linguistic apparatus is well-described as an "applied" Post system. Considerable evidence shows ancient mastery of context-sensitive grammars, and a general ability to solve many complex problems. Frits Staal has written that "Panini is the Indian Euclid."

Other works

Two literary works are attributed to Pāṇini, though they are now lost.

Jāmbavati Vijaya is a lost work cited by one Rajashekhar in Jahlana's Sukti Muktāvalī. A fragment is to be found in Ramayukta's commentary on Namalinganushasana. From the title it may be inferred that the work dealt with Krishna's winning of Jambavati in the underworld as his bride. Rajashekhara in Jahlana's Sukti Muktāvalī:

नमः पाणिनये तस्मै यस्मादाविर भूदिह।आदौ व्याकरणं काव्यमनु जाम्बवतीजयम्॥namaḥ pāṇinaye tasmai yasmādāvirabhūdiha।ādau vyākaraṇaṃ kāvyamanu jāmbavatījayam॥

Ascribed to Pāṇini, Pātāla Vijaya is a lost work cited by Namisadhu in his commentary on Kavyalankara of Rudrata.

Āryabhatta

Born: 476 CE
Kusumapura (Pataliputra) (present day Patna)[1]

Died: 550 CE

Residence: India

Academic background

Influences : Surya Siddhanta

Academic work

Era: Gupta era

Main interests: Mathematics, astronomy

Notable works: Āryabhaṭīya, Arya-siddhanta

Notable ideas: Explanation of lunar eclipse and solar eclipse, rotation of Earth on its axis, reflection of light by moon, sinusoidal functions, solution of single variable quadratic equation, value of π correct to 4 decimal places, circumference of Earth to 99.8% accuracy, calculation of the length of sidereal year

Influenced: Lalla, Bhaskara I, Brahmagupta, Varahamihira.

Biography

Name

While there is a tendency to misspell his name as "Aryabhatta" by analogy with other names having the "bhatta" suffix, his name is properly spelled Aryabhata: every astronomical text spells his name thus,[7] including Brahmagupta's references to him "in more than a hundred places by name".[1] Furthermore, in most instances "Aryabhatta" would not fit the metre either.[7]

Time and place of birth

Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476.[5] Aryabhata called himself a native of Kusumapura or Pataliputra (present day Patna, Bihar).[1]

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one belonging to the Aśmaka country." During the Buddha's time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers in central India.[7][8]

It has been claimed that the aśmaka (Sanskrit for "stone") where Aryabhata originated may be the present day Kodungallur which was the historical capital city of Thiruvanchikkulamof ancient Kerala.[9] This is based on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, old records show that the city was actually Koṭum-kol-ūr ("city of strict governance"). Similarly, the fact that several commentaries on the Aryabhatiya have come from Kerala has been used to suggest that it was Aryabhata's main place of life and activity; however, many commentaries have come from outside Kerala, and the Aryasiddhanta was completely unknown in Kerala.[7] K. Chandra Hari has argued for the Kerala hypothesis on the basis of astronomical evidence.[10]

Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but his "Lanka" is an abstraction, standing for a point on the equator at the same longitude as his Ujjayini.[11]

Education

It is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time.[12] Both Hindu and Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna.[7] A verse mentions that Aryabhata was the head of an institution (kulapa) at Kusumapura, and, because the university of Nalanda was in Pataliputra at the time and had an astronomical observatory, it is speculated that Aryabhata might have been the head of the Nalanda university as well.[7] Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar.[13]

Works

Aryabhata is the author of several treatises on mathematics and astronomy, some of which are lost.

His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines.

The Arya-siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta and Bhaskara I. This work appears to be based on the older Surya Siddhanta and uses the midnight-day reckoning, as opposed to sunrise in Aryabhatiya. It also contained a description of several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular and circular (dhanur-yantra / chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.[8]

A third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of this work is not known.

Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, Abū Rayhān al-Bīrūnī.[8]

Aryabhatiya

Direct details of Aryabhata's work are known only from the Aryabhatiya. The name "Aryabhatiya" is due to later commentators. Aryabhata himself may not have given it a name. His disciple Bhaskara I calls it Ashmakatantra (or the treatise from the Ashmaka). It is also occasionally referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there are 108 verses in the text. It is written in the very terse style typical of sutra literature, in which each line is an aid to memory for a complex system. Thus, the explication of meaning is due to commentators. The text consists of the 108 verses and 13 introductory verses, and is divided into four pādas or chapters:

Gitikapada: (13 verses): large units of time—kalpa, manvantra, and yuga—which present a cosmology different from earlier texts such as Lagadha's Vedanga Jyotisha (c. 1st century BCE). There is also a table of sines (jya), given in a single verse. The duration of the planetary revolutions during a mahayuga is given as 4.32 million years.
Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuṭṭaka).
Kalakriyapada (25 verses): different units of time and a method for determining the positions of planets for a given day, calculations concerning the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week with names for the days of week.
Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, features of the ecliptic, celestial equator, node, shape of the earth, cause of day and night, rising of zodiacal signs on horizon, etc. In addition, some versions cite a few colophons added at the end, extolling the virtues of the work, etc.

The Aryabhatiya presented a number of innovations in mathematics and astronomy in verse form, which were influential for many centuries. The extreme brevity of the text was elaborated in commentaries by his disciple Bhaskara I (Bhashya, c. 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya, (1465 CE).

Mathematics

Place value system and zero

The place-value system, first seen in the 3rd-century Bakhshali Manuscript, was clearly in place in his work. While he did not use a symbol for zero, the French mathematician Georges Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients.[14]

However, Aryabhata did not use the Brahmi numerals. Continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in a mnemonic form.[15]

Approximation of π

Aryabhata worked on the approximation for pi (π), and may have come to the conclusion that π is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.
"Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached."
[16]

This implies that the ratio of the circumference to the diameter is ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures.

It is speculated that Aryabhata used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, because the irrationality of pi (π) was proved in Europe only in 1761 by Lambert.[17]

After Aryabhatiya was translated into Arabic (c. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra.[8]

Trigonometry

In Ganitapada 6, Aryabhata gives the area of a triangle as

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

that translates to: "for a triangle, the result of a perpendicular with the half-side is the area."[18]

Aryabhata discussed the concept of sine in his work by the name of ardha-jya, which literally means "half-chord". For simplicity, people started calling it jya. When Arabic writers translated his works from Sanskrit into Arabic, they referred it as jiba. However, in Arabic writings, vowels are omitted, and it was abbreviated as jb. Later writers substituted it with jaib, meaning "pocket" or "fold (in a garment)". (In Arabic, jiba is a meaningless word.) Later in the 12th century, when Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic jaib with its Latin counterpart, sinus, which means "cove" or "bay"; thence comes the English word sine.[19]

Indeterminate equations

A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to Diophantine equations that have the form ax + by = c. (This problem was also studied in ancient Chinese mathematics, and its solution is usually referred to as the Chinese remainder theorem.) This is an example from Bhāskara's commentary on Aryabhatiya:

Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7

That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations, such as this, can be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose more ancient parts might date to 800 BCE. Aryabhata's method of solving such problems, elaborated by Bhaskara in 621 CE, is called the kuṭṭaka (कुट्टक) method. Kuṭṭaka means "pulverizing" or "breaking into small pieces", and the method involves a recursive algorithm for writing the original factors in smaller numbers. This algorithm became the standard method for solving first-order diophantine equations in Indian mathematics, and initially the whole subject of algebra was called kuṭṭaka-gaṇita or simply kuṭṭaka.[20]

Algebra

In Aryabhatiya, Aryabhata provided elegant results for the summation of series of squares and cubes:[21]

$1^2 + 2^2 + \cdots + n^2 = {n(n + 1)(2n + 1) \over 6}$

and

$1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ (see squared triangular number)

Astronomy

Aryabhata's system of astronomy was called the audAyaka system, in which days are reckoned from uday, dawn at lanka or "equator". Some of his later writings on astronomy, which apparently proposed a second model (or ardha-rAtrikA, midnight) are lost but can be partly reconstructed from the discussion in Brahmagupta's Khandakhadyaka. In some texts, he seems to ascribe the apparent motions of the heavens to the Earth's rotation. He may have believed that the planet's orbits as elliptical rather than circular.[22][23]

Motions of the solar system

Aryabhata correctly insisted that the earth rotates about its axis daily, and that the apparent movement of the stars is a relative motion caused by the rotation of the earth, contrary to the then-prevailing view, that the sky rotated.[24] This is indicated in the first chapter of the Aryabhatiya, where he gives the number of rotations of the earth in a yuga,[25] and made more explicit in his gola chapter:[26]

In the same way that someone in a boat going forward sees an unmoving [object] going backward, so [someone] on the equator sees the unmoving stars going uniformly westward. The cause of rising and setting [is that] the sphere of the stars together with the planets [apparently?] turns due west at the equator, constantly pushed by the cosmic wind.

Aryabhata described a geocentric model of the solar system, in which the Sun and Moon are each carried by epicycles. They in turn revolve around the Earth. In this model, which is also found in the Paitāmahasiddhānta (c. CE 425), the motions of the planets are each governed by two epicycles, a smaller manda (slow) and a larger śīghra (fast). [27] The order of the planets in terms of distance from earth is taken as: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms."[8]

The positions and periods of the planets was calculated relative to uniformly moving points. In the case of Mercury and Venus, they move around the Earth at the same mean speed as the Sun. In the case of Mars, Jupiter, and Saturn, they move around the Earth at specific speeds, representing each planet's motion through the zodiac. Most historians of astronomy consider that this two-epicycle model reflects elements of pre-Ptolemaic Greek astronomy.[28] Another element in Aryabhata's model, the śīghrocca, the basic planetary period in relation to the Sun, is seen by some historians as a sign of an underlying heliocentric model.[29]

Eclipses

Solar and lunar eclipses were scientifically explained by Aryabhata. He states that the Moon and planets shine by reflected sunlight. Instead of the prevailing cosmogony in which eclipses were caused by Rahu and Ketu (identified as the pseudo-planetary lunar nodes), he explains eclipses in terms of shadows cast by and falling on Earth. Thus, the lunar eclipse occurs when the moon enters into the Earth's shadow (verse gola.37). He discusses at length the size and extent of the Earth's shadow (verses gola.38–48) and then provides the computation and the size of the eclipsed part during an eclipse. Later Indian astronomers improved on the calculations, but Aryabhata's methods provided the core. His computational paradigm was so accurate that 18th-century scientist Guillaume Le Gentil, during a visit to Pondicherry, India, found the Indian computations of the duration of the lunar eclipse of 30 August 1765 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.[8]

Sidereal periods

Considered in modern English units of time, Aryabhata calculated the sidereal rotation (the rotation of the earth referencing the fixed stars) as 23 hours, 56 minutes, and 4.1 seconds;[30] the modern value is 23:56:4.091. Similarly, his value for the length of the sidereal year at 365 days, 6 hours, 12 minutes, and 30 seconds (365.25858 days)[31] is an error of 3 minutes and 20 seconds over the length of a year (365.25636 days).[32]

Heliocentrism

As mentioned, Aryabhata advocated an astronomical model in which the Earth turns on its own axis. His model also gave corrections (the śīgra anomaly) for the speeds of the planets in the sky in terms of the mean speed of the sun. Thus, it has been suggested that Aryabhata's calculations were based on an underlying heliocentric model, in which the planets orbit the Sun,[33][34][35] though this has been rebutted.[36] It has also been suggested that aspects of Aryabhata's system may have been derived from an earlier, likely pre-Ptolemaic Greek, heliocentric model of which Indian astronomers were unaware,[37] though the evidence is scant.[38] The general consensus is that a synodic anomaly (depending on the position of the sun) does not imply a physically heliocentric orbit (such corrections being also present in late Babylonian astronomical texts), and that Aryabhata's system was not explicitly heliocentric.[39]

Legacy

Statue of Aryabhata on the grounds of IUCAA, Pune.[a]

India's first satellite named after Aryabhata

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Aryabhata's work was of great influence in the Indian astronomical tradition and influenced several neighbouring cultures through translations. The Arabic translation during the Islamic Golden Age (c. 820 CE), was particularly influential. Some of his results are cited by Al-Khwarizmi and in the 10th century Al-Biruni stated that Aryabhata's followers believed that the Earth rotated on its axis.

His definitions of sine (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) influenced the birth of trigonometry. He was also the first to specify sine and versine (1 − cos x) tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.

In fact, modern names "sine" and "cosine" are mistranscriptions of the words jya and kojya as introduced by Aryabhata. As mentioned, they were translated as jiba and kojiba in Arabic and then misunderstood by Gerard of Cremona while translating an Arabic geometry text to Latin. He assumed that jiba was the Arabic word jaib, which means "fold in a garment", L. sinus (c. 1150).[40]

Aryabhata's astronomical calculation methods were also very influential. Along with the trigonometric tables, they came to be widely used in the Islamic world and used to compute many Arabic astronomical tables (zijes). In particular, the astronomical tables in the work of the Arabic Spain scientist Al-Zarqali (11th century) were translated into Latin as the Tables of Toledo (12th century) and remained the most accurate ephemeris used in Europe for centuries.

Calendric calculations devised by Aryabhata and his followers have been in continuous use in India for the practical purposes of fixing the Panchangam (the Hindu calendar). In the Islamic world, they formed the basis of the Jalali calendar introduced in 1073 CE by a group of astronomers including Omar Khayyam,[41] versions of which (modified in 1925) are the national calendars in use in Iran and Afghanistan today. The dates of the Jalali calendar are based on actual solar transit, as in Aryabhata and earlier Siddhanta calendars. This type of calendar requires an ephemeris for calculating dates. Although dates were difficult to compute, seasonal errors were less in the Jalali calendar than in the Gregorian calendar.

Aryabhatta Knowledge University (AKU), Patna has been established by Government of Bihar for the development and management of educational infrastructure related to technical, medical, management and allied professional education in his honour. The university is governed by Bihar State University Act 2008.

India's first satellite Aryabhata and the lunar crater Aryabhata are named in his honour. An Institute for conducting research in astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Institute of Observational Sciences (ARIES) near Nainital, India. The inter-school Aryabhata Maths Competition is also named after him,[42] as is Bacillus aryabhata, a species of bacteria discovered in the stratosphere by ISRO scientists in 2009.

Varāhamihira

VārāhaMihira

Born: 505 CE

Died: 587 CE

Occupation: astronomer, mathematician, and astrologer.

Period: Gupta era

Subject: Astronomy, Astrology, Mathematics

Notable works: Pancha-Siddhāntikā, Brihat-Samhita, Brihat Jataka

Vārāhamihira pronunciation (help·info) (505–587 CE), also called Vārāha or Mihira, was an Indian astronomer, mathematician, and astrologer who lived in Ujjain. He was born in the Avanti region, roughly corresponding to modern-day Malwa, to Adityadasa, who was himself an astronomer. According to one of his own works, he was educated at Kapitthaka.[1] He is considered to be one of the "Nine Jewels" (Navaratnas) of the court of legendary ruler Yashodharman Vikramaditya of Malwa

Works:

Pancha-Siddhantika

Varahamihira's main work is the book Pañcasiddhāntikā (or Pancha-Siddhantika, "[Treatise] on the Five [Astronomical] Canons) dated ca. 575 CE gives us information about older Indian texts which are now lost. The work is a treatise on mathematical astronomy and it summarises five earlier astronomical treatises, namely the Surya Siddhanta, Romaka Siddhanta, Paulisa Siddhanta, Vasishtha Siddhanta and Paitamaha Siddhanta. It is a compendium of Vedanga Jyotisha as well as Hellenistic astronomy (including Greek, Egyptian and Roman elements).[4] Varahamihira was the first one to mention that the ayanamsa, or the shifting of the equinox is 50.32 seconds.[clarification needed]

"They [the Indians] have 5 Siddhāntas:

Sūrya-Siddhānta, ie. the Siddhānta of the Sun, thought to be composed by Lāṭadeva, but actually composed by Mayasura also known as Mamuni Mayan as stated in the text itself.
Vasishtha-siddhānta, so called from one of the stars of the Great Bear, composed by Vishnucandra,
Paulisa-siddhānta, so called from Pulisa, the Greek, from the city of Saintra, which is supposed to be Alexandria, composed by Pulisa.
Romaka-siddhānta, so called from the Rūm, ie. the subjects of the Roman Empire, composed by Śrīsheṇa.
Paitahama-siddhānta.

Brihat-Samhita

Another important contribution of Varahamihira is the encyclopedic Brihat-Samhita. It covers wide ranging subjects of human interest, including astrology, planetary movements, eclipses, rainfall, clouds, architecture, growth of crops, manufacture of perfume, matrimony, domestic relations, gems, pearls, and rituals. The volume expounds on gemstone evaluation criterion found in the Garuda Purana, and elaborates on the sacred Nine Pearls from the same text. It contains 106 chapters and is known as the "great compilation".

On Astrology

He was also an astrologer. He wrote on all the three main branches of Jyotisha astrology:

Brihat Jataka - is considered as one of the five main treatises on Hindu astrology on horoscopy.
Laghu Jataka - also known as 'Swalpa Jataka'
Samasa Samhita - also known as 'Lagu Samhita' or 'Swalpa Samhita'
Brihat Yogayatra - also known as 'Mahayatra' or 'Yakshaswamedhiya yatra'
Yoga Yatra - also known as 'Swalpa yatra'
Tikkani Yatra
Brihat Vivaha Patal
Lagu Vivaha Patal - also known as 'Swalpa Vivaha Patal'
Lagna Varahi
Kutuhala Manjari
Daivajna Vallabha (apocryphal)

His son Prithuyasas also contributed to Hindu astrology; his book Hora Sara is a famous book on horoscopy. Khana (also named Lilavati elsewhere), the medieval Bengali poet astrologer, is believed to be the daughter-in-law of Varahamihira.

Influences

The Romaka Siddhanta ("Doctrine of the Romans") and the Paulisa Siddhanta were two works of Western origin which influenced Varahamihira's thought, though this view is controversial as there is much evidence to suggest that it was actually Vedic thought indigenous to India which first influenced Western astrologers and subsequently came back to India reformulated.[5] Paulisa Siddhanta is often mistakenly thought to be a single work and attributed to Paul of Alexandria (c. 378 CE).[6] However, this notion has been rejected by other scholars in the field, notably by David Pingree who stated that "...the identification of Paulus Alexandrinus with the author of the Pauliśa Siddhānta is totally false".[7] Number of his writings share similarities with the earlier texts like Vedanga Jyotisha.[8]

A comment in the Brihat-Samhita by Varahamihira says: "The Greeks, though impure.,[9] must be honored since they have shown tremendous interest in our science....." ("mleccha hi yavanah tesu samyak shastram kdamsthitam/ rsivat te 'p i pujyante kim punar daivavid dvijah" (Brihat-Samhita 2.15)).[citation needed]

Contributions

Trigonometry

Varahamihira improved the accuracy of the sine tables of Aryabhata .

Combinatorics

He was among the first mathematicians to discover a version of what is now known as the Pascal's triangle. He used it to calculate the binomial coefficients.

Optics

Among Varahamihira's contribution to physics is his statement that reflection is caused by the back-scattering of particles and refraction (the change of direction of a light ray as it moves from one medium into another) by the ability of the particles to penetrate inner spaces of the material, much like fluids that move through porous objects.

1. ^ "the Pañca-siddhāntikā ("Five Treatises"), a compendium of Greek, Egyptian, Roman and Indian astronomy. Varāhamihira's knowledge of Western astronomy was thorough. In 5 sections, his monumental work progresses through native Indian astronomy and culminates in 2 treatises on Western astronomy, showing calculations based on Greek and Alexandrian reckoning and even giving complete Ptolemaic mathematical charts and tables. Encyclopædia Britannica (2007) s.v.Varahamihira ^

2. E. C. Sachau, Alberuni's India (1910), vol. I, p. 153.

Yativṛṣabha

Yativṛṣabha, also known as Jadivasaha, was a mathematician and Jain monk. He is believed to have lived during the 6th century, probably during 500-570. He lived and worked between the periods of two great Indian mathematicians, Aryabhata (476 – 550) and Brahmagupta (598-668). He wrote the book named Tiloyapannatti which describes cosmology from the point of view of Jain religion and philosophy. "The work also gives various units for measuring distances and time." Tiloya Panatti postulated different concepts about infinity.

Yativrsabha (or Jadivasaha) was a Jaina mathematician who studied under Arya Manksu and Nagahastin. We know nothing of Yativrsabha's dates except for a reference which he makes to the end of the Gupta dynasty which he says was after 231 years of ruling. This ended in 551 so we must assume that 551 AD is a date which occured during Yativrsabha's lifetime. This fits with the only other information regarding his dates which are that his work is referenced by Jinabhadra Ksamasramana in 609 and that Yativrsabha himself refers to a work written by Sarvanandin in 458.

Yativrsabha's work Tiloyapannatti gives various units for measuring distances and time and also describes the system of infinite time measures. It is a work which describes Jaina cosmology and gives a description of the universe which is of historical importance in understanding Jaina science and mathematics. The Jaina belief was in an infinite world, both infinite in space and in time. This led the Jainas to devise ways of measuring larger and larger distances and longer and longer intervals of time. It led them to consider different measures of infinity, and in this respect the Jaina mathematicians would appear to be the only ones before the time when Cantor developed the theory of infinite cardinals to envisage different magnitudes of infinity.

Brahmagupta

Born: c. 598 CE

Died: c. 668 CE

Residence

Known for

Scientific career

Fields: Mathematics, astronomy

Brahmagupta ( listen (help·info)) (born c. 598 CE, died c. 668 CE) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the Brāhmasphuṭasiddhānta (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical treatise, and the Khaṇḍakhādyaka ("edible bite", dated 665), a more practical text.

Brahmagupta was the first to give rules to compute with zero. The texts composed by Brahmagupta were composed in elliptical verse in Sanskrit, as was common practice in Indian mathematics. As no proofs are given, it is not known how Brahmagupta's results were derived.

Life and career

Brahmagupta was born in 598 CE according to his own statement. He lived in Bhillamala (modern Bhinmal) during the reign of the Chapa dynasty ruler, Vyagrahamukha. He was the son of Jishnugupta and was a Shaivite by religion.[3] Even though most scholars assume that Brahmagupta was born in Bhillamala, there is no conclusive evidence for it. However, he lived and worked there for a good part of his life. Prithudaka Svamin, a later commentator, called him Bhillamalacharya, the teacher from Bhillamala.[4] Sociologist G. S. Ghuryebelieved that he might have been from the Multan or Abu region.[5]

Bhillamala, called pi-lo-mo-lo by Xuanzang, was the apparent capital of the Gurjaradesa, the second largest kingdom of Western India, comprising southern Rajasthan and northern Gujarat in modern-day India. It was also a centre of learning for mathematics and astronomy. Brahmagupta became an astronomer of the Brahmapaksha school, one of the four major schools of Indian astronomy during this period. He studied the five traditional siddhanthas on Indian astronomy as well as the work of other astronomers including Aryabhata I, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin and Vishnuchandra.[4]

In the year 628, at an age of 30, he composed the Brāhmasphuṭasiddhānta (the improved treatise of Brahma) which is believed to be a revised version of the received siddhanta of the Brahmapaksha school. Scholars state that he incorporated a great deal of originality to his revision, adding a considerable amount of new material. The book consists of 24 chapters with 1008 verses in the ārya metre. A good deal of it is astronomy, but it also contains key chapters on mathematics, including algebra, geometry, trigonometry and algorithmics, which are believed to contain new insights due to Brahmagupta himself.[4][6][7]

Later, Brahmagupta moved to Ujjain, which was also a major centre for astronomy. At the age of 67, he composed his next well known work Khanda-khādyaka, a practical manual of Indian astronomy in the karana category meant to be used by students.[8]

Brahmagupta lived beyond 665 CE. He is believed to have died in Ujjain.[citation needed]

Controversy

Brahmagupta directed a great deal of criticism towards the work of rival astronomers, and his Brahmasphutasiddhanta displays one of the earliest schisms among Indian mathematicians. The division was primarily about the application of mathematics to the physical world, rather than about the mathematics itself. In Brahmagupta's case, the disagreements stemmed largely from the choice of astronomical parameters and theories.[9] Critiques of rival theories appear throughout the first ten astronomical chapters and the eleventh chapter is entirely devoted to criticism of these theories, although no criticisms appear in the twelfth and eighteenth chapters.[9]

Reception

The historian of science George Sarton called him "one of the greatest scientists of his race and the greatest of his time."[8] Brahmagupta's mathematical advances were carried on further by Bhāskara II, a lineal descendant in Ujjain, who described Brahmagupta as the ganaka-chakra-chudamani (the gem of the circle of mathematicians). Prithudaka Svaminwrote commentaries on both of his works, rendering difficult verses into simpler language and adding illustrations. Lalla and Bhattotpala in the 8th and 9th centuries wrote commentaries on the Khanda-khadyaka.[10] Further commentaries continued to be written into the 12th century.[8]

A few decades after the death of Brahmagupta, Sindh came under the Arab Caliphate in 712 CE. Expeditions were sent into Gurjaradesa. The kingdom of Bhillamala seems to have been annihilated but Ujjain repulsed the attacks. The court of Caliph Al-Mansur (754–775) received an embassy from Sindh, including an astrologer called Kanaka, who brought (possibly memorised) astronomical texts, including those of Brahmagupta. Brahmagupta's texts were translated into Arabic by Muhammad al-Fazari, an astronomer in Al-Mansur's court under the names Sindhind and Arakhand. An immediate outcome was the spread of the decimal number system used in the texts. The mathematician Al-Khwarizmi (800–850 CE) wrote a text called al-Jam wal-tafriq bi hisal-al-Hind (Addition and Subtraction in Indian Arithmetic), which was translated into Latin in the 13th century as Algorithmi de numero indorum. Through these texts, the decimal number system and Brahmagupta's algorithms for arithmetic have spread throughout the world. Al-Khwarizmi also wrote his own version of Sindhind, drawing on Al-Fazari's version and incorporating Ptolemaic elements. Indian astronomic material circulated widely for centuries, even passing into medieval Latin texts.[11][12][13]

Mathematics

Algebra

Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphutasiddhanta,

The difference between rupas, when inverted and divided by the difference of the unknowns, is the unknown in the equation. The rupas are [subtracted on the side] below that from which the square and the unknown are to be subtracted.[14]

which is a solution for the equation $bx + c = dx + e$ equivalent to $x = e - c / b - d$ , where rupas refers to the constants $c$ and $e$ . He further gave two equivalent solutions to the general quadratic equation

18.44. Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. [The result is] the middle [number].
18.45. Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.[14]

which are, respectively, solutions for the equation $ax 2 + bx = c$ equivalent to,

$x={\frac {{\sqrt {4ac+b^{2}}}-b}{2a}}$

and

$x={\frac {{\sqrt {ac+{\tfrac {b^{2}}{4}}}}-{\tfrac {b}{2}}}{a}}.$

He went on to solve systems of simultaneous indeterminate equations stating that the desired variable must first be isolated, and then the equation must be divided by the desired variable's coefficient. In particular, he recommended using "the pulverizer" to solve equations with multiple unknowns.

18.51. Subtract the colors different from the first color. [The remainder] divided by the first [color's coefficient] is the measure of the first. [Terms] two by two [are] considered [when reduced to] similar divisors, [and so on] repeatedly. If there are many [colors], the pulverizer [is to be used].[14]

Like the algebra of Diophantus, the algebra of Brahmagupta was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms.[15] The extent of Greek influence on this syncopation, if any, is not known and it is possible that both Greek and Indian syncopation may be derived from a common Babylonian source.[15]

Arithmetic

The four fundamental operations (addition, subtraction, multiplication, and division) were known to many cultures before Brahmagupta. This current system is based on the Hindu Arabic number system and first appeared in Brahmasphutasiddhanta. Brahmagupta describes the multiplication as thus “The multiplicand is repeated like a string for cattle, as often as there are integrant portions in the multiplier and is repeatedly multiplied by them and the products are added together. It is multiplication. Or the multiplicand is repeated as many times as there are component parts in the multiplier”. [16][page needed] Indian arithmetic was known in Medieval Europe as "Modus Indoram" meaning method of the Indians. In Brahmasphutasiddhanta, multiplication was named Gomutrika. In the beginning of chapter twelve of his Brahmasphutasiddhanta, entitled Calculation, Brahmagupta details operations on fractions. The reader is expected to know the basic arithmetic operations as far as taking the square root, although he explains how to find the cube and cube-root of an integer and later gives rules facilitating the computation of squares and square roots. He then gives rules for dealing with five types of combinations of fractions: $a / c + b / c$ ; $a / c \times b / d$ ; $a / 1 + b / d$ ; $a / c + b / d \times a / c = a (d + b) / cd$ ; and $a / c - b / d \times a / c = a (d - b) / cd$ .[17]

Series

Brahmagupta then goes on to give the sum of the squares and cubes of the first $n$ integers.

12.20. The sum of the squares is that [sum] multiplied by twice the [number of] step[s] increased by one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these with identical balls [can also be computed].[18]

Here Brahmagupta found the result in terms of the sum of the first $n$ integers, rather than in terms of $n$ as is the modern practice.[19]

He gives the sum of the squares of the first $n$ natural numbers as $n (n + 1)(2 n + 1) / 6$ and the sum of the cubes of the first n natural numbers as $(n (n + 1) / 2) 2$ .

Zero

Brahmagupta's Brahmasphuṭasiddhanta is the first book that provides rules for arithmetic manipulations that apply to zero and to negative numbers.[20] The Brahmasphutasiddhantais the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans. In chapter eighteen of his Brahmasphutasiddhanta, Brahmagupta describes operations on negative numbers. He first describes addition and subtraction,

18.30. [The sum] of two positives is positives, of two negatives negative; of a positive and a negative [the sum] is their difference; if they are equal it is zero. The sum of a negative and zero is negative, [that] of a positive and zero positive, [and that] of two zeros zero.

[...]
18.32. A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added.[14]

He goes on to describe multiplication,

18.33. The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.[14]

But his description of division by zero differs from our modern understanding:

18.34. A positive divided by a positive or a negative divided by a negative is positive; a zero divided by a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative.
18.35. A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or of a positive is positive; [the square] of zero is zero. That of which [the square] is the square is [its] square-root.[14]

Here Brahmagupta states that 0/0 = 0 and as for the question of $a$ /0 where $a$ ≠ 0 he did not commit himself.[21] His rules for arithmetic on negative numbers and zero are quite close to the modern understanding, except that in modern mathematics division by zero is left undefined.

Diophantine analysis

Pythagorean triples

In chapter twelve of his Brahmasphutasiddhanta, Brahmagupta provides a formula useful for generating Pythagorean triples:

12.39. The height of a mountain multiplied by a given multiplier is the distance to a city; it is not erased. When it is divided by the multiplier increased by two it is the leap of one of the two who make the same journey.[22]

Or, in other words, if $d = mx / x + 2$ , then a traveller who "leaps" vertically upwards a distance $d$ from the top of a mountain of height $m$ , and then travels in a straight line to a city at a horizontal distance $mx$ from the base of the mountain, travels the same distance as one who descends vertically down the mountain and then travels along the horizontal to the city.[22] Stated geometrically, this says that if a right-angled triangle has a base of length $a = mx$ and altitude of length $b = m + d$ , then the length, $c$ , of its hypotenuse is given by $c = m (1 + x) - d$ . And, indeed, elementary algebraic manipulation shows that $a 2 + b 2 = c 2$ whenever $d$ has the value stated. Also, if $m$ and $x$ are rational, so are $d$ , $a$ , $b$ and $c$ . A Pythagorean triple can therefore be obtained from $a$ , $b$ and $c$ by multiplying each of them by the least common multiple of their denominators.

Pell's equation

Brahmagupta went on to give a recurrence relation for generating solutions to certain instances of Diophantine equations of the second degree such as $Nx 2 + 1 = y 2$ (called Pell's equation) by using the Euclidean algorithm. The Euclidean algorithm was known to him as the "pulverizer" since it breaks numbers down into ever smaller pieces.[23]

The nature of squares:
18.64. [Put down] twice the square-root of a given square by a multiplier and increased or diminished by an arbitrary [number]. The product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed.
18.65. The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additive rupas.[14]

The key to his solution was the identity,

$(x_{1}^{2}-Ny_{1}^{2})(x_{2}^{2}-Ny_{2}^{2})=(x_{1}x_{2}+Ny_{1}y_{2})^{2}-N(x_{1}y_{2}+x_{2}y_{1})^{2}$

which is a generalisation of an identity that was discovered by Diophantus,

$(x_{1}^{2}-y_{1}^{2})(x_{2}^{2}-y_{2}^{2})=(x_{1}x_{2}+y_{1}y_{2})^{2}-(x_{1}y_{2}+x_{2}y_{1})^{2}.$

Using his identity and the fact that if $(x 1, y 1)$ and $(x 2, y 2)$ are solutions to the equations $x 2 - Ny 2 = k 1$ and $x 2 - Ny 2 = k 2$ , respectively, then $(x 1 x 2 + Ny 1 y 2, x 1 y 2 + x 2 y 1)$ is a solution to $x 2 - Ny 2 = k 1 k 2$ , he was able to find integral solutions to Pell's equation through a series of equations of the form $x 2 - Ny 2 = ki$ . Brahmagupta was not able to apply his solution uniformly for all possible values of $N$ , rather he was only able to show that if $x 2 - Ny 2 = k$ has an integer solution for $k$ = ±1, ±2, or ±4, then $x 2 - Ny 2 = 1$ has a solution. The solution of the general Pell's equation would have to wait for Bhaskara II in c. 1150 CE.[24]

Geometry

Brahmagupta's formula

Diagram for reference

Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area,

12.21. The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral. The accurate [area] is the square root from the product of the halves of the sums of the sides diminished by [each] side of the quadrilateral.[18]

So given the lengths $p$ , $q$ , $r$ and $s$ of a cyclic quadrilateral, the approximate area is $p + r / 2 \cdot q + s / 2$ while, letting $t = p + q + r + s / 2$ , the exact area is

$\sqrt (t - p)(t - q)(t - r)(t - s)$ .

Although Brahmagupta does not explicitly state that these quadrilaterals are cyclic, it is apparent from his rules that this is the case.[25] Heron's formula is a special case of this formula and it can be derived by setting one of the sides equal to zero.

Triangles

Brahmagupta dedicated a substantial portion of his work to geometry. One theorem gives the lengths of the two segments a triangle's base is divided into by its altitude:

12.22. The base decreased and increased by the difference between the squares of the sides divided by the base; when divided by two they are the true segments. The perpendicular [altitude] is the square-root from the square of a side diminished by the square of its segment.[18]

Thus the lengths of the two segments are $1 / 2 (b \pm c 2 - a 2 / b)$ .

He further gives a theorem on rational triangles. A triangle with rational sides $a$ , $b$ , $c$ and rational area is of the form:

$a={\frac {1}{2}}\left({\frac {u^{2}}{v}}+v\right),\ \ b={\frac {1}{2}}\left({\frac {u^{2}}{w}}+w\right),\ \ c={\frac {1}{2}}\left({\frac {u^{2}}{v}}-v+{\frac {u^{2}}{w}}-w\right)$

for some rational numbers $u$ , $v$ , and $w$ .[26]

Brahmagupta's theorem

Brahmagupta's theorem states that AF = FD.

Brahmagupta continues,

12.23. The square-root of the sum of the two products of the sides and opposite sides of a non-unequal quadrilateral is the diagonal. The square of the diagonal is diminished by the square of half the sum of the base and the top; the square-root is the perpendicular [altitudes].[18]

So, in a "non-unequal" cyclic quadrilateral (that is, an isosceles trapezoid), the length of each diagonal is $\sqrt pr + qs$ .

He continues to give formulas for the lengths and areas of geometric figures, such as the circumradius of an isosceles trapezoid and a scalene quadrilateral, and the lengths of diagonals in a scalene cyclic quadrilateral. This leads up to Brahmagupta's famous theorem,

12.30-31. Imaging two triangles within [a cyclic quadrilateral] with unequal sides, the two diagonals are the two bases. Their two segments are separately the upper and lower segments [formed] at the intersection of the diagonals. The two [lower segments] of the two diagonals are two sides in a triangle; the base [of the quadrilateral is the base of the triangle]. Its perpendicular is the lower portion of the [central] perpendicular; the upper portion of the [central] perpendicular is half of the sum of the [sides] perpendiculars diminished by the lower [portion of the central perpendicular].[18]

Pi

In verse 40, he gives values of $π$ ,

12.40. The diameter and the square of the radius [each] multiplied by 3 are [respectively] the practical circumference and the area [of a circle]. The accurate [values] are the square-roots from the squares of those two multiplied by ten.[18]

So Brahmagupta uses 3 as a "practical" value of $π$ , and ${\sqrt {10}}\approx 3.1622\ldots$ as an "accurate" value of $π$ . The error in this "accurate" value is less than 1%.

Measurements and constructions

In some of the verses before verse 40, Brahmagupta gives constructions of various figures with arbitrary sides. He essentially manipulated right triangles to produce isosceles triangles, scalene triangles, rectangles, isosceles trapezoids, isosceles trapezoids with three equal sides, and a scalene cyclic quadrilateral.

After giving the value of pi, he deals with the geometry of plane figures and solids, such as finding volumes and surface areas (or empty spaces dug out of solids). He finds the volume of rectangular prisms, pyramids, and the frustum of a square pyramid. He further finds the average depth of a series of pits. For the volume of a frustum of a pyramid, he gives the "pragmatic" value as the depth times the square of the mean of the edges of the top and bottom faces, and he gives the "superficial" volume as the depth times their mean area.[27]

Trigonometry

Sine table

In Chapter 2 of his Brahmasphutasiddhanta, entitled Planetary True Longitudes, Brahmagupta presents a sine table:

2.2-5. The sines: The Progenitors, twins; Ursa Major, twins, the Vedas; the gods, fires, six; flavors, dice, the gods; the moon, five, the sky, the moon; the moon, arrows, suns [...][28]

Here Brahmagupta uses names of objects to represent the digits of place-value numerals, as was common with numerical data in Sanskrit treatises. Progenitors represents the 14 Progenitors ("Manu") in Indian cosmology or 14, "twins" means 2, "Ursa Major" represents the seven stars of Ursa Major or 7, "Vedas" refers to the 4 Vedas or 4, dice represents the number of sides of the tradition die or 6, and so on. This information can be translated into the list of sines, 214, 427, 638, 846, 1051, 1251, 1446, 1635, 1817, 1991, 2156, 2312, 1459, 2594, 2719, 2832, 2933, 3021, 3096, 3159, 3207, 3242, 3263, and 3270, with the radius being 3270.[29]

Interpolation formula

In 665 Brahmagupta devised and used a special case of the Newton–Stirling interpolation formula of the second-order to interpolate new values of the sine function from other values already tabulated.[30] The formula gives an estimate for the value of a function $f$ at a value $a + xh$ of its argument (with $h > 0$ and $-1 \leq x \leq 1$ ) when its value is already known at $a - h$ , $a$ and $a + h$ .

The formula for the estimate is:

$f(a+xh)\approx f(a)+x\left({\frac {\Delta f(a)+\Delta f(a-h)}{2}}\right)+{\frac {x^{2}\Delta ^{2}f(a-h)}{2!}}.$

where $Δ$ is the first-order forward-difference operator, i.e.

$\Delta f(a)\ {\stackrel {\mathrm {def} }{=}}\ f(a+h)-f(a).$

Astronomy

This section needs expansionwith: Astronomical details reflecting his substantial astronomical work. You can help byadding to it. (July 2016)

Some of the important contributions made by Brahmagupta in astronomy are his methods for calculating the position of heavenly bodies over time (ephemerides), their rising and setting, conjunctions, and the calculation of solar and lunar eclipses.[31]

In chapter seven of his Brahmasphutasiddhanta, entitled Lunar Crescent, Brahmagupta rebuts the idea that the Moon is farther from the Earth than the Sun, an idea which had been suggested by Vedic scripture.[clarification needed] He does this by explaining the illumination of the Moon by the Sun.[32]

7.1. If the moon were above the sun, how would the power of waxing and waning, etc., be produced from calculation of the [longitude of the] moon? the near half [would be] always bright.

7.2. In the same way that the half seen by the sun of a pot standing in sunlight is bright, and the unseen half dark, so is [the illumination] of the moon [if it is] beneath the sun.
7.3. The brightness is increased in the direction of the sun. At the end of a bright [i.e. waxing] half-month, the near half is bright and the far half dark. Hence, the elevation of the horns [of the crescent can be derived] from calculation. [...][33]

He explains that since the Moon is closer to the Earth than the Sun, the degree of the illuminated part of the Moon depends on the relative positions of the Sun and the Moon, and this can be computed from the size of the angle between the two bodies.[32]

Further work exploring the longitudes of the planets, diurnal rotation, lunar and solar eclipses, risings and settings, the moon's crescent and conjunctions of the planets, are discussed in his treatise Khandakhadyaka.

Bhaskar-1

Bhāskara (c. 600 – c. 680) (commonly called Bhaskara I to avoid confusion with the 12th century mathematician Bhāskara II) was a 7th-century mathematician, who was the first to write numbers in the Hindu decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhatta's work.[1] This commentary, Āryabhaṭīyabhāṣya, written in 629 CE, is among the oldest known prose works in Sanskrit on mathematics and astronomy. He also wrote two astronomical works in the line of Aryabhata's school, the Mahābhāskarīya and the Laghubhāskarīya.[2]

On 7 June 1979 the Indian Space Research Organisation launched Bhaskara I honouring the mathematician.

Biography

Little is known about Bhāskara's life. He was probably a Marathi astronomer.[4] He was born at Bori, in Parbhani district of Maharashtra state in India in 7th century.[citation needed]

His astronomical education was given by his father. Bhaskara is considered the most important scholar of Aryabhata's astronomical school. He and Brahmagupta are two of the most renowned Indian mathematicians who made considerable contributions to the study of fractions.

Representation of numbers

Bhaskara's probably most important mathematical contribution concerns the representation of numbers in a positional system. The first positional representations were known to Indian astronomers about 500 years ago. However, the numbers were not written in figures, but in words or allegories, and were organized in verses. For instance, the number 1 was given as moon, since it exists only once; the number 2 was represented by wings, twins, or eyes, since they always occur in pairs; the number 5 was given by the (5) senses. Similar to our current decimal system, these words were aligned such that each number assigns the factor of the power of ten corresponding to its position, only in reverse order: the higher powers were right from the lower ones.

His system is truly positional, since the same words representing, can also be used to represent the values 40 or 400.[5] Quite remarkably, he often explains a number given in this system, using the formula ankair api ("in figures this reads"), by repeating it written with the first nine Brahmi numerals, using a small circle for the zero . Contrary to his word number system, however, the figures are written in descending valuedness from left to right, exactly as we do it today. Therefore, at least since 629 the decimal system is definitely known to the Indian scientists. Presumably, Bhaskara did not invent it, but he was the first having no compunctions to use the Brahmi numerals in a scientific contribution in Sanskrit.

Further contributions

Bhaskara wrote three astronomical contributions. In 629 he annotated the Aryabhatiya, written in verses, about mathematical astronomy. The comments referred exactly to the 33 verses dealing with mathematics. There he considered variable equations and trigonometric formulae.

His work Mahabhaskariya divides into eight chapters about mathematical astronomy. In chapter 7, he gives a remarkable approximation formula for sin x, that is

$\sin x \approx \frac{16x (\pi - x)}{5 \pi^2 - 4x (\pi - x)}, \qquad (0 \leq x \leq \frac{\pi}{2} )$

which he assigns to Aryabhata. It reveals a relative error of less than 1.9% (the greatest deviation $\frac{16}{5\pi} - 1 \approx 1.859\%$ at $x=0$ ). Moreover, relations between sine and cosine, as well as between the sine of an angle >90° >180° or >270° to the sine of an angle <90° are given. Parts of Mahabhaskariya were later translated into Arabic.

Bhaskara already dealt with the assertion that if p is a prime number, then 1 + (p–1)! is divisible by p.[dubious – discuss][citation needed] It was proved later by Al-Haitham, also mentioned by Fibonacci, and is now known as Wilson's theorem.

Moreover, Bhaskara stated theorems about the solutions of today so called Pell equations. For instance, he posed the problem: "Tell me, O mathematician, what is that square which multiplied by 8 becomes - together with unity - a square?" In modern notation, he asked for the solutions of the Pell equation $8x^2 + 1 = y^2$ . It has the simple solution x = 1, y = 3, or shortly (x,y) = (1,3), from which further solutions can be constructed, e.g., (x,y) = (6,17).

Sridhar acharya

(Bengali: শ্রীধর আচার্য; c. 750 CE – c. 930 CE) was an Indian mathematician, Sanskrit pandit and philosopher. He was born in Bhurishresti (Bhurisristi or Bhurshut) village in South Radha (at present day Hughli) in the 8th Century AD. His father's name was Baladev Acharya and his mother's name was Acchoka bai. His father was a Sanskrit pandit.

Works

He was known for 2 treatises: Trisatika (nit sometimes called the Patiganitasara) and the Patiganita. His major work Patiganitasara was named Trisatika because it was written in three hundred slokas. The book discusses counting of numbers, measures, natural number, multiplication, division, zero, squares, cubes, fraction, rule of three, interest- calculation, joint business or partnership and mensurations.

He gave an exposition on the zero. He wrote, "If zero is added to any number, the sum is the same number; if zero is subtracted from any number, the number remains unchanged; if zero is multiplied by any number, the product is zero".
In the case of dividing a fraction he has found out the method of multiplying the fraction by the reciprocal of the divisor.
He wrote on the practical applications of algebra
He separated algebra from arithmetic
He was one of the first to give a formula for solving quadratic equations.

Derivation:

$ax^{2}+bx+c=0$ Multiply both sides by 4a, $4a^{2}x^{2}+4abx+4ac=0$ Subtract 4ac from both sides, $4a^{2}x^{2}+4abx=-4ac$ Add $b^{2}$ to both sides, $4a^{2}x^{2}+4abx+b^{2}=-4ac+b^{2}$ Since $(m+n)^{2}=m^{2}+2mn+n^{2}$ Complete the square on the left side, $(2ax+b)^{2}=b^{2}-4ac={D}$ Take square roots, $2ax+b=\pm {\sqrt {D}}$ $2ax=-b\pm {\sqrt {D}}$ and, divide by 2a, $x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.$

Mahāvīra

(or Mahaviracharya, "Mahavira the Teacher") was a 9th-century Jain mathematician from Karnataka, India.[1][2][3] He was the author of Gaṇitasārasan̄graha (or Ganita Sara Samgraha, c. 850), which revised the Brāhmasphuṭasiddhānta.[1] He was patronised by the Rashtrakuta king Amoghavarsha.[4] He separated astrology from mathematics. It is the earliest Indian text entirely devoted to mathematics.[5] He expounded on the same subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly. His work is a highly syncopated approach to algebra and the emphasis in much of his text is on developing the techniques necessary to solve algebraic problems.[6] He is highly respected among Indian mathematicians, because of his establishment of terminology for concepts such as equilateral, and isosceles triangle; rhombus; circle and semicircle.[7] Mahāvīra's eminence spread in all South India and his books proved inspirational to other mathematicians in Southern India.[8] It was translated into Telugu language by Pavuluri Mallana as Saar Sangraha Ganitam.[9]

He discovered algebraic identities like a3 = a (a + b) (a − b) + b2 (a − b) + b3.[3] He also found out the formula for nCr as
[n (n − 1) (n − 2) ... (n − r + 1)] / [r (r − 1) (r − 2) ... 2 * 1].[10] He devised a formula which approximated the area and perimeters of ellipses and found methods to calculate the square of a number and cube roots of a number.[11] He asserted that the square root of a negative number did not exist.

Rules for decomposing fractions

Mahāvīra's Gaṇita-sāra-saṅgraha gave systematic rules for expressing a fraction as the sum of unit fractions.[13] This follows the use of unit fractions in Indian mathematics in the Vedic period, and the Śulba Sūtras' giving an approximation of √2 equivalent to $1+{\tfrac 13}+{\tfrac 1{3\cdot 4}}-{\tfrac 1{3\cdot 4\cdot 34}}$ .[13]

In the Gaṇita-sāra-saṅgraha (GSS), the second section of the chapter on arithmetic is named kalā-savarṇa-vyavahāra (lit. "the operation of the reduction of fractions"). In this, the bhāgajāti section (verses 55–98) gives rules for the following:[13]

To express 1 as the sum of n unit fractions (GSS kalāsavarṇa 75, examples in 76):[13]

rūpāṃśakarāśīnāṃ rūpādyās triguṇitā harāḥ kramaśaḥ /

dvidvitryaṃśābhyastāv ādimacaramau phale rūpe //

When the result is one, the denominators of the quantities having one as numerators are [the numbers] beginning with one and multiplied by three, in order. The first and the last are multiplied by two and two-thirds [respectively].

$1={\frac {1}{1\cdot 2}}+{\frac {1}{3}}+{\frac {1}{3^{2}}}+\dots +{\frac {1}{3^{n-2}}}+{\frac {1}{{\frac {2}{3}}\cdot 3^{n-1}}}$

To express 1 as the sum of an odd number of unit fractions (GSS kalāsavarṇa 77):[13]

$1={\frac {1}{2\cdot 3\cdot 1/2}}+{\frac {1}{3\cdot 4\cdot 1/2}}+\dots +{\frac {1}{(2n-1)\cdot 2n\cdot 1/2}}+{\frac {1}{2n\cdot 1/2}}$

To express a unit fraction $1/q$ as the sum of n other fractions with given numerators $a_{1},a_{2},\dots ,a_{n}$ (GSS kalāsavarṇa 78, examples in 79):

${\frac {1}{q}}={\frac {a_{1}}{q(q+a_{1})}}+{\frac {a_{2}}{(q+a_{1})(q+a_{1}+a_{2})}}+\dots +{\frac {a_{n-1}}{(q+a_{1}+\dots +a_{n-2})(q+a_{1}+\dots +a_{n-1})}}+{\frac {a_{n}}{a_{n}(q+a_{1}+\dots +a_{n-1})}}$

To express any fraction $p/q$ as a sum of unit fractions (GSS kalāsavarṇa 80, examples in 81):[13]

Choose an integer i such that ${\tfrac {q+i}{p}}$ is an integer r, then write ${\frac {p}{q}}={\frac {1}{r}}+{\frac {i}{r\cdot q}}$ and repeat the process for the second term, recursively. (Note that if i is always chosen to be the smallest such integer, this is identical to the greedy algorithm for Egyptian fractions.)

To express a unit fraction as the sum of two other unit fractions (GSS kalāsavarṇa 85, example in 86):[13]

${\frac {1}{n}}={\frac {1}{p\cdot n}}+{\frac {1}{\frac {p\cdot n}{n-1}}}$ where $p$ is to be chosen such that ${\frac {p\cdot n}{n-1}}$ is an integer (for which $p$ must be a multiple of $n-1$ ). ${\frac {1}{a\cdot b}}={\frac {1}{a(a+b)}}+{\frac {1}{b(a+b)}}$

To express a fraction $p/q$ as the sum of two other fractions with given numerators $a$ and $b$ (GSS kalāsavarṇa 87, example in 88):[13]

${\frac {p}{q}}={\frac {a}{{\frac {ai+b}{p}}\cdot {\frac {q}{i}}}}+{\frac {b}{{\frac {ai+b}{p}}\cdot {\frac {q}{i}}\cdot {i}}}$ where $i$ is to be chosen such that $p$ divides $ai+b$

Some further rules were given in the Gaṇita-kaumudi of Nārāyaṇa in the 14th century.

Pavuluri Mallana

Pavuluri Mallana, who followed Adikavi Nannaya, was a mathematician of the 11th century. He was a contemporary of King Rajaraja Narendra (1022–1063 AD). He translated Ganitasara Samgraham, a mathematical treatise of Mahivaracharya, into Telugu as Sara Sangraha Ganitamu.[1] He also wrote Bhadradri Rama Satakamu published by Vavilla Ramaswamy Sastrulu and Sons in 1916.[2]

Rajaraja Narendra has donated Navakhandavada agraharam near Pithapuram named Mallana. His grandson, also named Mallana, was a famous writer. Pavuluru village is in Parchuru Mandal of Prakasam district.

Hemachandra

Acharya Hemachandra was a Jain scholar, poet, and polymath who wrote on grammar, philosophy, prosody, and contemporary history. Noted as a prodigy by his contemporaries, he gained the title kalikālasarvajña, "the all-knowing of the Kali Yuga".

Early life

Hemachandra was born in Dhandhuka, in present-day Gujarat, on Kartika Sud Purnima (the full moon day of Kartika month). His date of birth differs according to sources but 1088 is generally accepted.[note 1][1] His father, Chachiga-deva was a Modh Bania Vaishnava. His mother, Pahini, was a Jain.[2][3] Hemchandra's original given name was Changadeva. In his childhood, the Jain monk Devachandra Suri visited Dhandhuka and was impressed by the young Hemachandra's intellect. His mother and maternal uncle concurred with Devachandra, in opposition to his father, that Hemachandra be a disciple of his. Devachandra took Hemachandra to Khambhat, where Hemachandra was placed under the care of the local governor Udayana. Chachiga came to Udayana's place to take his son back, but was so overwhelmed by the kind treatment he received, that he decided to willingly leave his son with Devachandra.[4]

Some years later, Hemachandra was initiated a Jain monk on Magha Sud Chauth (4th day of the bright half of Magha month) and was given a new name, Somchandra. Udayana helped Devchandra Suri in the ceremony.[2][3] He was trained in religious discourse, philosophy, logic and grammar and became well versed in Jain and non–Jain scriptures. At the age of 21, he was ordained an acharyaof the Śvētāmbara school of Jainism at Nagaur in present-day Rajasthan. At this time, he was named Hemachandra Suri.[2][3][5][6]

Hemachandra and Siddharaja

At the time, Gujarat was ruled by the Chaulukya dynasty from Anhilavada (Patan). It is not certain when Hemachandra visited Patan for the first time. As Jain monks are mendicants for eight months and stay at one place during Chaturmas, the four monsoon months, he started living at Patan during these periods and produced the majority of his works there.[2][3]

Probably around 1125, he was introduced to the Jayasimha Siddharaja (fl. 1092–1141) and soon rose to prominence in the Chaulukya royal court.[3] According to the Prabhavaka Charita of Chandraprabha, the earliest biography of Hemachandra, Jayasimha spotted Hemachandra while passing through the streets of his capital. The king was impressed with an impromptu verse uttered by the young monk.[7]

In 1135, when the Siddharaja conquered Malwa, he brought the works of Bhoja from Dhar along with other things. One day Siddhraja came across the manuscript of Sarasvati-Kanthabharana (also known as the Lakshana Prakash), a treatise on Sanskrit grammar. He was so impressed by it that he told the scholars in his court to produce a grammar that was as easy and lucid. Hemachandra requested Siddharaja to find the eight best grammatical treatises from Kashmir. He studied them and produced a new grammar work in the style of Pāṇini's Aṣṭādhyāyī.[2][3] He named his work Siddha-Hema-Śabdanuśāśana after himself and the king. Siddharaja was so pleased with the work that he ordered it to be placed on the back of an elephant and paraded through the streets of Anhilwad Patan.[8]Hemachandra also composed the Dvyashraya Kavya, an epic on the history of the Chaulukya dynasty, to illustrate his grammar.[3]

Hemachandra and Kumarapala

Idol of Hemachandra at Jain Center of New Jersey, US

According to the Prabhachandra, there was an incident where Siddharaja wanted to kill his nephew Kumarapala because it was prophesied that the kingdom would meet its demise at Kumarapala's hands. Hemachandra hid Kumarapala under a pile of manuscripts to save him.[2]However, such motifs are common in Indian folk literature, so it is unlikely it was an actual historical event. Also, many sources differ on Siddharaja's motives.[2]

Hemachandra became the advisor to Kumarapala.[2][3] During Kumarapala's reign, Gujarat became a center of culture. Using the Jain approach of Anekantavada, Hemchandra is said to have displayed a broad-minded attitude, which pleased Kumarapala.[5] Kumarapala was a Shaiva and ordered the rebuilding of Somnath at Prabhas Patan. Some people who were jealous of Hemachandra's rising popularity with the Kumarapala complained that Hemachandra was a very arrogant person, that he did not respect the devas and that he refused to bow down to Shiva. When called upon to visit the temple on the inauguration with Kumarapala, Hemachandra readily bowed before the lingam but said:

Bhava Bijankaura-janana Ragadyam Kshayamupagata Yasya, Brahma va Vishnu va Haro Jino va Namastasmai.

I bow down to him who has destroyed the passions like attachment and malice which are the cause of the cycle of birth and death; whether he is Brahma, Vishnu, Shiva or Jina.[5][9]

Ultimately, the king became a devoted follower of Hemachandra and a champion of Jainism.[2][5]

Starting in 1121, Hemachandra was involved in the construction of the Jain temple at Taranga. His influence on Kumarapala resulted in Jainism becoming the official religion of Gujarat and animal slaughter was banned in the state. The tradition of animal sacrifice in the name of religion was completely uprooted in Gujarat. As a result, even almost 900 years after Hemchandra, Gujarat still continues to be a predominantly lacto-vegetarian state, despite having an extensive coastline.[2][3]

Death

He announced about his death six months in advance and fasted in his last days, a Jain practice called sallekhana. He died at Anhilavad Patan. The year of death differs according to sources but 1173 is generally accepted.[1]

Works

Instruction by Monks, Folio from the Siddhahemashabdanushasana

Worship of Parshvanatha, Folio from the Siddhahemashabdanushasana

A prodigious writer, Hemachandra wrote grammars of Sanskrit and Prakrit, poetry, prosody, lexicons, texts on science and logic and many branches of Indian philosophy. It is said that Hemachandra composed 3.5 crore verses in total, many of which are now lost.[citation needed]

Jain philosophy

His systematic exposition of the Jain path in the Yogaśāstra and its auto-commentary is a very influential text in Jain thought. According to Olle Quarnström it is "the most comprehensive treatise on Svetambara Jainism known to us".[10]

Grammar

Siddha-Hema-Śabdanuśāśana

This Sanskrit grammar was written in the style of Pāṇini. It has seven chapters with each chapter having four sections, similar to that of the grammar of Bhoja. The Siddha-Hema-Śabdanuśāśana also includes six Prakrit languages: the "standard" Prakrit (virtually Maharashtri Prakrit), Shauraseni, Magahi, Paiśācī, the otherwise-unattested Cūlikāpaiśācī and Apabhraṃśa (virtually Gurjar Apabhraṃśa, prevalent in the area of Gujarat and Rajasthan at that time and the precursor of Gujarati language). He gave a detailed grammar of Apabhraṃśa and also illustrated it with the folk literature of the time for better understanding. It is the only known Apabhraṃśa grammar.[3]

Poetry

Dvyashraya Kavya

To illustrate the grammar, he produced the epic poetry Dvyashraya Kavya on the history of Chaulukya dynasty. It is an important source of history of region of the time.[3]

Trishashthi-Shalaka-Purusha

The epic poem Trīṣaṣṭiśalākāpuruṣacharitra or "Lives of Sixty-Three Great Men" is a hagiographical treatment of the twenty four tirthankaras and other important persons instrumental in defining the Jain philosophical position, collectively called the "śalākāpuruṣa", their asceticism and eventual liberation from the cycle of death and rebirth, as well as the legendary spread of the Jain influence. It still serves as the standard synthesis of source material for the early history of Jainism.[3] The appendix to this work, the Pariśiṣṭaparvanor Sthavirāvalīcarita,[11] contains his own commentary and is in itself a treatise of considerable depth[3] It has been translated into English as The Lives of the Jain Elders.[12]

Other

His Kavyanuprakasha follows the model of Kashmiri rhetorician Mammata's Kavya-prakasha. He quoted other scholars like Anandavardhana and Abhinavagupta in his works.[3]

Lexicography

Abhidhan-Chintamani (IAST abhidhāna-cintāmaṇi-kośa) is a lexicon while Anekarth Kosha is a lexicon of words bearing multiple meanings. Deshi-Shabda-Sangraho or Desi-nama-mala is the lexicon of local or non-Sanskrit origin. Niganthu Sesa is a botanical lexicon.[3]

Mathematics

Hemachandra, following the earlier Gopala, presented an earlier version of the Fibonacci sequence. It was presented around 1150, about fifty years before Fibonacci (1202). He was considering the number of cadences of length n, and showed that these could be formed by adding a short syllable to a cadence of length n − 1, or a long syllable to one of n − 2. This recursion relation F(n) = F(n − 1) + F(n − 2) is what defines the Fibonacci sequence.[13][14]

Other works

His other works are Chandanushasana (prosody), commentary in rhetoric work Alankara Chudamani, Abhidhana-chintamani,[2][15] Pramana-mimansa (logic), Vitaraga-Stotra(prayers).

Bhāskara[1] (also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I) (1114–1185), was an Indian mathematician and astronomer. He was born in Bijapur in Karnataka.[2]

Bhāskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. He has been called the greatest mathematician of medieval India.[3] His main work Siddhānta Shiromani, (Sanskrit for "Crown of Treatises")[4] is divided into four parts called Lilāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya,[5] which are also sometimes considered four independent works.[6] These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karaṇa Kautūhala.[6]

Bhāskara's work on calculus predates Newton and Leibniz by over half a millennium.[7][8] He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. While Newton and Leibniz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhāskara was a pioneer in some of the principles of differential calculus. He was perhaps the first to conceive the differential coefficient and differential calculus.[9]

On 20 November 1981 the Indian Space Research Organisation launched the Bhaskara II satellite honouring the mathematician and astronomer.

Date, place, and family

Bhāskara gives his date of birth, and date of composition of his major work, in a verse in the Āryā metre:[6]

rasa-guṇa-pūrṇa-mahīsama
śhaka-nṛpa samaye 'bhavat mamotpattiḥ /
rasa-guṇa-varṣeṇa mayā
siddhānta-śiromaṇī racitaḥ //

This reveals that he was born in 1036 of the Śhaka era (1114 CE), and that he composed the Siddhānta Śiromaṇī when he was 36 years old.[6] He also wrote another work called the Karaṇa-kutūhala when he was 69 (in 1183).[6] His works show the influence of Brahmagupta, Sridhara, Mahāvīra, Padmanābha and other predecessors.[6]

He was born near Vijjadavida (believed to be Bijjaragi of Vijayapur in modern Karnataka). Bhāskara is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical center of medieval India. He lived in the Sahyadri region (Patnadevi, in Jalgaon district, Maharashtra).[1]

History records his great-great-great-grandfather holding a hereditary post as a court scholar, as did his son and other descendants. His father Mahesvara[1] (Maheśvaropādhyāya[6]) was a mathematician, astronomer[6] and astrologer, who taught him mathematics, which he later passed on to his son Loksamudra. Loksamudra's son helped to set up a school in 1207 for the study of Bhāskara's writings.He died in 1185 CE

The Siddhanta-Shiromani

Lilavati

The first section Līlāvatī (also known as pāṭīgaṇita or aṅkagaṇita) it is the name of his daughter. consists of 277 verses.[6] It covers calculations, progressions, measurement, permutations, and other topics.[6]

Bijaganita

The second section Bījagaṇita has 213 verses.[6] It discusses zero, infinity, positive and negative numbers, and indeterminate equations including (the now called) Pell's equation, solving it using a kuṭṭaka method.[6] In particular, he also solved the $61x^{2}+1=y^{2}$ case that was to elude Fermat and his European contemporaries centuries later.[6]

Grahaganita

In the third section Grahagaṇita, while treating the motion of planets, he considered their instantaneous speeds.[6] He arrived at the approximation:[11]

$\sin y'-\sin y\approx (y'-y)\cos y$ for $y'$ close to $y$ , or in modern notation:[11] ${\frac {d}{dy}}\sin y=\cos y$ .

In his words:[11]

bimbārdhasya koṭijyā guṇastrijyāhāraḥ phalaṃ dorjyāyorantaram

This result had also been observed earlier by Muñjalācārya (or Mañjulācārya) in 932, in his astronomical work 'Laghu-mānasam, in the context of a table of sines.[11]

Bhāskara also stated that at its highest point a planet's instantaneous speed is zero.[11]

Mathematics

Some of Bhaskara's contributions to mathematics include the following:

A proof of the Pythagorean theorem by calculating the same area in two different ways and then canceling out terms to get a2 + b2 = c2.[12]
In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations are explained.[13]
Solutions of indeterminate quadratic equations (of the type ax2 + b = y2).
Integer solutions of linear and quadratic indeterminate equations (Kuṭṭaka). The rules he gives are (in effect) the same as those given by the Renaissance European mathematicians of the 17th century
A cyclic Chakravala method for solving indeterminate equations of the form ax2 + bx + c = y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the chakravala method.
The first general method for finding the solutions of the problem x2 − ny2 = 1 (so-called "Pell's equation") was given by Bhaskara II.[14]
Solutions of Diophantine equations of the second order, such as 61x2 + 1 = y2. This very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in the 18th century.[13]
Solved quadratic equations with more than one unknown, and found negative and irrational solutions.[citation needed]
Preliminary concept of mathematical analysis.
Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus.[15]
Conceived differential calculus, after discovering an approximation of the derivative and differential coefficient.
Stated Rolle's theorem, a special case of one of the most important theorems in analysis, the mean value theorem. Traces of the general mean value theorem are also found in his works.
Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)
In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric results. (See Trigonometry section below.)

Arithmetic

Bhaskara's arithmetic text Leelavati covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, methods to solve indeterminate equations, and combinations.

Lilavati is divided into 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and measurement. More specifically the contents include:

Definitions.
Properties of zero (including division, and rules of operations with zero).
Further extensive numerical work, including use of negative numbers and surds.
Estimation of π.
Arithmetical terms, methods of multiplication, and squaring.
Inverse rule of three, and rules of 3, 5, 7, 9, and 11.
Problems involving interest and interest computation.
Indeterminate equations (Kuṭṭaka), integer solutions (first and second order). His contributions to this topic are particularly important,[citation needed] since the rules he gives are (in effect) the same as those given by the renaissance European mathematicians of the 17th century, yet his work was of the 12th century. Bhaskara's method of solving was an improvement of the methods found in the work of Aryabhata and subsequent mathematicians.

His work is outstanding for its systemisation, improved methods and the new topics that he has introduced. Furthermore, the Lilavati contained excellent recreative problems and it is thought that Bhaskara's intention may have been that a student of 'Lilavati' should concern himself with the mechanical application of the method.[citation needed]

Algebra

His Bijaganita ("Algebra") was a work in twelve chapters. It was the first text to recognize that a positive number has two square roots (a positive and negative square root).[16] His work Bijaganita is effectively a treatise on algebra and contains the following topics:

Positive and negative numbers.
Zero.
The 'unknown' (includes determining unknown quantities).
Determining unknown quantities.
Surds (includes evaluating surds).
Kuṭṭaka (for solving indeterminate equations and Diophantine equations).
Simple equations (indeterminate of second, third and fourth degree).
Simple equations with more than one unknown.
Indeterminate quadratic equations (of the type ax2 + b = y2).
Solutions of indeterminate equations of the second, third and fourth degree.
Quadratic equations.
Quadratic equations with more than one unknown.
Operations with products of several unknowns.

Bhaskara derived a cyclic, chakravala method for solving indeterminate quadratic equations of the form ax2 + bx + c = y.[16] Bhaskara's method for finding the solutions of the problem Nx2 + 1 = y2 (the so-called "Pell's equation") is of considerable importance.[14]

Trigonometry

The Siddhānta Shiromani (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also developed spherical trigonometry, along with other interesting trigonometrical results. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskara, results found in his works include computation of sines of angles of 18 and 36 degrees, and the now well known formulae for $\sin \left(a+b\right)$ and $\sin \left(a-b\right)$ .

Calculus

His work, the Siddhānta Shiromani, is an astronomical treatise and contains many theories not found in earlier works.[citation needed] Preliminary concepts of infinitesimal calculus and mathematical analysis, along with a number of results in trigonometry, differential calculus and integral calculus that are found in the work are of particular interest.

Evidence suggests Bhaskara was acquainted with some ideas of differential calculus.[16] Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of 'infinitesimals'.[17]

There is evidence of an early form of Rolle's theorem in his work
- If $f\left(a\right)=f\left(b\right)=0$ then $f'\left(x\right)=0$ for some $\ x$ with $\ a<x<b$
He gave the result that if {\displaystyle x\approx y} $x\approx y$ then {\displaystyle \sin(y)-\sin(x)\approx (y-x)\cos(y)} $\sin(y)-\sin(x)\approx (y-x)\cos(y)$ , thereby finding the derivative of sine, although he never developed the notion of derivatives.[18]
- Bhaskara uses this result to work out the position angle of the ecliptic, a quantity required for accurately predicting the time of an eclipse.
In computing the instantaneous motion of a planet, the time interval between successive positions of the planets was no greater than a truti, or a 1⁄33750 of a second, and his measure of velocity was expressed in this infinitesimal unit of time.
He was aware that when a variable attains the maximum value, its differential vanishes.
He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero.[citation needed] In this result, there are traces of the general mean value theorem, one of the most important theorems in analysis, which today is usually derived from Rolle's theorem. The mean value theorem was later found by Parameshvara in the 15th century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati.

Madhava (1340–1425) and the Kerala School mathematicians (including Parameshvara) from the 14th century to the 16th century expanded on Bhaskara's work and further advanced the development of calculus in India.

Astronomy

Using an astronomical model developed by Brahmagupta in the 7th century, Bhaskara accurately defined many astronomical quantities, including, for example, the length of the sidereal year, the time that is required for the Earth to orbit the Sun, as 365.2588 days which is the same as in Suryasiddhanta.[citation needed] The modern accepted measurement is 365.2563 days, a difference of just 3.5 minutes.[citation needed]

His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first part on mathematical astronomy and the second part on the sphere.

The twelve chapters of the first part cover topics such as:

Mean longitudes of the planets.
True longitudes of the planets.
The three problems of diurnal rotation.(Diurnal motion is an astronomical term referring to the apparent daily motion of stars around the Earth, or more precisely around the two celestial poles. It is caused by the Earth's rotation on its axis, so every star apparently moves on a circle, that is called the diurnal circle.)
Syzygies.
Lunar eclipses.
Solar eclipses.
Latitudes of the planets.
Sunrise equation
The Moon's crescent.
Conjunctions of the planets with each other.
Conjunctions of the planets with the fixed stars.
The paths of the Sun and Moon.

The second part contains thirteen chapters on the sphere. It covers topics such as:

Praise of study of the sphere.
Nature of the sphere.
Cosmography and geography.
Planetary mean motion.
Eccentric epicyclic model of the planets.
The armillary sphere.
Spherical trigonometry.
Ellipse calculations.[citation needed]
First visibilities of the planets.
Calculating the lunar crescent.
Astronomical instruments.
The seasons.
Problems of astronomical calculations.

Engineering

The earliest reference to a perpetual motion machine date back to 1150, when Bhāskara II described a wheel that he claimed would run forever.[19]

Bhāskara II used a measuring device known as Yaṣṭi-yantra. This device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale.[20]

Legends

In his book Lilavati, he reasons: "In this quantity also which has zero as its divisor there is no change even when many quantities have entered into it or come out [of it], just as at the time of destruction and creation when throngs of creatures enter into and come out of [him, there is no change in] the infinite and unchanging [Vishnu]".[21]

"Behold!"

It has been stated, by several authors, that Bhaskara II proved the Pythagorean theorem by drawing a diagram and providing the single word "Behold!".[22][23] Sometimes Bhaskara's name is omitted and this is referred to as the Hindu proof, well known by schoolchildren.[24]

However, as mathematics historian Kim Plofker points out, after presenting a worked out example, Bhaskara II states the Pythagorean theorem:

Hence, for the sake of brevity, the square-root of the sum of the squares of the arm and upright is the hypotenuse: thus it is demonstrated.[25]

This is followed by:

And otherwise, when one has set down those parts of the figure there [merely] seeing [it is sufficient].[25]

Plofker suggests that this additional statement may be the ultimate source of the widespread "Behold!" legend.

Narayana Pandit

From Wikipedia, the free encyclopedia

Narayana Pandita (Bengali: নারায়ণ পণ্ডিত; Sanskrit: नारायण पण्डित) (1340–1400[citation needed]) was a major mathematician of India. Plofker writes that his texts were the most significant Sanskrit mathematics treatises after those of Bhaskara II, other than the Kerala school.[1]:52 He wrote the Ganita Kaumudi (lit "Moonlight of mathematics"[2]) in 1356[2]about mathematical operations. The work anticipated many developments in combinatorics. About his life, the most that is known is that:[1]

His father’s name was Nṛsiṃha or Narasiṃha, and the distribution of the manuscripts of his works suggests that he may have lived and worked in the northern half of India.

Narayana Pandit had written two works, an arithmetical treatise called Ganita Kaumudi and an algebraic treatise called Bijganita Vatamsa. Narayanan is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavati, titled Karmapradipika (or Karma-Paddhati).[3] Although the Karmapradipika contains little original work, it contains seven different methods for squaring numbers, a contribution that is wholly original to the author, as well as contributions to algebra and magic squares.[3]

Narayana's other major works contain a variety of mathematical developments, including a rule to calculate approximate values of square roots, investigations into the second order indeterminate equation nq2 + 1 = p2 (Pell's equation), solutions of indeterminate higher-order equations, mathematical operations with zero, several geometrical rules, and a discussion of magic squares and similar figures.[3] Evidence also exists that Narayana made minor contributions to the ideas of differential calculus found in Bhaskara II's work. Narayana has also made contributions to the topic of cyclic quadrilaterals.[4] Narayana is also credited with developing a method for systematic generation of all permutations of a given sequence.

Narayana's cows is an integer sequence created by considering a cow, which begins to have one baby a year, beginning in its fourth year, and all its children do the same. A000930: 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, …

Madhava of Sangamagrama

Madhava of Sangamagrama (c. 1340 – c. 1425), was a mathematician and astronomer from the town of Sangamagrama (believed to be present-day Aloor, Irinjalakuda in Thrissur District), Kerala, India. He is considered the founder of the Kerala school of astronomy and mathematics. He was the first to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity".[1] One of the greatest mathematician-astronomers of the Middle Ages, Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra.

Some scholars have also suggested that Madhava's work, through the writings of the Kerala school, may have been transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Muziris at the time. As a result, it may have had an influence on later European developments in analysis and calculus.

Name

Madhava was born as Irińńaŗappiļļy or Iriññinavaļļi Mādhava . He had written that his house name was related to the Vihar where a plant called "bakuļam" was planted. According to Achyuta Pisharati, (who wrote a commentary on Veṇvāroha written by Madhava) bakuļam was locally known as "iraňňi". Dr. K.V. Sarma, an authority on Madhava has the opinion that the house name is either Irińńāŗappiļļy or Iriññinavaļļy'.[citation needed]

Irinjalakuda was once known as 'Irińńāţikuţal'. Sangamagrāmam (lit. sangamam = union, grāmam = village) is a rough translation to Sanskrit from Dravidian word 'Irińńāţikuţal', which means 'iru (two) ańńāţi (market) kǖţal (union)' or the union of two markets.

Historiography

Although there is some evidence of mathematical work in Kerala prior to Madhava (e.g., Sadratnamala c. 1300, a set of fragmentary results[6]), it is clear from citations that Madhava provided the creative impulse for the development of a rich mathematical tradition in medieval Kerala. However, most of Madhava's original work (except a couple of them) is lost. He is referred to in the work of subsequent Kerala mathematicians, particularly in Nilakantha Somayaji's Tantrasangraha (c. 1500), as the source for several infinite series expansions, including sinθ and arctanθ. The 16th-century text Mahajyānayana prakāra (Method of Computing Great Sines) cites Madhava as the source for several series derivations for π. In Jyeṣṭhadeva's Yuktibhāṣā (c. 1530),[7] written in Malayalam, these series are presented with proofs in terms of the Taylor series expansions for polynomials like 1/(1+x2), with x = tanθ, etc.

Thus, what is explicitly Madhava's work is a source of some debate. The Yukti-dipika (also called the Tantrasangraha-vyakhya), possibly composed by Sankara Variyar, a student of Jyeṣṭhadeva, presents several versions of the series expansions for sinθ, cosθ, and arctanθ, as well as some products with radius and arclength, most versions of which appear in Yuktibhāṣā. For those that do not, Rajagopal and Rangachari have argued, quoting extensively from the original Sanskrit,[1] that since some of these have been attributed by Nilakantha to Madhava, possibly some of the other forms might also be the work of Madhava.

Others have speculated that the early text Karanapaddhati (c. 1375–1475), or the Mahajyānayana prakāra might have been written by Madhava, but this is unlikely.[3]

Karanapaddhati, along with the even earlier Keralese mathematics text Sadratnamala, as well as the Tantrasangraha and Yuktibhāṣā, were considered in an 1834 article by Charles Matthew Whish, which was the first to draw attention to their priority over Newton in discovering the Fluxion (Newton's name for differentials).[6] In the mid-20th century, the Russian scholar Jushkevich revisited the legacy of Madhava,[8] and a comprehensive look at the Kerala school was provided by Sarma in 1972.[9]

Lineage

Explanation of the sine rule in Yuktibhāṣā

There are several known astronomers who preceded Madhava, including Kǖţalur Kizhār (2nd century),[10] Vararuci (4th century), and Sankaranarayana (866 AD). It is possible that other unknown figures may have preceded him. However, we have a clearer record of the tradition after Madhava. Parameshvara was a direct disciple. According to a palmleaf manuscript of a Malayalam commentary on the Surya Siddhanta, Parameswara's son Damodara (c. 1400–1500) had both Nilakantha Somayaji as his disciples. Jyeshtadevan was the disciple of Nilakanda. Achyuta Pisharati of Trikkantiyur is mentioned as a disciple of Jyeṣṭhadeva, and the grammarian Melpathur Narayana Bhattathiri as his disciple.[7]

Contributions

If we consider mathematics as a progression from finite processes of algebra to considerations of the infinite, then the first steps towards this transition typically come with infinite series expansions. It is this transition to the infinite series that is attributed to Madhava. In Europe, the first such series were developed by James Gregory in 1667. Madhava's work is notable for the series, but what is truly remarkable is his estimate of an error term (or correction term).[11] This implies that the limit nature of the infinite series was quite well understood by him. Thus, Madhava may have invented the ideas underlying infinite series expansions of functions, power series, trigonometric series, and rational approximations of infinite series.[12]

However, as stated above, which results are precisely Madhava's and which are those of his successors, are somewhat difficult to determine. The following presents a summary of results that have been attributed to Madhava by various scholars.

Infinite series

Among his many contributions, he discovered the infinite series for the trigonometric functions of sine, cosine, tangent and arctangent, and many methods for calculating the circumference of a circle. One of Madhava's series is known from the text Yuktibhāṣā, which contains the derivation and proof of the power series for inverse tangent, discovered by Madhava.[13] In the text, Jyeṣṭhadeva describes the series in the following manner:

“The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.[14]”

This yields:

$r\theta ={{\frac {r\sin \theta }{\cos \theta }}}-(1/3)\,r\,{{\frac {\left(\sin \theta \right)^{{3}}}{\left(\cos \theta \right)^{{3}}}}}+(1/5)\,r\,{{\frac {\left(\sin \theta \right)^{{5}}}{\left(\cos \theta \right)^{{5}}}}}-(1/7)\,r\,{{\frac {\left(\sin \theta \right)^{{7}}}{\left(\cos \theta \right)^{{7}}}}}+\cdots$

or equivalently:

$\theta =\tan \theta -{\frac {\tan ^{3}\theta }{3}}+{\frac {\tan ^{5}\theta }{5}}-{\frac {\tan ^{7}\theta }{7}}+\cdots$

This series is Gregory's series (named after James Gregory, who rediscovered it three centuries after Madhava). Even if we consider this particular series as the work of Jyeṣṭhadeva, it would pre-date Gregory by a century, and certainly other infinite series of a similar nature had been worked out by Madhava. Today, it is referred to as the Madhava-Gregory-Leibniz series.[14][15]

Trigonometry

Madhava composed an accurate table of sines. Marking a quarter circle at twenty-four equal intervals, he gave the lengths of the half-chord (sines) corresponding to each of them. It is believed that he may have computed these values based on the series expansions:[4]

sin q = q – q3/3! + q5/5! – q7/7! +...cos q = 1 – q2/2! + q4/4! – q6/6! +...

The value of π (pi)

Madhava's work on the value of the mathematical constant Pi is cited in the Mahajyānayana prakāra ("Methods for the great sines").[citation needed] While some scholars such as Sarma[7] feel that this book may have been composed by Madhava himself, it is more likely the work of a 16th-century successor.[4] This text attributes most of the expansions to Madhava, and gives the following infinite series expansion of π, now known as the Madhava-Leibniz series:[16][17]

${\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +{\frac {(-1)^{n}}{2n+1}}+\cdots$

which he obtained from the power series expansion of the arc-tangent function. However, what is most impressive is that he also gave a correction term, Rn, for the error after computing the sum up to n terms. Madhava gave three expressions for the correction term Rn,[4] namely

Rn = 1/(4n), orRn = n/ (4n2 + 1), orRn = (n2 + 1) / (4n3 + 5n).

where the third correction leads to highly accurate computations of π.

It is not clear how Madhava might have found these correction terms.[18]

He also gave a more rapidly converging series by transforming the original infinite series of π, obtaining the infinite series

$\pi ={\sqrt {12}}\left(1-{1 \over 3\cdot 3}+{1 \over 5\cdot 3^{2}}-{1 \over 7\cdot 3^{3}}+\cdots \right)$

By using the first 21 terms to compute an approximation of π, he obtains a value correct to 11 decimal places (3.14159265359).[19] The value of 3.1415926535898, correct to 13 decimals, is sometimes attributed to Madhava,[20] but may be due to one of his followers. These were the most accurate approximations of π given since the 5th century (see History of numerical approximations of π).

The text Sadratnamala, usually considered as prior to Madhava, appears to give the astonishingly accurate value of π =3.14159265358979324 (correct to 17 decimal places). Based on this, R. Gupta has argued that this text may also have been composed by Madhava.[3][19]

Algebra

[citation needed]

Madhava also carried out investigations into other series for arc lengths and the associated approximations to rational fractions of π, found methods of polynomial expansion, discovered tests of convergence of infinite series, and the analysis of infinite continued fractions.[3] He also discovered the solutions of transcendental equations by iteration, and found the approximation of transcendental numbers by continued fractions.[3]

Calculus

Madhava laid the foundations for the development of calculus, which were further developed by his successors at the Kerala school of astronomy and mathematics.[12][21] (It should be noted that certain ideas of calculus were known to earlier mathematicians.) Madhava also extended some results found in earlier works, including those of Bhāskara II.

Madhava developed some components of calculus such as differentiation, term-by-term integration, iterative methods for solutions of non-linear equations, and the theory that the area under a curve is its integral.[citation needed]

Madhava's works

K.V. Sarma has identified Madhava as the author of the following works:[22][23]

Golavada
Madhyamanayanaprakara
Mahajyanayanaprakara (Method of Computing Great Sines)
Lagnaprakarana (लग्नप्रकरण)
Venvaroha (वेण्वारोह)[24]
Sphutacandrapti (स्फुटचन्द्राप्ति)
Aganita-grahacara (अगणित-ग्रहचार)
Chandravakyani (चन्द्रवाक्यानि) (Table of Moon-mnemonics)

Kerala School of Astronomy and Mathematics

The Kerala school of astronomy and mathematics flourished for at least two centuries beyond Madhava. In Jyeṣṭhadeva we find the notion of integration, termed sankalitam, (lit. collection), as in the statement:

ekadyekothara pada sankalitam samam padavargathinte pakuti,[15]

which translates as the integration a variable (pada) equals half that variable squared (varga); i.e. The integral of x dx is equal to x2 / 2. This is clearly a start to the process of integral calculus. A related result states that the area under a curve is its integral. Most of these results pre-date similar results in Europe by several centuries. In many senses, Jyeshthadeva's Yuktibhāṣā may be considered the world's first calculus text.[6][12][21]

The group also did much other work in astronomy; indeed many more pages are developed to astronomical computations than are for discussing analysis related results.[7]

The Kerala school also contributed much to linguistics (the relation between language and mathematics is an ancient Indian tradition, see Katyayana). The ayurvedic and poetic traditions of Kerala can also be traced back to this school. The famous poem, Narayaneeyam, was composed by Narayana Bhattathiri.

Influence

Madhava has been called "the greatest mathematician-astronomer of medieval India",[3] or as "the founder of mathematical analysis; some of his discoveries in this field show him to have possessed extraordinary intuition."[25] O'Connor and Robertson state that a fair assessment of Madhava is that he took the decisive step towards modern classical analysis.[4]

Possible propagation to Europe

The Kerala school was well known in the 15th and 16th centuries, in the period of the first contact with European navigators in the Malabar Coast. At the time, the port of Muziris, near Sangamagrama, was a major center for maritime trade, and a number of Jesuit missionaries and traders were active in this region. Given the fame of the Kerala school, and the interest shown by some of the Jesuit groups during this period in local scholarship, some scholars, including G. Joseph of the U. Manchester have suggested[26] that the writings of the Kerala school may have also been transmitted to Europe around this time, which was still about a century before Newton.

Vatasseri Parameshvara Nambudiri (c. 1380–1460)

was a major Indian mathematician and astronomer of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama. He was also an astrologer. Parameshvara was a proponent of observational astronomy in medieval India and he himself had made a series of eclipse observations to verify the accuracy of the computational methods then in use. Based on his eclipse observations, Parameshvara proposed several corrections to the astronomical parameters which had been in use since the times of Aryabhata. The computational scheme based on the revised set of parameters has come to be known as the Drgganita or Drig system. Parameshvara was also a prolific writer on matters relating to astronomy. At least 25 manuscripts have been identified as being authored by Parameshvara.

Biographical details

Parameshvara was a Hindu of Bhrgugotra following the Ashvalayanasutra of the Rgveda. Parameshvara's family name (Illam) was Vatasseri (also called Vatasreni) and his family resided in the village of Alathiyur (Sanskritised as Asvatthagrama) in Tirur, Kerala. Alathiyur is situated on the northern bank of the river Nila (river Bharathappuzha) at its mouth in Kerala. He was a grandson of a disciple of Govinda Bhattathiri (1237–1295 CE), a legendary figure in the astrological traditions of Kerala.

Parameshvara studied under teachers Rudra and Narayana, and also under Sangamagrama Madhava (c. 1350 – c. 1425) the founder of the Kerala school of astronomy and mathematics. Damodara, another prominent member of the Kerala school, was his son and also his pupil. Parameshvara was also a teacher of Nilakantha Somayaji (1444–1544) the author of the celebrated Tantrasamgraha.

Work

Parameshvara wrote commentaries on many mathematical and astronomical works such as those by Bhaskara I and Aryabhata. He made a series of eclipse observations over a 55-year period, and constantly attempted to compare these with the theoretically computed positions of the planets. He revised planetary parameters based on his observations.

One of Parameshvara's more significant contributions was his mean value type formula for the inverse interpolation of the sine.[2]

He was the first mathematician to give the radius of circle with an inscribed quadrilateral.[3] The expression is sometimes attributed to Lhuilier (1782), 350 years later. With the sides of the cyclic quadrilateral being a, b, c, and d, the radius R of the circumscribed circle is:

$R = \sqrt {\frac{(ab + cd) (ac + bd) (ad + bc)}{(a + b + c - d) (b + c + d - a) (c + d + a - b) (d + a + b - c)}}.$

Works by Parameshvara

The following works of Parameshvara are well-known.[4] A complete list of all manuscripts attributed to Parameshvara is available in Pingree.[1]

Bhatadipika – Commentary on Āryabhaṭīya of Āryabhaṭa I
Karmadipika – Commentary on Mahabhaskariya of Bhaskara I
Paramesvari – Commentary on Laghubhaskariya of Bhaskara I
Sidhantadipika – Commentary on Mahabhaskariyabhashya of Govindasvāmi
Vivarana – Commentary on Surya Siddhanta and Lilāvati
Drgganita – Description of the Drig system (composed in 1431 CE)
Goladipika – Spherical geometry and astronomy (composed in 1443 CE)
Grahanamandana – Computation of eclipses (Its epoch is 15 July 1411 CE.)
Grahanavyakhyadipika – On the rationale of the theory of eclipses
Vakyakarana – Methods for the derivation of several astronomical tables.

Nilakantha was born into a Namputiri Brahmin family which came from South Malabar in Kerala. The Nambudiri is the main caste of Kerala. It is an orthodox caste whose members consider themselves descendants of the ancient Vedic religion.

He was born in a house called Kelallur which it is claimed coincides with the present Etamana in the village of Trkkantiyur near Tirur in south India. His father was Jatavedas and the family belonged to the Gargya gotra, which was a Indian caste that prohibits marriage to anyone outside the caste. The family followed the Ashvalayana sutra which was a manual of sacrificial ceremonies in the Rigveda, a collection of Vedic hymns. He worshipped the personified deity Soma who was the "master of plants" and the healer of disease. This explains the name Somayaji which means he was from a family qualified to conduct the Soma ritual.

Nilakantha studied astronomy and Vedanta, one of the six orthodox systems of Indian Hindu philosophy, under the teacher Ravi. He was also taught by Damodra who was the son of Paramesvara. Paramesvara was a famous Indian astronomer and Damodra followed his father's teachings. This led Nilakantha also to become a follower of Paramesvara. A number of texts on mathematical astronomy written by Nilakantha have survived. In all he wrote about ten treatises on astronomy.

The Tantrasamgraha is his major astronomy treatise written in 1501. It consists of 432 Sanskrit verses divided into 8 chapters, and it covers various aspects of Indian astronomy. It is based on the epicyclic and eccentric models of planetary motion. The first two chapters deal with the motions and longitudes of the planets. The third chapter Treatise on shadow deals with various problems related with the sun's position on the celestial sphere, including the relationships of its expressions in the three systems of coordinates, namely ecliptic, equatorial and horizontal coordinates.

The fourth and fifth chapters are Treatise on the lunar eclipse and On the solar eclipse and these two chapters treat various aspects of the eclipses of the sun and the moon. The sixth chapter is On vyatipataand deals with the complete deviation of the longitudes of the sun and the moon. The seventh chapter On visibility computation discusses the rising and setting of the moon and planets. The final chapter On elevation of the lunar cusps examines the size of the part of the moon which is illuminated by the sun and gives a graphical representation of it.

The Tantrasamgraha is very important in terms of the mathematics Nilakantha uses. In particular he uses results discovered by Madhava and it is an important source of the remarkable mathematical results which he discovered. However, Nilakantha does not just use Madhava's results, he extends them and improves them. An anonymous commentary entitled Tantrasangraha-vakhya appeared and, somewhat later in about 1550, Jyesthadeva published a commentary entitled Yuktibhasa that contained proofs of the earlier results by Madhava and Nilakantha. This is quite unusual for an Indian text in giving mathematical proofs.

The series π/4 = 1 - 1/3 + 1/5 - 1/7 + ... is a special case of the series representation for arctan, namely

tan-1x = x - x3/3 + x5/5 - x7/7 + ...

It is well known that one simply puts x = 1 to obtain the series for π/4. The author of [4] reports on the appearance of these series in the work of Leibniz and James Gregory from the 1670s. The contributions of the two European mathematicians to this series are well known but in [4] the results on this series in the work of Madhava nearly three hundred years earlier as presented by Nilakantha in the Tantrasamgraha is also discussed.

Nilakantha derived the series expansion

tan-1x = x - x3/3 + x5/5 - x7/7 + ...

by obtaining an approximate expression for an arc of the circumference of a circle and then considering the limit. An interesting feature of his work was his introduction of several additional series for π/4 that converged more rapidly than

π/4 = 1 - 1/3 + 1/5 - 1/7+ ... .

The author of [4] provides a reconstruction of how he may have arrived at these results based on the assumption that he possessed a certain continued fraction representation for the tail series

1/(n+2) - 1/(n+4) + 1/(n+6) - 1/(n+8) + .... .

The Tantrasamgraha is not the only work of Nilakantha of which we have the text. He also wrote Golasara which is written in fifty-six Sanskrit verses and shows how mathematical computations are used to calculate astronomical data. The Siddhanta Darpana is written in thirty-two Sanskrit verses and describes a planetary model. The Candracchayaganita is written in thirty-one Sanskrit verses and explains the computational methods used to calculate the moon's zenith distance.

The head of the Nambudiri caste in Nilakantha's time was Netranarayana and he became Nilakantha's patron for another of his major works, namely the Aryabhatiyabhasya Ⓣ which is a commentary on the Aryabhatiya Ⓣ of Aryabhata I. In this work Nilakantha refers to two eclipses which he observed, the first on 6 March 1467 and the second on 28 July 1501 at Anantaksetra. Nilakantha also refers in the Aryabhatiyabhasya Ⓣ to other works which he wrote such as the Grahanirnaya on eclipses which have not survived.

Kelallur Nilakantha Somayaji (also referred to as Kelallur Comatiri;[1] 14 June 1444 – 1544) was a major mathematician and astronomer of the Kerala school of astronomy and mathematics in India. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501. He had also composed an elaborate commentary on Aryabhatiya called the Aryabhatiya Bhasya. In this Bhasya, Nilakantha had discussed infinite series expansions of trigonometric functions and problems of algebra and spherical geometry. Grahapareeksakrama is a manual on making observations in astronomy based on instruments of the time.

Biographical details

Nilakantha Somayaji was one of the very few authors of the scholarly traditions of India who had cared to record details about his own life and times. So fortunately a few accurate particulars about Nilakantha Somayaji are known.[2][3]

In one of his works titled Siddhanta-star and also in his own commentary on Siddhanta-darpana, Nilakantha Somayaji has stated that he was born on Kali-day 1,660,181 which works out to 14 June 1444 CE. A contemporary reference to Nilakantha Somayaji in a Malayalam work on astrology implies that Somayaji lived to a ripe old age even to become a centenarian. Sankara Variar, a pupil of Nilakantha Somayaji, in his commentary on Tantrasamgraha titled Tantrasamgraha-vyakhya, points out that the first and last verses of Tantrasamgraha contain chronograms specifying the Kali-days of the commencement (1,680,548) and of completion (1,680,553) of Somayaji's magnum opus Tantrasamgraha. Both these days occur in 1500 CE.

In Aryabhatiya-bhashya, Nilakantha Somayaji has stated that he was the son of Jatavedas and he had a brother named Sankara. Somayaji has further stated that he was a Bhatta belonging to the Gargya gotra and was a follower of Asvalayana-sutra of Rigveda. References in his own Laghuramayana indicate that Nilakantha Somayaji was a member of the Kelallur family (Sanskritised as Kerala-sad-grama) residing at Kundagrama, now known as Trikkandiyur in modern Tirur, Kerala. His wife was named Arya and he had two sons Rama and Dakshinamurti.

Nilakantha Somayaji studied vedanta and some aspects of astronomy under one Ravi. However, It was Damodara, son of Kerala-drgganita author Paramesvara, who initiated him into the science of astronomy and instructed him in the basic principles of mathematical computations. The great Malayalam poet Thunchaththu Ramanujan Ezhuthachan is said to have been a student of Nilakantha Somayaji.

The epithet Somayaji is a title assigned to or assumed by a Namputiri who has performed the vedic ritual of Somayajna.[4] So it could be surmised that Nilakantha Somayaji had also performed a Somayajna ritual and assumed the title of a Somayaji in later life. In colloquial Malayalam usage the word Somayaji has been corrupted to Comatiri.

Nilakantha Somayaji as a polymath

Nilakantha's writings substantiate his knowledge of several branches of Indian philosophy and culture. It is said that he could refer to a Mimamsa authority to establish his view-point in a debate and with equal felicity apply a grammatical dictum to the same purpose. In his writings he refers to a Mimamsa authority, quotes extensively from Pingala's chandas-sutra, scriptures, Dharmasastras, Bhagavata and Vishnupurana also. Sundararaja, a contemporary Tamil astronomer, refers to Nilakantha as sad-darshani-parangata, one who had mastered the six systems of Indian philosophy.[2]

Astronomy

In his Tantrasangraha, Nilakantha revised Aryabhata's model for the planets Mercury and Venus. His equation of the centre for these planets remained the most accurate until the time of Johannes Kepler in the 17th century.[5]

In his Aryabhatiyabhasya, a commentary on Aryabhata's Aryabhatiya, Nilakantha developed a computational system for a partially heliocentric planetary model in which Mercury, Venus, Mars, Jupiter and Saturn orbit the Sun, which in turn orbits the Earth, similar to the Tychonic system later proposed by Tycho Brahe in the late 16th century. Most astronomers of the Kerala school who followed him accepted this planetary model.[5][6]

Works of Nilakantha Somayaji

The following is a brief description of the works by Nilakantha Somayaji dealing with astronomy and mathematics.[2][7]

Tantrasamgraha
Golasara : Description of basic astronomical elements and procedures
Sidhhantadarpana : A short work in 32 slokas enunciating the astronomical constants with reference to the Kalpa and specifying his views on astronomical concepts and topics.
Candrachayaganita : A work in 32 verses on the methods for the calculation of time from the measurement of the shadow of the gnomon cast by the moon and vice versa.
Aryabhatiya-bhashya : Elaborate commentary on Aryabhatiya.
Sidhhantadarpana-vyakhya : Commentary on his own Siddhantadarapana.
Chandrachhayaganita-vyakhya : Commentary on his own Chandrachhayaganita.
Sundaraja-prasnottara : Nilakantha's answers to questions posed by Sundaraja, a Tamil Nadu based astronomer.
Grahanadi-grantha : Rationale of the necessity of correcting old astronomical constants by observations.
Grahapariksakrama : Description of the principles and methods for verifying astronomical computations by regular observations.
Jyotirmimamsa : Analysis of astronomy.

Raghunatha Siromani

Raghunatha Shiromani (Bengali: রঘুনাথ শিরোমণি, IAST: Raghunātha Śiromaṇi) (c. 1477–1547[1]) was an Indian philosopher and logician. He was born at Nabadwip in present-day Nadia district of West Bengal state. He was the grandson of Śulapāṇi (c. 14th century CE), a noted writer on Smṛti from his mother's side. He was a pupil of Vāsudeva Sārvabhauma. He brought the new school of Nyaya, Navya Nyāya, representing the final development of Indian formal logic, to its zenith of analytic power.

Raghunatha's analysis of relations revealed the true nature of number, inseparable from the abstraction of natural phenomena, and his studies of metaphysics dealt with the negation or nonexistence of a complex reality. His most famous work in logic was the Tattvacintāmaṇidīdhiti, a commentary on the Tattvacintāmaṇi of Gangeśa Upādhyāya, founder of the Navya Nyāya school.

A descriptive information of Raghunatha with some controversial issues (his connection with Mahaprabhu Shri Chaitanya) and bibliography are to be found at Raghunatha: A Name of Negatives. The contemporary deployment of a new category, svatva ( endowment, possessed-ness, entitlement, my-ness), introduced by Raghunatha, is discussed in Language: From I-dentity to My-dentity.

Mahendra Sūri

Mahendra Sūri is the 14th century Jain astronomer who wrote the Yantraraja, the first Indian treatise on the astrolabe.[1]

Suri was a Jain. Jainism began around the sixth century BC and the religion had a strong influence on mathematics particularly in the last couple of centuries BC. By the time of Mahendra Suri, however, Jainism had lost support as a national religion and was much less vigorous. It had been influenced by Islam and in particular Islamic astronomy came to form a part of the background. However, Pingree in [4] writes that this filtering of Islamic astronomy into Indian culture was:- ... not allowed to affect in any way the structure of the traditional science. Mahendra Suri was a pupil of Madana Suri. He is famed as the first person to write a Sanskrit treatise on the astrolabe. Ohashi writes in [3] of the early history of the astrolabe in the Delhi Sultanate in India:

The astrolabe was introduced into India at the time of Firuz Shah Tughluq (reign AD 1351 - 88), and Mahendra Suri wrote the first Sanskrit treatise on the astrolabe entitled Yantraraja (AD 1370). The Delhi Sultanate was established around 1200 and from that time on Muslim culture flourished in India. The ideas of Islamic astronomy began to appear in works in the Sanskrit language and it is the Islamic ideas on the astrolabe which Mahendra Suri wrote on in his famous text. It is clear from the various references in the text and also from the particular values that Mahendra Suri uses for the angle of the ecliptic etc. that his work is based on Islamic rather than traditional Indian astronomy works.

Mahendra Suri (1340-1410) is the 14th century Jain astronomer who wrote the Yantraraja, the first Indian treatise on the astrolabe.

Astrolobe [Universe within one's palm] is a higly sophisticated astromical intrument of the pre-modern times. It is a versatile observational and computational instrument. As an observational instrument, it was employed for measuring the altitudes of heavenly bodies and for measuring the heights and distances in land survery. As an computational device, it can simulate the motion of the heavens at any given locality and time. It was also an analog computer for solving numerous problems in sphercial trignometry.

He was a pupil of Madana Suri. Mahendra Suri acted as a mediator between the Islamic and sanskritic tradition of learning.

The Yantraraja or "the king of astronomical instruments" is divided into five chapters
Chapter 1: Ganitadhyaya provides trigonometical parameters needed for the construction of astrolabe.
Chapter 2: Yantraghatanadhaya enumerates the different parts of astrolabe
Chapter 3: Yantraracanadhyaya construciton of common northern astrolabe and other variants
Chapter 4: Yantrasodhanadhyaya the method of verifying whether the astrolabe is properly constructed or not
Chapter 5: Yantravicaranadhyaya the use of astrolabe as an observational and computational instrument and dwells on the various problems in astronomy and spherical trinometry that can be solved using astrolabe.

Sankara Variar

hankara Variyar (IAST: Śaṅkara Vāriyar; c. 1500 – c. 1560[1]) was an astronomer-mathematician of the Kerala school of astronomy and mathematics. His family were employed as temple-assistants in the Shiva temple at Tṛkkuṭaveli near modern Ottapalam.[2]

Mathematical lineage

He was taught mainly by Nilakantha Somayaji (1444–1544), the author of the Tantrasamgraha and Jyesthadeva (1500–1575), the author of Yuktibhāṣā. Other teachers of Shankara include Netranarayana, the patron of Nilakantha Somayaji and Chitrabhanu, the author of an astronomical treaties dated to 1530 and a small work with solutions and proofs for algebraic equations.[2]

Works

The known works of Shankara Variyar are the following:[2]

Yukti-dipika - an extensive commentary in verse on Tantrasamgraha based on Yuktibhāṣā.
Laghu-vivrti - a short commentary in prose on Tantrasamgraha.
Kriya-kramakari - a lengthy prose commentary on Lilavati of Bhaskara II.
An astronomical commentary dated 1529 CE.
An astronomical handbook completed around 1554 CE.

Jyeṣṭhadeva

Jyeṣṭhadeva (Malayalam: ജ്യേഷ്ഠദേവന്) (c. 1500 – c. 1575)[1][2] was an astronomer-mathematician of the Kerala school of astronomy and mathematics founded by Sangamagrama Madhava (c. 1350 – c. 1425). He is best known as the author of Yuktibhāṣā, a commentary in Malayalam of Tantrasamgraha by Nilakantha Somayaji (1444–1544). In Yuktibhāṣā, Jyeṣṭhadeva had given complete proofs and rationale of the statements in Tantrasamgraha. This was unusual for traditional Indian mathematicians of the time. An analysis of the mathematics content of Yuktibhāṣā has prompted some scholars to call it "the first textbook of calculus".[3] Jyeṣṭhadeva also authored Drk-karana a treatise on astronomical observations.

Life period of Jyeṣṭhadeva

There are a few references to Jyeṣṭhadeva scattered across several old manuscripts.[1] From these manuscripts, one can deduce a few bare facts about the life of Jyeṣṭhadeva. He was a Nambudiri belonging to the Parangngottu family (Sanskrtised as Parakroda) born about the year 1500 CE. He was a pupil of Damodara and a younger contemporary of Nilakantha Somayaji. Achyuta Pisharati was a pupil of Jyeṣṭhadeva. In the concluding verse of his work titled Uparagakriyakrama, completed in 1592, Achyuta Pisharati has referred to Jyeṣṭhadeva as his aged benign teacher. From a few references in Drkkarana, a work believed to be of Jyeṣṭhadeva, one may conclude that Jyeṣṭhadeva lived up to about 1610 CE.

Parangngottu, the family house of Jyeṣṭhadeva, still exists in the vicinity of Trikkandiyur and Alathiyur.[1] There are also several legends connected with members of Parangngottu family.

Mathematical lineage

Little is known about the mathematical traditions in Kerala prior to Sangamagrama Madhava. Vatasseri Paramesvara was a direct disciple of Madhava. Damodara was a son of Paramesvara. Nilakantha Somayaji and Jyeshthadeva were pupils of Damodara. Jyeṣṭhadeva's pupil was Achyuta Pisharati and Melpathur Narayana Bhattathiri was Achyuta Pisharati's student.

Jyeshthadeva's works

Jyeṣṭhadeva is known to have composed only two works, namely, Yuktibhāṣā and Drkkarana. The former is commentary with rationales of Tantrasamgraha of Nilakantha Somayajiand the latter is a treatise on astronomical computations.

Three factors make Yuktibhāṣā unique in the history of the development of mathematical thinking in the Indian subcontinent:

It is composed in the spoken language of the local people, namely, the Malayalam language. This is in contrast to the centuries-old Indian tradition of composing scholarly works in the Sanskrit language which was the language of the learned.
The work is in prose, again in contrast to the prevailing style of writing even technical manuals in verse. All the other notable works of the Kerala school are in verse.
Most importantly, Yuktibhāṣā was composed intentionally as a manual of proofs. The very purpose of writing the book was to record in full detail the rationales of the various results discovered by mathematicians-astronomers of the Kerala school, especially of Nilakantha Somayaji. This book is proof enough to establish that the concept of proof was not unknown to Indian mathematical traditions.

Achyuta Pisharati

Achyutha Pisharodi (c. 1550 at Trikkandiyur (aka Kundapura), Tirur, Kerala, India – 7 July 1621 in Kerala) was a Sanskrit grammarian, astrologer, astronomer and mathematicianwho studied under Jyeṣṭhadeva and was a member of Madhava of Sangamagrama's Kerala school of astronomy and mathematics. He is remembered mainly for his part in the composition of his student Melpathur Narayana Bhattathiri's devotional poem, Narayaneeyam.

Works

He discovered the techniques of 'the reduction of the ecliptic'. He authored Sphuta-nirnaya, Raasi-gola-sphuta-neeti (raasi meaning zodiac, gola meaning sphere and neeti roughly meaning rule), Karanottama (1593) and a four- chapter treatise Uparagakriyakrama on lunar and solar eclipses.

PraveśakaAn introduction to Sanskrit grammar.
KaraṇottamaAstronomical work dealing with the computation of the mean and true longitudes of the planets, with eclipses, and with the vyatūpātas of the sun and moon.
Uparāgakriyākrama (1593)Treatise on lunar and solar eclipses.
SphuṭanirṇayaAstronomical text.
ChāyāṣṭakaAstronomical text.
UparāgaviṃśatiManual on the computation of eclipses.
RāśigolasphuṭānūtiWork concerned with the reduction of the moon’s true longitude in its own orbit to the ecliptic.
VeṇvārohavyākhyāMalayalam commentary on the Veṇvāroha of Mādhava of Saṅgamagrāma (ca. 1340–1425) written at the request of Netranārāyaṇa.
HorāsāroccayaAn adaptation of the Jātakapaddhati of Śrīpati.

Narayaneeyam

Pisharati is known to have scolded and provoked an errant Narayana to take up the Brahmin's duties of prayer and religious practices. He accepted Narayana as his student. Later when Pisharati was struck with paralysis (or rheumatism by another account), Narayana, unable to bear the pain of his dear guru, by way of Gurudakshina took the disease upon himself. As a result, Pisharati is said to have been cured, but no medicine could cure Narayana. As a last resort, Narayana went to Guruvayur and requested Thunchaththu Ramanujan Ezhuthachan, a great devotee of Guruvayoorappan, to suggest a remedy for his disease. Ramajunan Ezhuthachan advised him to compose a poetical work on the Avatars (incarnations) of Lord Vishnu beginning with that of Matsya (Fish). Narayana composed beautiful slokas in praise of Lord Guruvayurappan and recited them before the deity. He was soon cured of his disease.

The book of slokas written by Narayana were named Narayaneeyam. The day on which Narayana dedicated his Narayaneeyam to Sri Guruvayurappan is celebrated as "Narayaneeyam Dinam" every year at Guruvayur.

Kamalakara

Kamalakara (1616–1700), an Indian astronomer and mathematician, came from a learned family of scholars from Golagrāma, a village on the northern bank of the river Godāvarī. His father was Nrsimha who was born in 1586.[1] Two of Kamalakara's three brothers were also astronomer and mathematicians: Divakara, who was the eldest of the brothers born in 1606, and Ranganatha who was youngest. Kamalākara learnt astronomy from his elder brother Divākara, who compiled five works on astronomy. His family later moved to Vārāṇasī.

Major works

Kamalākara's major work, "Siddhāntatattvaviveka", was compiled in Varanasi at about 1658 and has been published by Sudhakar Dwivedi in the Vārāṇasī series. This work consists of 13 chapters in 3,024 verses. It deals with the topics of: units of time measurement; mean motions of the planets; true longitudes of the planets; the three problems of diurnal rotation; diameters and distances of the planets; the earth's shadow; the moon's crescent; risings and settings; syzygies; lunar eclipses, solar eclipses; planetary transits across the sun's disk; the patas of the moon and sun; the "great problems"; along a conclusion. His other works include Śeṣavāsanā and Sauravāsanā. Kamalākara was bitterly opposed to Munishvara, the author of Siddhāntasārvabhauma.

It is wrongly believed by some moderners that Kamalākara discovered the idea that the pole star we see at present is not exactly at the pole. But this ideas was first expressed in Brahmaanda Purana and Matsya Purana by sage Veda Vyaasa: "uttAnapAda-putro-asau meDhibhooto dhruvo divi | sa hi bhraman bhtaamayate nityam chandraadityau grahaiH saha ||". The meaning of this expression is "Uttanapada's son Dhruva is fixed like a pole in the Heaven, but it is moving itself and is making all the planets together with Sun and Moon move".

Kamalākara's contribution was to rejuvenate this forgotten idea.

Contributions

He combined traditional Indian astronomy with Aristotelian physics and Ptolemaic astronomy as presented by Islamic scientists.
In the third chapter of the Siddhanta-tattva-viveka Kamalakara used the addition and subtraction theorems for the sine and the cosine to give trigonometric formulae for the sines and cosines of double, triple, quadruple and quintuple angles. In particular he gives formulae for sin(A/2) and sin(A/4) in terms of sin(A) and iterative formulae for sin(A/3) and sin(A/5).
According to David Pingree, he presents the only Sanskrit treatise on geometrical optics.[3]
He has assumed a value of 60 units for the radius of the Earth and gives values for sines at 1° intervals.
Kamalākara also gives a table for finding the right ascension of a planet from its longitude.
Kamalakara was an Indian astronomer and mathematician who came from a family of famous astronomers. Kamalakara's father was Nrsimha who was born in 1586. Two of Kamalakara's three brothers were also famous astronomer/ mathematicians, these being Divakara, who was the eldest of the brothers born in 1606, and Ranganatha who was younger than Kamalakara.
As was common throughout the classical period of Indian mathematics, members of the family acted as teachers to other family members. In particular Kamalakara was taught by his elder brother Divakara while Divakara himself had been taught by their uncle Siva. Pingree writes in [1]:-

[Kamalakara] combined traditional Indian astronomy with Aristotelian physics and Ptolemaic astronomy as presented by Islamic scientists (especially Ulugh Beg). Following his family's tradition he wrote a commentary, Manorama, on Ganesa's Grahalaghava and, like his father, Nrsimha, another commentary on the Suryasiddhanta, called the Vasanabhasya ...
Kamalakara's most famous work, the Siddhanta-tattva-viveka, was commented on by Kamalakara himself. The work was completed in 1658. It is a work of fifteen chapters covering standard topics for Indian astronomy texts at this time. It deals with the topics of: units of time measurement; mean motions of the planets; true longitudes of the planets; the three problems of diurnal rotation; diameters and distances of the planets; the earth's shadow; the moon's crescent; risings and settings; syzygies; lunar eclipses, solar eclipses; planetary transits across the sun's disk; the patas of the moon and sun; the "great problems"; and a final chapter which forms a conclusion.
The third chapter of the Siddhanta-tattva-viveka contains some of the most interesting mathematical results. In that chapter Kamalakara used the addition and subtraction theorems for the sine and the cosine to give trigonometric formulae for the sines and cosines of double, triple, quadruple and quintuple angles. In particular he gives formulae for sin(A/2) and sin(A/4) in terms of sin(A) and iterative formulae for sin(A/3) and sin(A/5). See for example [7] and [8] for a discussion of the details of Kamalakara's work in this area.

The Siddhanta-tattva-viveka is a Sanskrit text and in it Kamalakara makes frequent use of the place-value number system with Sanskrit numerals. This and many other aspects of the work are discussed in [3].

Jagannatha Samrat

Paṇḍita Jagannātha Samrāṭ (1652–1744) was an Indian astronomer and mathematician who served in the court of Jai Singh II of Amber, and was also his guru.

Jagannātha, whose father's name was Gaṇeśa,[1][2] and grandfather's Viṭṭhala[2] was from a Vedic family[1] originally from Maharashtra.[2]

At the suggestion of Jai Singh, he learned Arabic and Persian, in order to study Islamic astronomy.[1][2] Having become proficient in these languages, he translated texts in these languages into Sanskrit.[1][2] These translations include:

Rekhā-gaṇita, a translation of Euclid's Elements made from Nasir al-Din al-Tusi's Arabic recension of the same. For this work, he had to coin more than a hundred Sanskrit mathematical terms[1]
Siddhānta-sāra-kaustubha, a translation of Ptolemy's Almagest from Nasir al-Din al-Tusi's Arabic version[1][2]

His original works include:

‍ Siddhānta-samrāṭ, which describes astronomical instruments, their design and construction, and observations. It also describes the use of these observations in correcting parameters and preparing almanacs. It mentions how J‌ai Singh, who earlier used astronomical instruments (such as the astrolabe) made of metal, later switched to huge outdoor observatories (such as the Jantar Mantar), as they were more precise; also they were made of stone and mortar rather than brick, to diminish the effects of wear-and-tear and climate.[1]
Yantra-prakāra, which describes astronomical instruments, measurements, computations, etc. in more detail, and also observations made by him.[1]

Jagannātha held that when theory and observation differed, observation was the true pramāṇa and overruled theory.[2] While he used and described a number of astronomical instruments, telescopes were not one of them.

Radhanath Sikdar

Radhanath Sikdar (Oct 1813 – 17 May 1870) was an Indian [Bengali রাধানাথ শিকদার] mathematician who, among many other things, calculated the height of Mount Everest in the Himalaya and showed it to be the tallest mountain above sea level.

Great Trigonometric Survey

When in 1831 Surveyor General of India George Everest was searching for a brilliant young mathematician with particular proficiency in spherical trigonometry, the Hindu College maths teacher Tytler superlatively recommended his pupil Radhanath, then only 19. Radhanath joined the Great Trigonometric Survey in 1831 December as a "computor" at a salary of thirty rupees per month. Soon he was sent to Sironj near Dehradun where he excelled in geodetic surveying. Apart from mastering the usual geodetic processes, he invented quite a few of his own. Everest was extremely impressed by his performance, so much so that when Sikdar wanted to leave GTS and be a Deputy Collector, Everest intervened, proclaiming that no government officer can change over to another department without the approval of his boss. Everest retired in 1843 and Col. Andrew Scott Waughbecame the Director.

After 20 years in the North, Sikdar was transferred to Calcutta in 1851 as the Chief Computer. Here apart from his duties of the GTS, he also served as the Superintendent of the Meteorological department. Here he introduced quite a few innovations that were to remain as standard procedures for many decades to come. The most notable was the formula for conversion of barometric readings taken at different temperatures to 32 degrees Fahrenheit.[2]

At the order of Col. Waugh he started measuring the snow-capped mountains near Darjeeling. Compiling data about Peak XV from six different observations, he eventually came to the conclusion the Peak XV was the tallest in the world. He gave a full report to Waugh who was cautious enough not to announce this discovery before checking with other data. When after some years, he was convinced, only then did he publicly announce the same. The norm, strictly followed by Everest himself, was that while naming a peak, the local name should be preferred. But in this case, Waugh made an exception. He paid a tribute to his ex-boss by proposing that the peak be named after Everest. Everest agreed, and Sikdar was conveniently forgotten.[3][4]

Other

It appears that while Everest and Waugh both extolled him for his exceptional mathematical abilities, his relations with the colonial administration were far from cordial. Two specific instances are on record.

In 1851 a voluminous Survey Manual (Eds. Capt. H. L. Thullier and Capt. F. Smyth) was published by the Survey Department. The preface to the Manual clearly and specifically mentioned that the more technical and mathematical chapters of the Manual were written by Babu Radhanath Sikdar. The Manual proved to be immensely useful to surveyors. However, the third edition, published in 1875 (i.e., after Sikdar's death) did not contain that preface, so that Sikdar's memorable contribution was de-recognized. The incident was condemned by a section of British surveyors. The paper Friend of India in 1876 called it 'robbery of the dead'.[5]

It is also on record that Sikdar was fined a sum of 200 rupees by the British court in 1843 for having vehemently protested against the unlawful exploitation of survey department workers by the Magistrate Vansittart. The incident was reported in detail in The Bengal Spectator edited by another great Derozian Ramgopal Ghosh.[5]

In 1854, he along with his Derozian friend Peary Chand Mitra started the Bengali journal Masik Patrika, for the education and empowerment of women. He used to write in a simple and uncluttered style that was rather atypical for the age.[6]

Sikdar had retired from service in 1862, and was later appointed as teacher of mathematics at the General Assembly's Institution (now Scottish Church College).[7][8]

He died on 17 May 1870 at Gondalpara, Chandannagar in his villa by the side of the Ganga.

Recognition

In recognition of Sikdar's mathematical genius, the German Philosophical Society made him a Corresponding Member in 1864, a very rare honour those days.[9][vague]

The Department of Posts, Government of India, launched a postal stamp on 27 June 2004, commemorating the establishment of the Great Trigonometric Survey in Chennai, India on 10 April 1802. The stamps feature Radhanath Sikdar and Nain Singh, two significant contributors to society. The Great Arc refers to the systematic exploration and recording of the entire topography of the Indian subcontinent which was spearheaded by the Great Trigonometric Survey.

Ramchundra

Ramchundra (Ramachandra Lal) (Devanagari,रामचन्द्र लाल) (1821 – 1880) was a British Indian mathematician. His book, Treatise on Problems of Maxima and Minima, was promoted by the prominent mathematician Augustus De Morgan.

In his introduction to Ramchundra's book, De Morgan says that he was born in 1821 in Panipat to Sunder Lal, a Kayasth of Delhi. De Morgan came to know of Ramchundra when, in 1850, he was sent by a friend to work on maxima and minima by the 29-year-old self-taught mathematician. Ramchundra had published his book at his own expense in Calcutta in that year. De Morgan arranged for the book to be republished in London under his own supervision.

De Morgan was so impressed that he undertook to bring Ramchundra's work to the notice of scientific men of Europe.

Charles Muses, in an article in the Mathematical Intelligencer (1998) called Ramchundra "De Morgan's Ramanujan". He was mystified why, in spite of De Morgan's efforts to make this "remarkable Hindu algebraist known, he does not appear in most texts on history of mathematics."

Ramchundra was teacher of science in Delhi College for some time. In 1858, he was native head master in Thomason Civil Engineering College (now Indian Institute of Technology, Roorkee) at Roorkee. Later that year, he was appointed head master of a school in Delhi.

Pathani Samanta

Mahamahopadhyaya Chandrasekhar Singh Harichandan Mohapatra Samanta (Odia: ମହାମହୋପାଧ୍ୟାୟ ଚନ୍ଦ୍ରଶେଖର ସିଂହ ହରିଚନ୍ଦନ ମହାପାତ୍ର ସାମନ୍ତ), popularly known as Pathani Samanta (Odia: ପଠାଣି ସାମନ୍ତ) (13 December 1835 - 11 June 1904) was an Indian astrologer and scholar who measured the distance from earth with a bamboo pipe and many other traditional instruments that he built. This earned him the "Mahamahopadhyaya" award in 1893. His observations, research and calculations were compiled into a book Siddhanta Darpana which has verses written in Sanskrit in Odia script.

Early life

Samanta was born in the princely state of Khandapada in Nayagarh districtin the Indian state of Odisha.[2] He was an eminent naked eye astronomer. He studied in Sanskrit and researched traditional Indian astronomy. Pathani Samanta was born on 13-12-1835 Sunday (Purnimanta Pousha Krishna Ashtami) and he died on 11-06-1904 Saturday (Purnimanta Adhika Jyeshtha Krishna Trayodashi) The wrong information given in this site should be corrected.

Instrument maker

During his research, he built many instruments using available materials such as wooden sticks and bamboo. His knowledge of astronomy ensured that these instruments had great accuracy. His findings were recorded in his book titled Siddhanta Darpana. This book found mentions in the European and American press in 1899. Samanta’s calculations are used in the preparation of almanacs in Odisha.

Ashutosh Mukherjee

Sir Ashutosh Mukherjee CSI, FASB, FRSE, FRAS, MRIA [1] (anglicised, originally Āśutōṣh Mukhōpādhyāẏa (Bengali: আশুতোষ মুখোপাধ্যায়)[1], also anglicised to Asutosh Mookerjee) (29 June 1864 – 25 May 1924) was a prolific Bengali educator, jurist, barrister and mathematician. He was the first student to be awarded a dual degree (MA in Mathematics and Physics) from Calcutta University. Perhaps the most emphatic figure of Indian education, he was a man of great personality, high self-respect, courage and towering administrative ability. The second Indian Vice-Chancellor of the University of Calcutta for four consecutive two-year terms (1906–1914) and a fifth two-year term (1921–23), Mukherjee was responsible for the foundation of the Bengal Technical Institute in 1906 and the College of Science of the Calcutta University in 1914.

Mukherjee also played a vital role in the founding of the University College of Law. The Calcutta Mathematical Society was also founded by Mukherjee in 1908 and he served as the president of the Society from 1908 to 1923.[2][3] He was also the president of the inaugural session of the Indian Science Congress in 1914. The Ashutosh College was also founded under his stewardship in 1916, when he was Vice-chancellor of University of Calcutta.

He was often called "Banglar Bagh" ("Tiger of Bengal") for his high self-esteem, courage, academic integrity and a general intransigent attitude towards the British Government.

Early life

Ashutosh Mukherjee was born on 29 June 1864 at Bowbazar, Kolkata to Jagattarini Devi and Ganga Prasad Mukhopadhyaya, a well-known doctor who founded the South Sub Urban School in Calcutta. Among his ancestors were several distinguished Sanskrit scholars, including Pandit Ramchandra Tarkalankar, a professor of nyaya who had been appointed by Warren Hastings to that chair at the Sanskrit College in Kolkata.[4] Brought up in an atmosphere of science and literature at home, young Asutosh went to the Sisu Vidyalaya at Chakraberia, Bhowanipore and showed an early aptitude for mathematics. When he was young, he met Ishwar Chandra Vidyasagar who was a major influence on him. He was a student of Madhusudan Das.[5]

Accomplished mathematician

In November 1879, at the age of fifteen, Mukherjee passed the entrance exam of the Calcutta University in which he stood second and received a first grade scholarship.[4] In the year 1880, he took admission at the Presidency College in Kolkata where he met P.C. Rayand Narendranath Dutta who would later become famous as Swami Vivekananda. Later that year, though only a first-year undergraduate, he published his first mathematical paper, on a new proof of the 25th proposition of Euclid's first book.[4]

In 1883, Mukherjee topped the BA examination at Calcutta University [6] to complete a postgraduate degree in mathematics. In 1883 S.N. Banerjee wrote an article in the newspaper Bengalee against the orders of the Calcutta High Court and he was arrested in contempt of court. Protests and hartals erupted across Bengal and other cities, led by a group of students headed by Mukherjee at Calcutta high court. In 1884, he won the Harishchandra Prize for academic achivements, and completed an M.A. with first-class honours in mathematics in 1885.[4] In 1886, he was awarded a second Masters in Natural Sciences, making him the first student to be awarded a dual degree from Calcutta University.[4] In the same year he was married to Jogomaya Devi, and also published his third mathematical paper, "A Note on Elliptic Functions." The paper was praised by the distinguished British mathematician and Fellow of the Royal Society Arthur Cayley as a contribution of "outstanding merit."[4] Mukherjee was recognised for his achievements by the grant of the Premchand Roychand Fellowship in Mathematics and Physics, Pure and Applied.[1] Still only aged 22, he was further recognised by his election as a Fellow of the Royal Society of Edinburgh (FRSE).[1] By 1888, Mukherjee was a lecturer in mathematics for the recently established Indian Association for the Cultivation of Science (IACS).[7]

Mukherjee continued publishing scholarly papers on mathematics and physics into his 30s. By 1893, aged 29, Mukherjee had been further elected to the fellowships of the Physical Society of France and the Mathematical Society of Palermo, and was a member of the Royal Irish Academy. He subsequently became a member of the London Mathematical Society, the Paris Mathematical Society and the American Mathematical Society (1900).[1][4] Although after 1893 he largely abandoned his mathematical pursuits for a legal career, Mukherjee has been recognised as the first modern Indian mathematician to enter the field of mathematical research, and founded the Calcutta Mathematical Society in 1908. Among his mathematical contributions, Mukherjee determined several crucial derivations of Gaspare Mainardi's answer to determining the oblique trajectory of a system of confocal ellipses. He also made lasting contributions in differential geometry, developing analytical methods of simplifying Gaspard Monge's interpretation of his general differential equation for conics.[4][1]

Lawyer, jurist and educationist

At the age of 24, Mukherjee became a Fellow of the Calcutta University. Turning down a job offer in the Department of Public Instruction in order to complete his Bachelor of Law degree, he received his degree in 1888 and enrolled as a vakil of the Calcutta High Court. By 1897, he had received an LL.D. and was appointed the Tagore Professor of Law of the Calcutta University in that year. In 1904, he was appointed a puisne judge of the High Court, and subsequently served as its acting Chief Justice for a couple of years.[4]

Mukherjee was influential in the University affairs throughout his life. From the age of 25, he was a member of its Syndicate, serving on the University Senate and Syndicate for the next 16 years. He served as President of the Board of Studies in Mathematics for 11 years, and represented his university in the Bengal Legislative Council from 1899 to 1903. He was appointed Vice-Chancellor from 1906 to 1914 and again from 1921 to 1923.[4] He was instrumental in discovering the talents of C. V. Raman and S. Radhakrishnan.

The French scholar Sylvain Lévi commented :

Had this Bengal Tiger been born in France, he would have exceeded even Georges Clemenceau, the French Tiger. Ashutosh had no peer in the whole of Europe.

Academic career and later life

Ashutosh Mukherjee had a vision of the kind of education he wanted young people to have, and he had the acumen and courage to extract it from his colonial masters. He set up several new academic graduate programs at the Calcutta University: comparative literature, anthropology, applied psychology, industrial chemistry, ancient Indian history and culture as well as Islamic culture. He also made arrangements for postgraduate teaching and research in Bengali, Hindi, Pali and Sanskrit. Scholars from all over India, irrespective of race, caste, and gender, came to study and teach there. He even persuaded European scholars to teach at his university. He was one of the first persons to recognise the work of Srinivasa Ramanujan. He also established Asutosh College in South Kolkata in 1916. He laid the foundry stone of Santragachi Kedarnath Institution.

Curzon's education mission in 1902 identified the universities including the Calcutta University, as centres of sedition where young people formed networks of resistance to colonial domination.[8] The cause of this was thought to be the unwise granting of autonomy to these universities in the nineteenth century. Thus in the period of 1905 to 1935, the colonial administration tried to reinstate government control of education.

In 1910, he was appointed the President of the Imperial (now National) Library Council to which he donated his personal collection of 80,000 books which are arranged in a separate section. He was the president of the inaugural session of the Indian Science Congress in 1914. Mukherjee was a member of the 1917–1919 Sadler Commission, presided over by Michael Ernest Sadler, which inquired into the state of Indian education. He was thrice elected as the president of The Asiatic Society. Having served as a fellow and subsequently as a vice-president of the Indian Association for the Cultivation of Science since the 1890s, in 1922 he was elected President of the IACS and held the office until his death.[9]

After serving five terms as Vice-Chancellor of Calcutta University, Mukherjee declined to be reappointed to a sixth term in 1923 when the university's Chancellor, Governor of Bengalthe Earl of Lytton, tried to impose conditions on his reappointment. Shortly thereafter, he also resigned his judgeship on the Calcutta High Court and resumed his private practice of law. While arguing a case in Patna the following year, Mukherjee died suddenly on 25 May 1924, a month before his sixtieth birthday. His body was returned to Kolkata and cremated at a funeral service which drew crowds of mourners.[4]

Recognition and legacy

Mukherjee was a polyglot learned in Pali, French and Russian. Apart from his fellowships and memberships in several international academic bodies, he was recognised by an award of the titles of Saraswati and Shastravachaspati by the pandits of Bengal for his service to Indian education. Mukherjee was appointed a Companion of the Order of the Star of India (CSI) in June 1909,[10] and knighted in December 1911.[11]

The Government of India issued a stamp in 1964 to commemorate Sir Ashutosh Mukherjee for his contribution to education.

The epitaph beneath his marble bust at the Ashutosh Museum of Arts at the University of Calcutta reads:

His noblest achievement, surest of them all/ A place for his mother tongue --- in step mother's hall.

Personal life

Mukherjee married Jogamaya Devi in 1885. The couple had seven children, Kamala (born 1895), Rama Prasad (1896-1983)[12], Syama Prasad (1901-1953), Uma Prasad (born 1902), Amala (born 1905), Bama Prasad (born 1906) and Ramala (born 1908). His son Syama Prasad Mookerjee, the most notable of his children, founded the Bharatiya Jana Sangh. Rama Prasad became a judge in the High Court of Calcutta while Uma became famed as a trekker and a travel writer.

Ganesh Prasad

Ganesh Prasad (15 November 1876 – 9 March 1935) was an Indian mathematician who specialised in the theory of potentials, theory of functions of a real variable, Fourier series and the theory of surfaces. He was trained at the Universities of Cambridge and Göttingenand on return to India he helped develop the culture of mathematical research in India. The mathematical community of India considers Ganesh Prasad as the Father of Mathematical Research in India.[1] He was also an educator taking special interest in the advancement of primary education in the rural areas of India.

Early days

Ganesh Prasad was born on 15 November 1876 at Ballia, Uttar Pradesh. He obtained the B.A. degree from Muir Central College, Allahabad, M.A. degree from the Universities in Allahabad and Calcutta and the D.Sc. degree from Allahabad University. After teaching at the Kayasth Pathshala, Allahabad, and at the Muir Central College, Allahabad, for about two years, he proceeded to Cambridge for higher studies and research. While at Cambridge he became acquainted with mathematicians like E.W. Hobson and Andrew Forsyth. He also sat, though unsuccessfully, for the Adams prize competition.

Later he moved to Göttingen where he was associated with Arnold Sommerfeld, David Hilbert and Georg Cantor. In Göttingen, Prasad showed his paper titled On the constitution of matter and the analytical theories of heat, the one he had submitted for the Adams prize competition, to Felix Klein, who appreciated it very much and got it published in the Göttingen Abhandllingen. Ganesh Prasad spent altogether about five years in Europe.

Mathematical career

Prasad returned to India from Europe in 1904 and was appointed professor of mathematics at the Muir Central College, Allahabad. Within a year of his appointment at Allahabad, Prasad was sent to the Queen's College, Banaras and he continued there till 1914 when he was invited to head the mathematics department of Calcutta University. Ganesh Prasad was the Ras Behari Ghosh Chair of Applied Mathematics of Calcutta University (he was the first person to occupy this Chair[2]) from 1914 to 1917 and Hardinge Professor of Mathematics in the same University from 1923 till his death on 9 March 1935. In between these two assignments he served Banaras Hindu University as professor of mathematics (1917–1923). While at Banaras, he helped found the Banaras Mathematical Society. Ganesh Prasad was elected President of the Calcutta Mathematical Society and the Vice-President of the Indian Association for Advancement of Science, Calcutta in 1924 and continued in the same position till his death. He was a founder member of the National Institute of Sciences, India, which has now been rechristened as the Indian National Science Academy. Ganesh Prasad authored 11 books including A Treatise on Spherical Harmonics and the Functions of Bessel and Lame and over fifty research papers in mathematics.

Other areas of work

Ganesh Prasad worked hard for the promotion of education in general in the rural areas of Uttar Pradesh. He was instrumental in the introduction of compulsory primary education in villages in Uttar Pradesh. He donated from his private savings an amount of Rs. 22,000 for the education of girls in Ballia. He also donated an amount of Rupees two hundred thousand for establishing prizes for the toppers at the M.A. and MSc examinations of the Agra University. He donated large amounts of money to the Allahabad and BanarasUniversities also.

Bharati Krishna Tirtha

Swāmī Bhāratī Krishna Tīrtha (March 1884 – February 2, 1960) was the Śankarācārya of Govardhana matha in Puri, Orissā (now Odishā) from 1925 to 1960. He is particularly known for his book Vedic Mathematics.

Early life

Venkatarāman Shastrī was born in March 1884 to an orthodox Tamil Brahmin family. His father was P. Narasimha Shastrī, originally a tehsildar at Tirunelveli in Madras Presidencywho later became the Deputy Collector of the Presidency. His uncle, Chandrasekhar Shastrī, was the Principal of the Mahārāja's college in Vizianagaram, while his great-grandfather, Justice C. Ranganāth Shastrī was a judge in the Madras High Court.[2]

Education

Venkatarāman Shastrī joined National College in Trichinopoly. After that he moved to the Church Missionary Society College and eventually the Hindu College, both in Tirunelveli. Shastri passed his matriculation examination from Madras University in January 1899, where he also finished the first.[3][4]

Although Venkatarāman always scored high in subjects like mathematics, sciences and humanities, he was also proficient in languages and particularly good at Sanskrit. According to his own testimonials, Sanskrit and oratory were his favourite subjects. Due to his skill at the language, that he was awarded the title "Saraswati" by the Madras Sanskrit Association in July 1899 at the age of 16. At about that time, Venkatraman was profoundly influenced by his Sanskrit guru Vedam Venkatrai Shastri.[5]

Venkatarāman passed B.A. examination in 1902. He then appeared for the M.A. Examination for the American College of Sciences in Rochester, New York from the Bombay centre in 1903. He also contributed to W. T. Stead's Review of Reviews on diverse topics like religion and science. During his college days, he also wrote extensively on history, sociology, philosophy, politics, and literature.[4]

Early public life

Venkatarāman Shastrī worked under Gopal Krishna Gokhale in 1905 for the National Education Movement and the South African Indian problems. However, his inclination towards Hindu studies led him to study the ancient Indian holy scriptures, Adhyātma-Vidyā. In 1908, he joined the Sringeri Matha in Mysore to study under Svāmī Satchidānanda Śivābhinava Nrsimha Bhāratī, the Śankarācārya of Sringeri. However, his spiritual practise was interrupted when he was pressured by nationalist leaders to head the newly started National College at Rajamahendri. Prof. Venkatarāman Shastrī taught at the college for three years. But in 1911, he suddenly left the college to go back to Sringeri Matha.[6]

Spiritual Path

Returning to Sringeri, Venkatarāman spent the next eight years studying advanced Vedanta philosophy with Satchidānanda Śivābhinava Nrsimha Swamigal.

He also practised vigorous meditation, Brahma-sadhana and Yoga-sādhāna, in the nearby forests during those years. It is believed that he attained spiritual self-realization during his years at Sringeri Matha. He would leave the material world and practice Yoga meditation in seclusion for many days. During those eight years, he also taught Sanskrit and philosophy at local schools and ashrams. He delivered a series of sixteen lectures on Śankarācārya's philosophy at Shankar Institute of Philosophy, Amalner (Khandesh). During that time, he also lectured as a guest professor at institutions in Mumbai, Pune and Khandesh.[7]

Initiation into Sannyāsa order

After Venkatarāman's eight-year period of spiritual practice and study of Vedānta and Vedic philosophy, he was initiated into the holy order of sannyāsa in Varanasi by Jagadguru Śankarācārya Svāmī Trivikrāma Tīrtha of Sharada Peetha, Sringeri, on July 4, 1919 and on this occasion he was given the title Swami and the new name "Svāmī Bhāratī Kṛṣṇa Tīrtha".[7]

Śankarācārya of Sharada Peetha

Svāmī Bhāratī Krishna Tīrtha was installed as Śankarācārya of Sharada Peetha in 1921 after just two years of sannyāsa. After assuming the pontificate, he was given another title, Jagadguru, as is the tradition. The Swami then toured India from corner to corner giving lectures on Sanātana Dharma, Vedic philosophy and Vedanta.

Śankarācārya of Govardhana Matha

Around the time the Svāmī became Śankarācārya of Sharada Peetha, the Śankarācārya of Govardhana matha, Svāmī Madhusudhana Tīrtha, was in failing health and was greatly impressed by Bhāratī Kṛishna Madhusudana requested Bhāratī Krishna to succeed him at Govardhana Matha. Bhāratī Krishna respectfully declined the offer. In 1925, however, Śankarācārya Svāmī Madhusudhana Tīrtha's health took a serious turn and Svāmī Bhāratī Kṛṣṇa Tīrtha had to accept the Govardhana Matha gaddi. In 1925, Svāmī Bhāratī Kṛṣṇa Tīrtha assumed the pontificate of Govardhana Matha, and relinquished the gaddi of Sharada Peetha. He installed Svāmī Svarupānanda as the new Śankarācārya of Sharada Peetha.[8]

Politics

In 1921, the Shankaracharya was one of the seven arrested in what became known as the "Karachi case". Maulanaa Mohammed Ali, Maulana Shaukat Ali, Dr. Kitchlu, Nissar Ahmed, Pir Ghulam Mujaddid, and Bharati Krishna Tirtha were charged with preaching in favour of a fatwa issued by the Muslim religious heads of India advocating all Muslims to not cooperate with the government. While the Shankaracharya was eventually acquitted, the others were sentenced to two years imprisonment.[9][10]

Jagadguru

After becoming the Śankarācārya of Govardhana Matha, Svāmī Bhāratī Krishna Tīrtha toured several countries for thirty-five years to spread the values of peace, harmony and brotherhood, and to spread the message of Sanātana Dharma. He took upon himself the task of contributing to the renaissance of Indian culture.[8]

While being a pontiff, he wrote a number of treatises and books on religion, sciences, mathematics, world peace, and social issues. In 1953, at Nagpur, he founded an organisation called the Sri Vishwa Punarnirmana Sangha (World Reconstruction Association). Initially, the administrative board consisted of Bhāratī Kṛṣṇa's disciples, devotees and admirers of his spiritual ideals for humanitarian service, but later distinguished people would also contribute to the mission. The Chief Justice of India, Justice B.P. Sinha, served as its President. Dr. C. D. Deshmukh, the ex-Finance Minister of India and ex-Chairman of the University Grants Commission served as its Vice-President.[11]

In February 1958, he went on a trans-oceanic tour to the United States to speak on world peace and Vedānta, staying for three months in Los Angeles, California, traveling via the United Kingdom. This was the first trip outside India by a Śankarācārya. The tour was sponsored by the Self-Realization Fellowship of Los Angeles, the Vedantic society founded by Paramahansa Yogananda in America.[12] At that time, Albert Rudolph, or "Rudi", became one of his students.

He attended national and international religious conferences and Yoga workshops. He believed in the Vedāntic ideal of Pūrnatva which, literally translated, means "all-round perfection and harmony". He remained the Śankarācārya of Govardhana Matha until his death in 1960.

In 1965, a Chair of Vedic Studies was founded at Banaras Hindu University by Arvind N. Mafatlal, a generous Mumbai business magnate and devotee of the late Śankarācārya.[13]

Mathematics

Bhāratī Krishna Tīrtha's book, Vedic Mathematics, is a list of sixteen terse sūtras, or "aphorisms", discussing strategies for mental calculation. Bhāratī Kṛṣṇa claimed that he found the sūtras after years of studying the Vedas, a set of sacred ancient Hindu scriptures.[14] [15][16]

For arithmetic, Bhāratī Krishna gives several algorithms for whole number multiplication and division, (flag or straight) division, fraction conversion to repeating decimal numbers, calculations with measures of mixed units, summation of a series, squares and square roots (duplex method), cubes and cube roots (with expressions for a digit schedule), and divisibility (by osculation).[17]

Several tests and techniques for factoring and solving certain algebraic equations with integer roots for quadratic, cubic, biquadratic, pentic equations, systems of linear equations, and systems of quadratic equations are demonstrated. For fractional expressions, a separation algorithm and fraction merger algorithms are given. Other techniques handle certain patterns of some special case algebraic equations. Just an introduction to differential and integral calculus is given.[18]

Geometric applications are reviewed for linear equations, analytic conics, the equation for the asymptotes, and the equation to the conjugate-hyperbola.[19] Five simple geometric proofs for the Pythagorean theorem are given.[20] A 5-line proof of Apollonius' theorem is given.[21]

Advanced topics promised included integral calculus (the centre of gravity of hemispheres, conics), trigonometry, astronomy (spherical triangles, earth's daily rotation, earth's annual rotation about the sun and eclipses), and engineering (dynamics, statics, hydrostatics, pneumatics and applied mechanics).[22]

In his final comments, he asserted that the names for "Arabic" numerals, the "Pythagorean" Theorem and the "Cartesian" co-ordinate system are historical misnomers; rather, according to Tirthaji, these mathematical insights were enumerated and formalised first by Indian mathematicians of the Hindu tradition, for whom credit ought to be acknowledged.

Srinivasa Ramanujan

Srinivasa Ramanujan FRS ( /ˈʃriːniˌvɑːsə ˈrɑːmɑːˌnʊdʒən/ ( listen), /-rɑːˈmɑːnʊdʒən/;[1] 22 December 1887 – 26 April 1920) was an Indian mathematician who lived during the British Rule in India. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems considered to be unsolvable. Ramanujan initially developed his own mathematical research in isolation; it was quickly recognized by Indian mathematicians. Seeking mathematicians who could better understand his work, in 1913 he began a postal partnership with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognizing the extraordinary work sent to him as samples, Hardy arranged travel for Ramanujan to Cambridge. In his notes, Ramanujan had produced groundbreaking new theorems, including some that Hardy stated had 'defeated [him and his colleagues] completely', in addition to rediscovering recently proven but highly advanced results.

During his short life, Ramanujan independently compiled nearly 3,900 results (mostly identities and equations).[2] Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae, and mock theta functions, have opened entire new areas of work and inspired a vast amount of further research.[3] Nearly all his claims have now been proven correct.[4] The Ramanujan Journal, a peer-reviewed scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan,[5] and his notebooks - containing summaries of his published and unpublished results - have been analyzed and studied for decades since his death as a source of new mathematical ideas. As late as 2011 and again in 2012, researchers continued to discover that mere comments in his writings about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle number theory results that remained unsuspected until nearly a century after his death and which relied on work published in 2006.[6][7] He became one of the youngest Fellows of the Royal Society and only the second Indian member, and the first Indian to be elected a Fellow of Trinity College, Cambridge. Of his original letters, Hardy stated that a 'single look' was enough to show they could only have been written by a mathematician of the highest calibre, comparing Ramanujan to other mathematical geniuses such as Euler and Jacobi.

In 1919, ill health – now believed to have been hepatic amoebiasis (a complication from episodes of dysentery many years previously) – compelled Ramanujan's return to India, where he died in 1920 at the age of 32. His last letters to Hardy, written January 1920, show that he was still continuing to produce new mathematical ideas and theorems. His "lost notebook", containing discoveries from the last year of his life, caused great excitement among mathematicians when it was rediscovered in 1976.

A deeply religious Hindu,[8] Ramanujan credited his substantial mathematical capacities to divinity, and stated that the mathematical knowledge he displayed was revealed to him by his family goddess. '"An equation for me has no meaning," he once said, "unless it expresses a thought of God."

Early life

Ramanujan's home on Sarangapani Sannidhi Street, Kumbakonam

Ramanujan (literally, "younger brother of Rama", a Hindu deity[10]:12) was born on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode, Madras Presidency (now Tamil Nadu), at the residence of his maternal grandparents.[10]:11 His father, K. Srinivasa Iyengar, originally from Thanjavur district, worked as a clerk in a sari shop.[10]:17-18 His mother, Komalatammal, was a housewife and also sang at a local temple.[11] They lived in a small traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam.[12] The family home is now a museum. When Ramanujan was a year and a half old, his mother gave birth to a son, Sadagopan, who died less than three months later. In December 1889, Ramanujan contracted smallpox, though he recovered, unlike 4,000 others who would die in a bad year in the Thanjavur district around this time. He moved with his mother to her parents' house in Kanchipuram, near Madras (now Chennai). His mother gave birth to two more children, in 1891 and 1894, both failing to reach their first birthdays.[10]:12

On 1 October 1892, Ramanujan was enrolled at the local school.[10]:13 After his maternal grandfather lost his job as a court official in Kanchipuram,[10]:19 Ramanujan and his mother moved back to Kumbakonam and he was enrolled in the Kangayan Primary School.[10]:14 When his paternal grandfather died, he was sent back to his maternal grandparents, then living in Madras. He did not like school in Madras, and tried to avoid attending. His family enlisted a local constable to make sure the boy attended school. Within six months, Ramanujan was back in Kumbakonam.[10]:14

Since Ramanujan's father was at work most of the day, his mother took care of the boy as a child. He had a close relationship with her. From her, he learned about tradition and puranas. He learned to sing religious songs, to attend pujas at the temple, and to maintain particular eating habits – all of which are part of Brahminculture.[10]:20 At the Kangayan Primary School, Ramanujan performed well. Just before turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography and arithmetic with the best scores in the district.[10]:25 That year, Ramanujan entered Town Higher Secondary School, where he encountered formal mathematics for the first time.[10]:25

By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book by S. L. Loney on advanced trigonometry.[13][14] He mastered this by the age of 13 while discovering sophisticated theorems on his own. By 14, he was receiving merit certificates and academic awards that continued throughout his school career, and he assisted the school in the logistics of assigning its 1200 students (each with differing needs) to its 35-odd teachers.[10]:27 He completed mathematical exams in half the allotted time, and showed a familiarity with geometry and infinite series. Ramanujan was shown how to solve cubic equations in 1902; he developed his own method to solve the quartic. The following year, Ramanujan tried to solve the quintic, not knowing that it could not be solved by radicals.

In 1903, when he was 16, Ramanujan obtained from a friend a library copy of a A Synopsis of Elementary Results in Pure and Applied Mathematics, G. S. Carr's collection of 5,000 theorems.[10]:39[15] Ramanujan reportedly studied the contents of the book in detail.[16] The book is generally acknowledged as a key element in awakening his genius.[16] The next year, Ramanujan independently developed and investigated the Bernoulli numbers and calculated the Euler–Mascheroni constant up to 15 decimal places.[10]:90 His peers at the time commented that they "rarely understood him" and "stood in respectful awe" of him.[10]:27

When he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding student who deserved scores higher than the maximum.[10] He received a scholarship to study at Government Arts College, Kumbakonam,[10]:28[10]:45 but was so intent on mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process.[10]:47 In August 1905, Ramanujan ran away from home, heading towards Visakhapatnam, and stayed in Rajahmundry [17] for about a month.[10]:47-48 He later enrolled at Pachaiyappa's College in Madras. There he passed in mathematics, choosing only to attempt questions that appealed to him and leaving the rest unanswered, but performed poorly in other subjects, such as English, physiology and Sanskrit.[18] Ramanujan failed his Fellow of Arts exam in December 1906 and again a year later. Without a FA degree, he left college and continued to pursue independent research in mathematics, living in extreme poverty and often on the brink of starvation.[10]:55-56

It was in 1910, after a meeting between the 23-year-old Ramanujan and the founder of the Indian Mathematical Society, V. Ramaswamy Aiyer, also known as Professor Ramaswami, that Ramanujan started to get recognition within the mathematics circles of Madras, subsequently leading to his inclusion as a researcher at the University of Madras.[19]

Adulthood in India

On 14 July 1909, Ramanujan married Janaki (Janakiammal) (21 March 1899 – 13 April 1994), a girl whom his mother had selected for him a year earlier and who was ten years old when they married.[20][21][10]:71 It was not unusual for marriages to be arranged with girls. She came from Rajendram, a village close to Marudur (Karur district) Railway Station. Ramanujan's father did not participate in the marriage ceremony.[22] As was common at that time, Janakiammal continued to stay at her maternal home for three years after marriage till she attained puberty. In 1912, she and Ramanujan's mother joined Ramanujan in Madras.[23]

After the marriage, Ramanujan developed a hydrocele testis.[10]:72 The condition could be treated with a routine surgical operation that would release the blocked fluid in the scrotal sac, but his family did not have the money for the operation. In January 1910, a doctor volunteered to do the surgery at no cost.[24]

After his successful surgery, Ramanujan searched for a job. He stayed at a friend's house while he went from door to door around Madras looking for a clerical position. To make money, he tutored students at Presidency College who were preparing for their F.A. exam.[10]:73

In late 1910, Ramanujan was sick again. He feared for his health, and told his friend R. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British professor Edward B. Ross, of the Madras Christian College."[10]:74-75 After Ramanujan recovered and retrieved his notebooks from Iyer, he took a train from Kumbakonam to Villupuram, a city under French control.[25][26] In 1912, Ramanujan moved to a house in Saiva Muthaiah Mudali street, George Town, Madras with his wife and mother where they lived for a few months.[27] In May 1913, upon securing a research position at Madras University, Ramanujan moved with his family to Triplicane.[28]

Pursuit of career in mathematics

Ramanujan met deputy collector V. Ramaswamy Aiyer, who had founded the Indian Mathematical Society.[10]:77 Wishing for a job at the revenue department where Aiyer worked, Ramanujan showed him his mathematics notebooks. As Aiyer later recalled:

I was struck by the extraordinary mathematical results contained in [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.[29]

Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras.[10]:77 Some of them looked at his work and gave him letters of introduction to R. Ramachandra Rao, the district collector for Nellore and the secretary of the Indian Mathematical Society.[30][31][32] Rao was impressed by Ramanujan's research but doubted that it was his own work. Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable Bombay mathematician, in which Saldhana expressed a lack of understanding of his work but concluded that he was not a phony.[10]:80 Ramanujan's friend C. V. Rajagopalachari tried to quell Rao's doubts about Ramanujan's academic integrity. Rao agreed to give him another chance, and listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series, which Rao said ultimately converted him to a belief in Ramanujan's brilliance.[10]:80 When Rao asked him what he wanted, Ramanujan replied that he needed work and financial support. Rao consented and sent him to Madras. He continued his research, with Rao's financial aid taking care of his daily needs. With Aiyer's help, Ramanujan had his work published in the Journal of the Indian Mathematical Society.[10]:86

One of the first problems he posed in the journal was:

${\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}}.$

He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied the solution to the problem himself. On page 105 of his first notebook, he formulated an equation that could be used to solve the infinitely nested radicals problem.

$x+n+a={\sqrt {ax+(n+a)^{2}+x{\sqrt {a(x+n)+(n+a)^{2}+(x+n){\sqrt {\cdots }}}}}}$

Using this equation, the answer to the question posed in the Journal was simply 3, obtained by setting $x = 2$ , $n = 1$ , and $a = 0$ .[10]:87 Ramanujan wrote his first formal paper for the Journal on the properties of Bernoulli numbers. One property he discovered was that the denominators (sequence A027642 in the OEIS) of the fractions of Bernoulli numbers were always divisible by six. He also devised a method of calculating $Bn$ based on previous Bernoulli numbers. One of these methods follows:

It will be observed that if n is even but not equal to zero,

$Bn$ is a fraction and the numerator of $Bn / n$ in its lowest terms is a prime number,
the denominator of $Bn$ contains each of the factors 2 and 3 once and only once,
$2 n (2 n - 1) Bn / n$ is an integer and $2(2 n - 1) Bn$ consequently is an odd integer.

In his 17-page paper, "Some Properties of Bernoulli's Numbers" (1911), Ramanujan gave three proofs, two corollaries and three conjectures.[10]:91 Ramanujan's writing initially had many flaws. As Journal editor M. T. Narayana Iyengar noted:

Mr. Ramanujan's methods were so terse and novel and his presentation so lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.[33]

Ramanujan later wrote another paper and also continued to provide problems in the Journal.[34] In early 1912, he got a temporary job in the Madras Accountant General's office, with a salary of 20 rupees per month. He lasted only a few weeks.[35] Toward the end of that assignment, he applied for a position under the Chief Accountant of the Madras Port Trust.

In a letter dated 9 February 1912, Ramanujan wrote:

Sir,
I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me.[36]

Attached to his application was a recommendation from E. W. Middlemast, a mathematics professor at the Presidency College, who wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics".[37] Three weeks after he had applied, on 1 March, Ramanujan learned that he had been accepted as a Class III, Grade IV accounting clerk, making 30 rupees per month.[10]:96 At his office, Ramanujan easily and quickly completed the work he was given, so he spent his spare time doing mathematical research. Ramanujan's boss, Sir Francis Spring, and S. Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, encouraged Ramanujan in his mathematical pursuits.

Contacting British mathematicians

In the spring of 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan's work to British mathematicians. M. J. M. Hill of University College London commented that Ramanujan's papers were riddled with holes.[10]:105 He said that although Ramanujan had "a taste for mathematics, and some ability," he lacked the educational background and foundation needed to be accepted by mathematicians.[38] Although Hill did not offer to take Ramanujan on as a student, he did give thorough and serious professional advice on his work. With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.[10]:106

The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan's papers without comment.[10]:170-171 On 16 January 1913, Ramanujan wrote to G. H. Hardy. Coming from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible fraud.[39] Hardy recognised some of Ramanujan's formulae but others "seemed scarcely possible to believe".[40]:494 One of the theorems Hardy found amazing was on the bottom of page three (valid for $0 < a < b + 1 / 2$ ):

${\begin{aligned}&\int \limits _{0}^{\infty }{\frac {1+{\dfrac {x^{2}}{(b+1)^{2}}}}{1+{\dfrac {x^{2}}{a^{2}}}}}\times {\frac {1+{\dfrac {x^{2}}{(b+2)^{2}}}}{1+{\dfrac {x^{2}}{(a+1)^{2}}}}}\times \cdots \,dx\\[6pt]={}&{\frac {\sqrt {\pi }}{2}}\times {\frac {\Gamma \left(a+{\frac {1}{2}}\right)\Gamma (b+1)\Gamma (b-a+1)}{\Gamma (a)\Gamma \left(b+{\frac {1}{2}}\right)\Gamma \left(b-a+{\frac {1}{2}}\right)}}.\end{aligned}}$

Hardy was also impressed by some of Ramanujan's other work relating to infinite series:

$1-5\left({\frac {1}{2}}\right)^{3}+9\left({\frac {1\times 3}{2\times 4}}\right)^{3}-13\left({\frac {1\times 3\times 5}{2\times 4\times 6}}\right)^{3}+\cdots ={\frac {2}{\pi }}$ $1+9\left({\frac {1}{4}}\right)^{4}+17\left({\frac {1\times 5}{4\times 8}}\right)^{4}+25\left({\frac {1\times 5\times 9}{4\times 8\times 12}}\right)^{4}+\cdots ={\frac {2{\sqrt {2}}}{{\sqrt {\pi }}\,\Gamma ^{2}\left({\frac {3}{4}}\right)}}.$

The first result had already been determined by G. Bauer in 1859. The second was new to Hardy, and was derived from a class of functions called hypergeometric series, which had first been researched by Leonhard Euler and Carl Friedrich Gauss. Hardy found these results "much more intriguing" than Gauss's work on integrals.[10]:167 After seeing Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy commented that the theorems "defeated me completely; I had never seen anything in the least like them before".[10]:168 He figured that Ramanujan's theorems "must be true, because, if they were not true, no one would have the imagination to invent them".[10]:168 Hardy asked a colleague, J. E. Littlewood, to take a look at the papers. Littlewood was amazed by Ramanujan's genius. After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I have received" and said that Ramanujan was "a mathematician of the highest quality, a man of altogether exceptional originality and power".[40]:494–495 One colleague, E. H. Neville, later remarked that "not one [theorem] could have been set in the most advanced mathematical examination in the world".[41]

On 8 February 1913, Hardy wrote Ramanujan a letter expressing his interest in his work, adding that it was "essential that I should see proofs of some of your assertions".[42] Before his letter arrived in Madras during the third week of February, Hardy contacted the Indian Office to plan for Ramanujan's trip to Cambridge. Secretary Arthur Davies of the Advisory Committee for Indian Students met with Ramanujan to discuss the overseas trip.[43] In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land".[10]:185 Meanwhile, he sent Hardy a letter packed with theorems, writing, "I have found a friend in you who views my labour sympathetically."[44]

To supplement Hardy's endorsement, Gilbert Walker, a former mathematical lecturer at Trinity College, Cambridge, looked at Ramanujan's work and expressed amazement, urging the young man to spend time at Cambridge.[10]:175 As a result of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an engineering college, invited Ramanujan's colleague Narayana Iyer to a meeting of the Board of Studies in Mathematics to discuss "what we can do for S. Ramanujan".[45] The board agreed to grant Ramanujan a research scholarship of 75 rupees per month for the next two years at the University of Madras.[46] While he was engaged as a research student, Ramanujan continued to submit papers to the Journal of the Indian Mathematical Society. In one instance, Narayana Iyer submitted some of Ramanujan's theorems on summation of series to the journal, adding, "The following theorem is due to S. Ramanujan, the mathematics student of Madras University." Later in November, British Professor Edward B. Ross of Madras Christian College, whom Ramanujan had met a few years before, stormed into his class one day with his eyes glowing, asking his students, "Does Ramanujan know Polish?" The reason was that in one paper, Ramanujan had anticipated the work of a Polish mathematician whose paper had just arrived in the day's mail.[47] In his quarterly papers, Ramanujan drew up theorems to make definite integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made to evaluate formerly unyielding integrals.[10]:183

Hardy's correspondence with Ramanujan soured after Ramanujan refused to come to England. Hardy enlisted a colleague lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan to England.[10]:184 Neville asked Ramanujan why he would not go to Cambridge. Ramanujan apparently had now accepted the proposal; as Neville put it, "Ramanujan needed no converting and that his parents' opposition had been withdrawn".[41] Apparently, Ramanujan's mother had a vivid dream in which the family goddess, the deity of Namagiri, commanded her "to stand no longer between her son and the fulfilment of his life's purpose".[41] Ramanujan voyaged to England by ship, leaving his wife to stay with his parents in India.

Life in England

Ramanujan (centre) and his colleague G. H. Hardy(extreme right), with other scientists, outside the Senate House, Cambridge, c.1914–19

Whewell's Court, Trinity College, Cambridge

Ramanujan departed from Madras aboard the S.S. Nevasa on 17 March 1914.[10]:196 When he disembarked in London on 14 April, Neville was waiting for him with a car. Four days later, Neville took him to his house on Chesterton Road in Cambridge. Ramanujan immediately began his work with Littlewood and Hardy. After six weeks, Ramanujan moved out of Neville's house and took up residence on Whewell's Court, a five-minute walk from Hardy's room.[10]:202 Hardy and Littlewood began to look at Ramanujan's notebooks. Hardy had already received 120 theorems from Ramanujan in the first two letters, but there were many more results and theorems in the notebooks. Hardy saw that some were wrong, others had already been discovered, and the rest were new breakthroughs.[48] Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I can believe that he's at least a Jacobi",[49] while Hardy said he "can compare him only with Euler or Jacobi."[50]

Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood, and published part of his findings there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs, and working styles. In the previous few decades, the foundations of mathematics had come into question and the need for mathematically rigorous proofs recognized. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas Ramanujan was a deeply religious man who relied very strongly on his intuition and insights. While in England, Hardy tried his best to fill the gaps in Ramanujan's education and to mentor him in the need for formal proofs to support his results, without hindering his inspiration - a conflict that neither found easy.

Ramanujan was awarded a Bachelor of Science degree by research (this degree was later renamed PhD) in March 1916 for his work on highly composite numbers, the first part of which was published as a paper in the Proceedings of the London Mathematical Society. The paper was more than 50 pages and proved various properties of such numbers. Hardy remarked that it was one of the most unusual papers seen in mathematical research at that time and that Ramanujan showed extraordinary ingenuity in handling it.[citation needed] On 6 December 1917, he was elected to the London Mathematical Society. In 1918 he was elected a Fellow of the Royal Society, the second Indian admitted to the Royal Society, following Ardaseer Cursetjee in 1841. At age 31 Ramanujan was one of the youngest Fellows in the history of the Royal Society. He was elected "for his investigation in Elliptic functions and the Theory of Numbers." On 13 October 1918, he was the first Indian to be elected a Fellow of Trinity College, Cambridge.[10]:299-300

Illness and death

Throughout his life, Ramanujan was plagued by health problems. His health worsened in England; possibly he was also less resilient due to the difficulty of keeping to the strict dietary requirements of his religion in England and wartime rationing during 1914–1918. He was diagnosed with tuberculosis and a severe vitamin deficiency at the time, and was confined to a sanatorium. In 1919 he returned to Kumbakonam, Madras Presidency, and soon thereafter, in 1920, died at the age of 32. After his death, his brother Tirunarayanan chronicled Ramanujan's remaining handwritten notes consisting of formulae on singular moduli, hypergeometric series and continued fractions and compiled them.[23] Ramanujan's widow, Smt. Janaki Ammal, moved to Bombay; in 1950 she returned to Madras, where she lived in Triplicane until her death in 1994.[22][23]

A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D. A. B. Young[51] concluded that his medical symptoms—including his past relapses, fevers and hepatic conditions—were much closer to those resulting from hepatic amoebiasis, an illness then widespread in Madras, rather than tuberculosis. He had two episodes of dysentery before he left India. When not properly treated, dysentery can lie dormant for years and lead to hepatic amoebiasis, whose diagnosis was not then well established.[52] Amoebiasis was a treatable and often curable disease at the time.[52][53]

Personality and spiritual life

Ramanujan has been described as a person of a somewhat shy and quiet disposition, a dignified man with pleasant manners.[54] He lived a simple life at Cambridge.[10]:234,241Ramanujan's first Indian biographers describe him as a rigorously orthodox Hindu. He credited his acumen to his family goddess, Mahalakshmi of Namakkal. He looked to her for inspiration in his work[10]:36 and said he dreamed of blood drops that symbolised her male consort, Narasimha. Afterward he would receive visions of scrolls of complex mathematical content unfolding before his eyes.[10]:281 He often said, "An equation for me has no meaning unless it represents a thought of God."[55]

Hardy cites Ramanujan as remarking that all religions seemed equally true to him.[10]:283 Hardy further argued that Ramanujan's religious belief had been romanticised by Westerners and overstated—in reference to his belief, not practice—by Indian biographers. At the same time, he remarked on Ramanujan's strict vegetarianism.[56]

Mathematical achievements

In mathematics, there is a distinction between insight and formulating or working through a proof. Ramanujan proposed an abundance of formulae that could be investigated later in depth. G. H. Hardy said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a byproduct of his work, new directions of research were opened up. Examples of the most interesting of these formulae include the intriguing infinite series for $π$ , one of which is given below:

${\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}.$

This result is based on the negative fundamental discriminant $d$ = −4 × 58 = −232 with class number $h (d)$ = 2. 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 3962 and is related to the fact that

${\textstyle e^{\pi {\sqrt {58}}}=396^{4}-104.000000177\dots .}$

This might be compared to Heegner numbers, which have class number 1 and yield similar formulae.

Ramanujan's series for $π$ converges extraordinarily rapidly (exponentially) and forms the basis of some of the fastest algorithms currently used to calculate $π$ . Truncating the sum to the first term also gives the approximation 9801√2/4412 for $π$ , which is correct to six decimal places. See also the more general Ramanujan–Sato series.

One of Ramanujan's remarkable capabilities was the rapid solution of problems, illustrated by the following anecdote about an incident in which P. C. Mahalanobis posed a problem:

"'Imagine that you are on a street with houses marked 1 through $n$ . There is a house in between ( $x$ ) such that the sum of the house numbers to the left of it equals the sum of the house numbers to its right. If $n$ is between 50 and 500, what are $n$ and $x$ ?' This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. 'It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind', Ramanujan replied."[57][58]

His intuition also led him to derive some previously unknown identities, such as

${\begin{aligned}&\left(1+2\sum _{n=1}^{\infty }{\frac {\cos(n\theta )}{\cosh(n\pi )}}\right)^{-2}+\left(1+2\sum _{n=1}^{\infty }{\frac {\cosh(n\theta )}{\cosh(n\pi )}}\right)^{-2}\\[6pt]&={\frac {2\Gamma ^{4}\left({\frac {3}{4}}\right)}{\pi }}={\frac {8\pi ^{3}}{\Gamma ^{4}\left({\frac {1}{4}}\right)}}\end{aligned}}$

for all $θ$ , where $Γ(z)$ is the gamma function, and related to a special value of the Dedekind eta function. Expanding into series of powers and equating coefficients of $θ 0$ , $θ 4$ , and $θ 8$ gives some deep identities for the hyperbolic secant.

In 1918 Hardy and Ramanujan studied the partition function $P (n)$ extensively. They gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. Hans Rademacher, in 1937, was able to refine their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae called the circle method.[59]

In the last year of his life, Ramanujan discovered mock theta functions.[60] For many years these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms.

The Ramanujan conjecture

Although there are numerous statements that could have borne the name Ramanujan conjecture, there is one that was highly influential on later work. In particular, the connection of this conjecture with conjectures of André Weil in algebraic geometry opened up new areas of research. That Ramanujan conjecture is an assertion on the size of the tau-function, which has as generating function the discriminant modular form Δ(q), a typical cusp form in the theory of modular forms. It was finally proven in 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures. The reduction step involved is complicated. Deligne won a Fields Medal in 1978 for that work.[61]

In his paper "On certain arithmetical functions", Ramanujan defined the so-called delta-function whose coefficients are called $τ (n)$ (the Ramanujan tau function).[62] He proved many congruences for these numbers such as $τ (p) \equiv 1 + p 11 mod 691$ for primes $p$ . This congruence (and others like it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) to conjecture that there is a theory of Galois representations which "explains" these congruences and more generally all modular forms. $Δ(z)$ is the first example of a modular form to be studied in this way. Pierre Deligne (in his Fields Medal-winning work) proved Serre's conjecture. The proof of Fermat's Last Theorem proceeds by first reinterpreting elliptic curves and modular forms in terms of these Galois representations. Without this theory there would be no proof of Fermat's Last Theorem.[63]

Ramanujan's notebooks

While still in Madras, Ramanujan recorded the bulk of his results in four notebooks of loose-leaf paper. They were mostly written up without any derivations. This is probably the origin of the misapprehension that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to prove most of his results, but chose not to.

This may have been for any number of reasons. Since paper was very expensive, Ramanujan would do most of his work and perhaps his proofs on slate, and then transfer just the results to paper. Using a slate was common for mathematics students in the Madras Presidency at the time. He was also quite likely to have been influenced by the style of G. S. Carr's book, which stated results without proofs. Finally, it is possible that Ramanujan considered his workings to be for his personal interest alone and therefore recorded only the results.[64]

The first notebook has 351 pages with 16 somewhat organised chapters and some unorganised material. The second notebook has 256 pages in 21 chapters and 100 unorganised pages, with the third notebook containing 33 unorganised pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself created papers exploring material from Ramanujan's work, as did G. N. Watson, B. M. Wilson, and Bruce Berndt.[64] A fourth notebook with 87 unorganised pages, the so-called "lost notebook", was rediscovered in 1976 by George Andrews.[52]

Hardy–Ramanujan number 1729

The number 1729 is known as the Hardy–Ramanujan number after a famous visit by Hardy to see Ramanujan at a hospital. In Hardy's words:[65]

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No", he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

Immediately before this anecdote, Hardy quoted Littlewood as saying, "Every positive integer was one of [Ramanujan's] personal friends."[66]

The two different ways are

1729 = 13 + 123 = 93 + 103.

Generalizations of this idea have created the notion of "taxicab numbers".

Mathematicians' views of Ramanujan

Hardy said: "He combined a power of generalization, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day. The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a doubly periodic function or of Cauchy's theorem, and had indeed but the vaguest idea of what a function of a complex variable was...".[67] When asked about the methods Ramanujan employed to arrive at his solutions, Hardy said that they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account."[68] He also stated that he had "never met his equal, and can compare him only with Euler or Jacobi."[68]

K. Srinivasa Rao has said,[69] "As for his place in the world of Mathematics, we quote Bruce C. Berndt: 'Paul Erdős has passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, J. E. Littlewood 30, David Hilbert 80 and Ramanujan 100.'" During a lecture at IIT Madras in May 2011, Berndt stated that over the last 40 years, as nearly all of Ramanujan's theorems have been proven right, there had been greater appreciation of Ramanujan's work and brilliance, and that Ramanujan's work was now pervading many areas of modern mathematics and physics.[60][70]

In his book Scientific Edge, the physicist Jayant Narlikar spoke of "Srinivasa Ramanujan, discovered by the Cambridge mathematician Hardy, whose great mathematical findings were beginning to be appreciated from 1915 to 1919. His achievements were to be fully understood much later, well after his untimely death in 1920. For example, his work on the highly composite numbers (numbers with a large number of factors) started a whole new line of investigations in the theory of such numbers."

Posthumous recognition

Bust of Ramanujan in the garden of Birla Industrial & Technological Museum

Ramanujan's home state of Tamil Nadu celebrates 22 December (Ramanujan's birthday) as 'State IT Day'. A stamp picturing Ramanujan was released by the Government of India in 1962 – the 75th anniversary of Ramanujan's birth – commemorating his achievements in the field of number theory,[71] and a new design was issued on 26 December 2011, by the India Post.[72][73]

Since Ramanujan's centennial year, his birthday, 22 December, has been annually celebrated as Ramanujan Day by the Government Arts College, Kumbakonam where he studied and at the IIT Madras in Chennai. A prize for young mathematicians from developing countries has been created in Ramanujan's name by the International Centre for Theoretical Physics (ICTP) in cooperation with the International Mathematical Union, which nominate members of the prize committee. The SASTRA University, based in the state of Tamil Nadu in South India, has instituted the SASTRA Ramanujan Prize of US$10,000 to be given annually to a mathematician not exceeding the age of 32 for outstanding contributions in an area of mathematics influenced by Ramanujan. Based on the recommendations of a high level committee appointed by the University Grants Commission (UGC), Government of India, Srinivasa Ramanujan Centre, established by SASTRA, has been declared as an OFF-CAMPUS CENTRE under the ambit of SASTRA University. House of Ramanujan Mathematics, a museum on life and works of the Mathematical prodigy, Srinivasa Ramanujan, also exists on this campus. SASTRA purchased the house where Srinivasa Ramanujan lived at Kumabakonam and renovated it.[74] Vasavi College of Engineering named its Department of Computer Science and Information Technology "Ramanujan Block".

In 2011, on the 125th anniversary of his birth, the Indian Government declared that 22 December will be celebrated every year as National Mathematics Day.[75] Then Indian Prime Minister Manmohan Singh also declared that the year 2012 would be celebrated as the National Mathematics Year [76].

In popular culture

The Man Who Knew Infinity: A Life of the Genius Ramanujan is a biography of Ramanujan, written in 1991 by Robert Kanigel and published by Washington Square Press.
The Man Who Knew Infinity is a 2015 film based on the book by Robert Kanigel. In the film, Ramanujan is portrayed by British actor Dev Patel.[77][78][79]
Ramanujan, an Indo-British collaboration film, chronicling the life of Ramanujan, was released in 2014 by the independent film company Camphor Cinema.[80] The cast and crew include director Gnana Rajasekaran, cinematographer Sunny Joseph and editor B. Lenin.[81][82] Popular Indian and English stars Abhinay Vaddi, Suhasini Maniratnam, Bhama, Kevin McGowan and Michael Lieber star in pivotal roles.[83]
The thriller novel The Steradian Trail by M. N. Krish weaves Ramanujan and his accidental discovery into its plot connecting religion, mathematics, finance and economics.[84][85]
Partition, a play by Ira Hauptman about Hardy and Ramanujan, first performed in 2013.[86][87][88][89]
A play, First Class Man by Alter Ego Productions,[90] was based on David Freeman's First Class Man. The play is centred around Ramanujan and his complex and dysfunctional relationship with Hardy. On 16 October 2011, it was announced that Roger Spottiswoode, best known for his James Bond film Tomorrow Never Dies, is working on the film version, starring actor Siddharth. Like the book and play it is also titled The First Class Man.[91]
A Disappearing Number is a recent British stage production by the company Complicite that explores the relationship between Hardy and Ramanujan.
The novel The Indian Clerk by David Leavitt explores in fiction the events following Ramanujan's letter to Hardy.[92][93]
Google honoured him on his 125th birth anniversary by replacing its logo with a doodle on its home page.[94][95]
Ramanujan was mentioned in the 1997 film Good Will Hunting, in a scene where professor Gerald Lambeau (Stellan Skarsgard) explains to Sean Maguire (Robin Williams) the genius of Will Hunting (Matt Damon) by comparing him to Ramanujan.
On 22 March 1988, the PBS series Nova aired a documentary about Ramanujan, "The Man Who Loved Numbers" (Season 15, Episode 19).[96]
In the book Hyperspace by Michio Kaku, Ramanujan's contributions to Superstring theory and a brief synopsis of his life are given in Part II Unification in Ten Dimensions in the chapter Superstrings under the sections Mystery of Modular Functions and Reinventing 100 Years of Mathematics.

Selected publications on Ramanujan and his work

Berndt, Bruce C. (1998). Butzer, P. L.; Oberschelp, W.; Jongen, H. Th., eds. Charlemagne and His Heritage: 1200 Years of Civilization and Science in Europe (PDF). Turnhout, Belgium: Brepols Verlag. pp. 119–146. ISBN 2-503-50673-9.
Berndt, Bruce C.; Andrews, George E. (2005). Ramanujan's Lost Notebook. Part I. New York: Springer. ISBN 0-387-25529-X.
Berndt, Bruce C.; Andrews, George E. (2008). Ramanujan's Lost Notebook. Part II. New York: Springer. ISBN 978-0-387-77765-8.
Berndt, Bruce C.; Andrews, George E. (2012). Ramanujan's Lost Notebook. Part III. New York: Springer. ISBN 978-1-4614-3809-0.
Berndt, Bruce C.; Andrews, George E. (2013). Ramanujan's Lost Notebook. Part IV. New York: Springer. ISBN 978-1-4614-4080-2.
Berndt, Bruce C.; Rankin, Robert A. (1995). Ramanujan: Letters and Commentary. 9. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-0287-9.
Berndt, Bruce C.; Rankin, Robert A. (2001). Ramanujan: Essays and Surveys. 22. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-2624-7.
Berndt, Bruce C. (2006). Number Theory in the Spirit of Ramanujan. 9. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-4178-5.
Berndt, Bruce C. (1985). Ramanujan's Notebooks. Part I. New York: Springer. ISBN 0-387-96110-0.
Berndt, Bruce C. (1999). Ramanujan's Notebooks. Part II. New York: Springer. ISBN 0-387-96794-X.
Berndt, Bruce C. (2004). Ramanujan's Notebooks. Part III. New York: Springer. ISBN 0-387-97503-9.
Berndt, Bruce C. (1993). Ramanujan's Notebooks. Part IV. New York: Springer. ISBN 0-387-94109-6.
Berndt, Bruce C. (2005). Ramanujan's Notebooks. Part V. New York: Springer. ISBN 0-387-94941-0.
Hardy, G. H. (March 1937). "The Indian Mathematician Ramanujan". The American Mathematical Monthly. 44 (3): 137–155. doi:10.2307/2301659. JSTOR 2301659.
Hardy, G. H. (1978). Ramanujan. New York: Chelsea Pub. Co. ISBN 0-8284-0136-5.
Hardy, G. H. (1999). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-2023-0.
Henderson, Harry (1995). Modern Mathematicians. New York: Facts on File Inc. ISBN 0-8160-3235-1.
Kanigel, Robert (1991). The Man Who Knew Infinity: a Life of the Genius Ramanujan. New York: Charles Scribner's Sons. ISBN 0-684-19259-4.
Kolata, Gina (19 June 1987). "Remembering a 'Magical Genius'". Science, New Series. American Association for the Advancement of Science. 236 (4808): 1519–1521. Bibcode:1987Sci...236.1519K. doi:10.1126/science.236.4808.1519.
Leavitt, David (2007). The Indian Clerk (paperback ed.). London: Bloomsbury. ISBN 978-0-7475-9370-6.
Narlikar, Jayant V. (2003). Scientific Edge: the Indian Scientist From Vedic to Modern Times. New Delhi, India: Penguin Books. ISBN 0-14-303028-0.
Ono, Ken; Aczel, Amir D. (2016-04-13). My Search for Ramanujan: How I Learned to Count. Springer. ISBN 978-3319255668.
Sankaran, T. M. (2005). "Srinivasa Ramanujan- Ganitha lokathile Mahaprathibha" (in Malayalam). Kochi, India: Kerala Sastra Sahithya Parishath.

Selected publications on works of Ramanujan

Ramanujan, Srinivasa; Hardy, G. H.; Seshu Aiyar, P. V.; Wilson, B. M.; Berndt, Bruce C. (2000). Collected Papers of Srinivasa Ramanujan. AMS. ISBN 0-8218-2076-1.

This book was originally published in 1927[97] after Ramanujan's death. It contains the 37 papers published in professional journals by Ramanujan during his lifetime. The third reprint contains additional commentary by Bruce C. Berndt.

S. Ramanujan (1957). Notebooks (2 Volumes). Bombay: Tata Institute of Fundamental Research.

These books contain photocopies of the original notebooks as written by Ramanujan.

S. Ramanujan (1988). The Lost Notebook and Other Unpublished Papers. New Delhi: Narosa. ISBN 3-540-18726-X.

This book contains photo copies of the pages of the "Lost Notebook".

Problems posed by Ramanujan, Journal of the Indian Mathematical Society.
S. Ramanujan (2012). Notebooks (2 Volumes). Bombay: Tata Institute of Fundamental Research.

This was produced from scanned and microfilmed images of the original manuscripts by expert archivists of Roja Muthiah Research Library, Chennai.

A. A. Krishnaswami Ayyangar

A. A. Krishnaswami Ayyangar (1892-1953)[1] was a mathematician from India. He got his M.A. in Mathematics at the age of 18 from Pachaiyappa's College and subsequently started teaching Mathematics there. In 1918 he joined the Mathematics Department of University of Mysore and retired from there in 1947. He died in June 1953. He is the father of the celebrated Kannada poet A. K. Ramanujan.

Works

Ayyangar wrote an article on the Chakravala method and showed how the method differs from the method of continued fractions. He recounted that this point was missed by Andre Weil, who thought that the Chakravala method was only an “experimental fact” to the Indians and attributed general proofs to Fermat and Lagrange.[2]

Professor Subhash Kak of Louisiana State University, Baton Rouge first noted that Ayyangar’s presentations of Indian works were unique, and was instrumental in bringing it to the notice of the scientific community.

Prasanta Chandra Mahalanobis

Prasanta Chandra Mahalanobis OBE, FRS [1] (29 June 1893 – 28 June 1972) was an Indian scientist and applied statistician. He is best remembered for the Mahalanobis distance, a statistical measure and for being one of the members of the first Planning Commission of free India. He made pioneering studies in anthropometry in India. He founded the Indian Statistical Institute, and contributed to the design of large-scale sample surveys.

Early life

Mahalanobis belonged to a family of Bengali landed gentry who lived in Bikrampur (now in Bangladesh). His grandfather Gurucharan (1833–1916) moved to Calcutta in 1854 and built up a business, starting a chemist shop in 1860. Gurucharan was influenced by Debendranath Tagore (1817–1905), father of the Nobel Prize–winning poet, Rabindranath Tagore. Gurucharan was actively involved in social movements such as the Brahmo Samaj, acting as its Treasurer and President. His house on 210 Cornwallis Street was the center of the Brahmo Samaj. Gurucharan married a widow, an action against social traditions.

His elder son Subodhchandra (1867–1953) became a distinguished educator after studying physiology at Edinburgh University. He was elected as a Fellow of the Royal Society of Edinburgh.[2] He was the Head of the Dept. of Physiology, University of Cardiff (the first Indian to occupy this post in a British university). In 1900, Subodhchandra returned to India, founding the Dept. of Physiology in the Presidency College, Calcutta. Subodhchandra also became a member of the Senate of the Calcutta University.

Gurucharan's younger son, Prabodh Chandra (1869-1942) was the father of P. C. Mahalanobis. Born in the house at 210 Cornwallis Street, Mahalanobis grew up in a socially active family surrounded by intellectuals and reformers.[1]

Mahalanobis received his early schooling at the Brahmo Boys School in Calcutta, graduating in 1908. He joined the Presidency College, Calcutta where he was taught by teachers who included Jagadish Chandra Bose, and Prafulla Chandra Ray. Others attending were Meghnad Saha, a year junior, and Subhas Chandra Bose, two years his junior at college.[7] Mahalanobis received a Bachelor of Science degree with honours in physics in 1912. He left for England in 1913 to join the University of London.

After missing a train, he stayed with a friend at King's College, Cambridge. He was impressed by King's College Chapel and his host's friend M. A. Candeth suggested that he could try joining there, which he did. He did well in his studies at King's, but also took an interest in cross-country walking and punting on the river. He interacted with the mathematical genius Srinivasa Ramanujan during the latter's time at Cambridge.[8] After his Tripos in physics, Mahalanobis worked with C. T. R. Wilson at the Cavendish Laboratory. He took a short break and went to India, where he was introduced to the Principal of Presidency College and was invited to take classes in physics.[1]

After returning to England, Mahalanobis was introduced to the journal Biometrika. This interested him so much that he bought a complete set and took them to India. He discovered the utility of statistics to problems in meteorology and anthropology, beginning to work on problems on his journey back to India.[1]

In Calcutta, Mahalanobis met Nirmalkumari, daughter of Herambhachandra Maitra, a leading educationist and member of the Brahmo Samaj. They married on 27 February 1923, although her father did not completely approve of the union. He was concerned about Mahalanobis's opposition to various clauses in the membership of the student wing of the Brahmo Samaj, including prohibitions against members' drinking alcohol and smoking. Sir Nilratan Sircar, P. C. Mahalanobis' maternal uncle, took part in the wedding ceremony in place of the father of the bride.[1]

Indian Statistical Institute

Mahalanobis memorial at ISI Delhi.

Many colleagues of Mahalanobis took an interest in statistics. An informal group developed in the Statistical Laboratory, which was located in his room at the Presidency College, Calcutta. On 17 December 1931 Mahalanobis called a meeting with Pramatha Nath Banerji (Minto Professor of Economics), Nikhil Ranjan Sen (Khaira Professor of Applied Mathematics) and Sir R. N. Mukherji. Together they established the Indian Statistical Institute (ISI), and formally registered on 28 April 1932 as a non-profit distributing learned society under the Societies Registration Act XXI of 1860.[1]

The Institute was initially in the Physics Department of the Presidency College; its expenditure in the first year was Rs. 238. It gradually grew with the pioneering work of a group of his colleagues, including S. S. Bose, J. M. Sengupta, R. C. Bose, S. N. Roy, K. R. Nair, R. R. Bahadur, Gopinath Kallianpur, D. B. Lahiri and C. R. Rao. The institute also gained major assistance through Pitamber Pant, who was a secretary to Prime Minister Jawaharlal Nehru. Pant was trained in statistics at the Institute and took a keen interest in its affairs.[1]

In 1933, the Institute founded the journal Sankhya, along the lines of Karl Pearson's Biometrika.[1]

The institute started a training section in 1938. Many of the early workers left the ISI for careers in the United States and with the government of India. Mahalanobis invited J. B. S. Haldane to join him at the ISI; Haldane joined as a Research Professor from August 1957, staying until February 1961. He resigned from the ISI due to frustrations with the administration and disagreements with Mahalanobis' policies. He was concerned with the frequent travels and absence of the director and complained that the "... journeyings of our Director define a novel random vector." Haldane helped the ISI develop in biometrics.[9]

In 1959, the institute was declared as an institute of national importance and a deemed university.[1]

Contributions to statistics

Mahalanobis distance

A chance meeting with Nelson Annandale, then the director of the Zoological Survey of India, at the 1920 Nagpur session of the Indian Science Congress led to Annandale asking him to analyse anthropometric measurements of Anglo-Indians in Calcutta. Mahalanobis had been influenced by the anthropometric studies published in the journal Biometrika and he chose to ask the questions on what factors influence the formation of European and Indian marriages. He wanted to examine if the Indian side came from any specific castes. He used the data collected by Annandale and the caste specific measurements made by Herbert Risley to come up with the conclusion that the sample represented a mix of Europeans mainly with people from Bengal and Punjab but not with those from the Northwest Frontier Provinces or from Chhota Nagpur. He also concluded that the intermixture more frequently involved the higher castes than the lower ones.[10][11] This analysis was described by his first scientific paper in 1922.[12] During the course of these studies he found a way of comparing and grouping populations using a multivariate distance measure. This measure, denoted "D2" and now eponymously named Mahalanobis distance, is independent of measurement scale.[1] Mahalanobis also took an interest in physical anthropology and in the accurate measurement of skull measurements for which he developed an instrument that he called the "profiloscope".[13]

Sample surveys

His most important contributions are related to large-scale sample surveys. He introduced the concept of pilot surveys and advocated the usefulness of sampling methods. Early surveys began between 1937 and 1944 and included topics such as consumer expenditure, tea-drinking habits, public opinion, crop acreage and plant disease. Harold Hotellingwrote: "No technique of random sample has, so far as I can find, been developed in the United States or elsewhere, which can compare in accuracy with that described by Professor Mahalanobis" and Sir R. A. Fisher commented that "The ISI has taken the lead in the original development of the technique of sample surveys, the most potent fact finding process available to the administration".[1]

He introduced a method for estimating crop yields which involved statisticians sampling in the fields by cutting crops in a circle of diameter 4 feet. Others such as P. V. Sukhatme and V. G. Panse who began to work on crop surveys with the Indian Council of Agricultural Research and the Indian Agricultural Statistics Research Institute suggested that a survey system should make use of the existing administrative framework. The differences in opinion led to acrimony and there was little interaction between Mahalanobis and agricultural research in later years.[14][15][16]

Later life

In later life, Mahalanobis was a member of the planning commission[17] contributed prominently to newly independent India's five-year plans starting from the second. In the second five-year plan he emphasised industrialisation on the basis of a two-sector model.[1] His variant of Wassily Leontief's Input-output model, the Mahalanobis model, was employed in the Second Five Year Plan, which worked towards the rapid industrialisation of India and with other colleagues at his institute, he played a key role in the development of a statistical infrastructure. He encouraged a project to assess deindustrialisation in India and correct some previous census methodology errors and entrusted this project to Daniel Thorner.[18]

Mahalanobis also had an abiding interest in cultural pursuits and served as secretary to Rabindranath Tagore, particularly during the latter's foreign travels, and also worked at his Visva-Bharati University, for some time. He received one of the highest civilian awards, the Padma Vibhushan from the Government of India for his contribution to science and services to the country.

Mahalanobis died on 28 June 1972, a day before his seventy-ninth birthday. Even at this age, he was still active doing research work and discharging his duties as the Secretary and Director of the Indian Statistical Institute and as the Honorary Statistical Advisor to the Cabinet of the Government of India.

Honours

Officer of the Order of the British Empire (Civil Division), 1942 New Year Honours list[19]
Weldon Memorial Prize from the University of Oxford (1944)
Fellow of the Royal Society, London (1945)[1][20]
President of Indian Science Congress (1950)
Fellow of the Econometric Society, USA (1951)
Fellow of the Pakistan Statistical Association (1952)
Honorary Fellow of the Royal Statistical Society, UK (1954)
Sir Deviprasad Sarvadhikari Gold Medal (1957)
Foreign member of the Academy of Sciences of the USSR (1958)
Honorary Fellow of King's College, Cambridge (1959)
Fellow of the American Statistical Association (1961)
Durgaprasad Khaitan Gold Medal (1961)
Padma Vibhushan (1968)
Srinivasa Ramanujan Gold Medal (1968)

The government of India decided in 2006 to celebrate his birthday, 29 June, as National Statistical Day.

Subbayya Sivasankaranarayana Pillai

Subbayya Sivasankaranarayana Pillai (1901–1950) was an Nagercoil native Indian mathematician specialising in number theory. His contribution to Waring's problem was described in 1950 by K. S. Chandrasekharan as "almost certainly his best piece of work and one of the very best achievements in Indian Mathematics since Ramanujan".[1]

Biography

Subbayya Sivasankaranarayana Pillai was born to parents Subbayya Pillai and Gomati Ammal who were natives of Nagercoil. His mother died a year after his birth and his father when Pillai was in his last year at school.[1]

Pillai did his Intermediate course in the Scott Christian College at Nagercoil [1] and managed to earn a B.A. degree from Maharaja's college, Trivandrum.[2]

In 1927, Pillai was awarded a research fellowship at the University of Madras to work among professors K. Ananda Rau and Ramaswamy S. Vaidyanathaswamy. He was from 1929 to 1941 at Annamalai University where he worked as a lecturer. It was in Annamalai University that he did his major work in Waring's problem.[2] In 1941 he went to the University of Travancore and a year later to the University of Calcutta as a lecturer (where he was at the invitation of Friedrich Wilhelm Levi).[3]

For his achievements he was invited in August 1950, for a year to visit the Institute for Advanced Study, Princeton, USA. He was also invited to participate in the International Congress of Mathematicians at Harvard University as a delegate of the Madras Universitybut he died during the crash of TWA Flight 903 in Egypt on the way to the conference.[4]

Contributions

He proved the Waring's problem for $k\ge 6$ in 1935[5] under the further condition of $(3^k + 1)/(2^k - 1)\le [1.5^k] + 1$ ahead of Leonard Eugene Dickson who around the same time proved it for $k\geq 7.$ [6]

He showed that $g(k)=2^{k}+l-2$ where $l$ is the largest natural number $\leq (3/2)^{k}$ and hence computed the precise value of $g(6)=73$ .[5]

The Pillai sequence 1, 4, 27, 1354, ..., is a quickly-growing integer sequence in which each term is the sum of the previous term and a prime number whose following prime gap is larger than the previous term. It was studied by Pillai in connection with representing numbers as sums of prime numbers.

Raj Chandra Bose

Raj Chandra Bose (19 June 1901 – 31 October 1987) was an Indian American mathematician and statistician best known for his work in design theory, finite geometry and the theory of error-correcting codes in which the class of BCH codes is partly named after him. He also invented the notions of partial geometry, strongly regular graph and started a systematic study of difference sets to construct symmetric block designs. He was notable for his work along with S. S. Shrikhande and E. T. Parker in their disproof of the famous conjecture made by Leonhard Euler dated 1782 that there do not exist two mutually orthogonal Latin squares of order 4n + 2 for every n.

Early life

Bose was born in Hoshangabad, India; he was the first of five children. His father was a physician and life was good until 1918 when his mother died in the influenza pandemic. His father died of a stroke the following year. Despite difficult circumstances, Bose continued to study securing first class in the M.A. examinations in pure mathematics at the University of Calcutta. His research was performed under the supervision of the geometry Professor Syamadas Mukhopadhyaya from Calcutta. Bose worked as a lecturer at Asutosh College, Calcutta. He published his work on the differential geometry of convex curves.

Academic life

Bose's course changed in December 1932 when P. C. Mahalanobis, director of the new (1931) Indian Statistical Institute, offered Bose a part-time job. Mahalanobis had seen Bose's geometrical work and wanted him to work on statistics. The day after Bose moved in, the secretary brought him all the volumes of Biometrika with a list of 50 papers to read and also Fisher's Statistical Methods for Research Workers. Mahalanobis told him, "You were saying that you do not know much statistics. You master the 50 papers ... and Fisher's book. This will suffice for your statistical education for the present." With Samarendra Nath Roy, who joined the ISI a little later, Bose was the chief mathematician at the Institute.

He first worked with multivariate analysis where he collaborated with Mahalanobis and Roy. In 1938–9 Fisher visited India and talked about the design of experiments. Roy had the idea of using the theory of finite fields and finite geometry to solve problems in design. The development of a mathematical theory of design would be Bose's main preoccupation until the mid-1950s.

In 1935 Bose had become full-time at the Institute. In 1940 joined the University of Calcutta where C. R. Rao and H. K. Nandi were in the first group of students he taught. In 1945 Bose became Head of the Department of Statistics. The university authorities told him he needed to have a doctorate. So he submitted his published papers on multivariate analysis and the design of experiments and was awarded a D. Litt. in 1947.

In 1947 Bose went to the United States as a visiting professor at Columbia University and the University of North Carolina at Chapel Hill. He received offers from American universities and he was also offered positions in India. The Indian jobs involved very heavy administration, which he saw as the end of his research work and in March 1949 he joined the University of North Carolina at Chapel Hill as Professor of Statistics.

In the years at Chapel Hill Bose made important discoveries on coding theory (with D.K. Ray-Chaudhuri) and constructed (with S. S. Shrikhande and E. T. Parker ) a Graeco-Latin square of size 10, a counterexample to Euler's conjecture that no Graeco-Latin square of size 4k + 2 exists. In 1971, he retired at the age of 70. He then accepted a chair at Colorado State University of Fort Collins from which he retired in 1980. His final doctoral student finished after this second retirement.

Bose died in Colorado, aged 86, in 1987. He is survived by two daughters. The elder, Purabi Schur, is retired from the Library of Congress and the younger, Sipra Bose Johnson, is retired as a professor of anthropology from the State University of New York at New Paltz.

Some articles by R. C. Bose

R. C. Bose, On the construction of balanced incomplete block designs, Annals of Eugenics. 9 (1939), 358–399.
R. C. Bose and K. R. Nair, Partially balanced incomplete block designs, Sankhya 4 (1939), 337–372.
Bose, Raj Chandra; Mesner, D. M. (1959). "On linear associative algebras corresponding to association schemes of partially balanced designs". Annals of Mathematical Statistics. 30 (1): 21–38. doi:10.1214/aoms/1177706356. JSTOR 2237117. MR 0102157.
R. C. Bose and S. S. Shrikhande, On the falsity of Euler's conjecture about the non-existence of two orthogonal Latin squares of order 4t + 2, Proceedings of the National Academy of Sciences USA, 45, (1959), 734–737.
R. C. Bose and D.K. Ray-Chaudhuri On a class of error-correcting binary codes, Information and control, 3, (1960), 68–79.

Autobiography

J. Gani (ed) (1982) The Making of Statisticians, New York: Springer-Verlag.

This has a chapter in which Bose tells the story of his life.

Discussions

Norman R. Draper (1990) Obituary: Raj Chandra Bose, Journal of the Royal Statistical Society Series A, Vol. 153, No. 1. pp. 98–99.
"Bose, Raj Chandra", pp. 183–184 in Leading Personalities in Statistical Sciences from the Seventeenth Century to the Present, (ed. N. L. Johnson and S. Kotz) 1997. New York: Wiley. Originally p

Tirukkannapuram Vijayaraghavan

Tirukkannapuram Vijayaraghavan (Tamil: திருக்கண்ணபுரம் விஜயராகவன்; 30 November 1902 – 20 April 1955) was an Indian mathematician from the Madras region. He worked with G. H. Hardy when he went to Oxford in mid-1920s on Pisot–Vijayaraghavan numbers. He was a fellow of Indian Academy of Sciences elected in the year 1934.

Vijayaraghavan was well versed in Sanskrit and Tamil. He was a close friend of André Weil. He served with him in Aligarh Muslim University. He later moved to the University of Dhaka in protest at Weil's firing from AMU.[1]

Vijayaraghavan proved a special case of Herschfeld's theorem on nested radicals:[2] For $a_{n}>0$

$\sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + \sqrt{a_4 + \cdots}}}}$

converges if and only if

$\overline{\lim} (\log a_n)/2^n < +\infty,$

where $\overline{\lim}$ denotes the limit superior.

D. R. Kaprekar

Dattathreya Ramchandra Kaprekar (1905–1986) was an Indian recreational mathematician who described several classes of natural numbers including the Kaprekar, Harshad and Self numbers and discovered the Kaprekar constant, named after him. Despite having no formal postgraduate training and working as a schoolteacher, he published extensively and became well known in recreational mathematics circles.

Biography

Kaprekar received his secondary school education in Thane and studied at Fergusson College in Pune. In 1927 he won the Wrangler R. P. Paranjpe Mathematical Prize for an original piece of work in mathematics.[2]

He attended the University of Mumbai, receiving his bachelor's degree in 1929. Having never received any formal postgraduate training, for his entire career (1930–1962) he was a schoolteacher at Nashik in Maharashtra, India. He published extensively, writing about such topics as recurring decimals, magic squares, and integers with special properties. He is also known as "Ganitanand" (गणितानंद)

Discoveries

Working largely alone, Kaprekar discovered a number of results in number theory and described various properties of numbers.[3] In addition to the Kaprekar constant and the Kaprekar numbers which were named after him, he also described self numbers or Devlali numbers, the Harshad numbers and Demlo numbers. He also constructed certain types of magic squares related to the Copernicus magic square.[4] Initially his ideas were not taken seriously by Indian mathematicians, and his results were published largely in low-level mathematics journals or privately published, but international fame arrived when Martin Gardner wrote about Kaprekar in his March 1975 column of Mathematical Games for Scientific American. Today his name is well-known and many other mathematicians have pursued the study of the properties he discovered.[1]

Kaprekar constant

In 1949, Kaprekar discovered an interesting property of the number 6174, which was subsequently named the Kaprekar constant.[5] He showed that 6174 is reached in the limit as one repeatedly subtracts the highest and lowest numbers that can be constructed from a set of four digits that are not all identical. Thus, starting with 1234, we have:

4321 − 1234 = 3087, then8730 − 0378 = 8352, and8532 − 2358 = 6174.

Repeating from this point onward leaves the same number (7641 − 1467 = 6174). In general, when the operation converges it does so in at most seven iterations.

A similar constant for 3 digits is 495.[6] However, in base 10 a single such constant only exists for numbers of 3 or 4 digits; for other digit lengths or bases other than 10, the Kaprekar's routine algorithm described above may in general terminate in multiple different constants or repeated cycles, depending on the starting value.[7]

Kaprekar number

Another class of numbers Kaprekar described are the Kaprekar numbers.[8] A Kaprekar number is a positive integer with the property that if it is squared, then its representation can be partitioned into two positive integer parts whose sum is equal to the original number (e.g. 45, since 452=2025, and 20+25=45, also 9, 55, 99 etc.) However, note the restriction that the two numbers are positive; for example, 100 is not a Kaprekar number even though 1002=10000, and 100+00 = 100. This operation, of taking the rightmost digits of a square, and adding it to the integer formed by the leftmost digits, is known as the Kaprekar operation.

Some examples of Kaprekar numbers in base 10, besides the numbers 9, 99, 999, …, are (sequence A006886 in the OEIS):

NumberSquareDecomposition703703² = 494209494+209 = 70327282728² = 7441984744+1984 = 272852925292² = 2800526428+005264 = 5292857143857143² = 734694122449734694+122449 = 857143

Devlali or Self number

In 1963, Kaprekar defined the property which has come to be known as self numbers,[9] which are integers that cannot be generated by taking some other number and adding its own digits to it. For example, 21 is not a self number, since it can be generated from 15: 15 + 1 + 5 = 21. But 20 is a self number, since it cannot be generated from any other integer. He also gave a test for verifying this property in any number. These are sometimes referred to as Devlali numbers (after the town where he lived); though this appears to have been his preferred designation,[9] the term self number is more widespread. Sometimes these are also designated Colombian numbers after a later designation.

Harshad number

Kaprekar also described the Harshad numbers which he named harshad, meaning "giving joy" (Sanskrit harsha, joy +da taddhita pratyaya, causative); these are defined by the property that they are divisible by the sum of their digits. Thus 12, which is divisible by 1 + 2 = 3, is a Harshad number. These were later also called Niven numbers after a 1977 lecture on these by the Canadian mathematician Ivan M. Niven. Numbers which are Harshad in all bases (only 1, 2, 4, and 6) are called all-Harshad numbers. Much work has been done on Harshad numbers, and their distribution, frequency, etc. are a matter of considerable interest in number theory today.

Demlo number

Kaprekar also studied the Demlo numbers,[10] named after a train station 30 miles from Bombay on the then G. I. P. Railway where he had the idea of studying them.[1] The best known of these are the Wonderful Demlo numbers 1, 121, 12321, …, which are the squares of the repunits 1, 11, 111, ….

Samarendra Nath Roy

Early life

Roy was the first of three children of Kali Nath Roy and Suniti Bala Roy.[1] His father, Kali Nath Roy was a freedom fighter and the Chief Editor of the newspaper The Tribune, then publishing from Lahore.[2] During the massacre of the Indians at the hands of the British in the infamous incident at Jallianwala Bagh in April 1919, The Tribune published a news report titled "Prayer at the Jama Masjid", on 6 April 1919. For this "offence" Kali Nath Roy was sentenced to rigorous imprisonment for two years along with a fine of one thousand rupees.[3][4][5][6]

Roy secured first division in the Matriculation Examination in 1923 from the Khulna District School.[6] He was the topper in the Intermediate Science (Higher Secondary) Examinations in 1925 from the Daulatpur Hindu Academy.[6] He obtained first class and was the topper in both the BSc Mathematics (Honours) from Presidency College of the University of Calcutta in 1928 and the MSc in Applied Mathematics (with the Theory of Relativity as the elective) from the University of Calcutta in 1931.[1][6]

Academic career

In 1931, when Roy joined the Department of Applied Mathematics at the University of Calcutta as a research associate, he used computing facilities at the newly established Indian Statistical Institute,[6] which was founded by Professor P. C. Mahalanobis. Roy along with several talented young scholars including J. M. Sengupta, H. C. Sinha, Raj Chandra Bose, K. R. Nair, K. Kishen and C. R. Rao, joined to form an active group of statisticians under Mahalanobis. Roy was one of the very early students of Mahalanobis, who initiated some of the early works in Statistics.[7] He was well known for his pioneering contribution to multivariate statistical analysis, mainly that of the Jacobians of complicated transformations for various exact distributions, rectangular coordinates and the Bartlett decomposition.[8] His dissertation included the Post master's work at the Indian Statistical Institute where he worked under Mahalanobis.[9]

It was Bose who first went to the United States as a visiting professor at Columbia University and then joined the University of North Carolina at Chapel Hill in 1947. Roy followed suit by later joining him at the University of North Carolina at Chapel Hill in the spring of 1950, after initially travelling to the United States to take up a Visiting Professorship of Statistics at Columbia University in New York in the spring of 1949. In between this Roy returned to India and became Head of the Department of Statistics at the University of Calcutta during the academic year 1949–50.[6] Roy joined Bose as full Professor of Statistics in the Statistics Department at the University of North Carolina at Chapel Hill. S. N. Roy had 15 doctorate students there from 1950 till 1963.[10] To commemorate his Birth Centenary an International Conference on "Multivariate Statistical Methods in the 21st Century: The Legacy of Prof. S.N. Roy" was held at Kolkata, India during 28–29 December 2006.[11] The Journal of Statistical Planning and Inference published a special Issue for celebrating the Centennial of Birth of S. N. Roy.[7]

Personal life

Roy was married to Bani Roy and had four children, Prabir, Subir, Tapon and Sunanda. He died while on holiday in Jasper, Canada. His eldest son, Prabir Roy was also a mathematician, who obtained his PhD from the University of North Carolina at Chapel Hill in 1962. His dissertation was on the "Separability of Metric Spaces".[12] He later went on to become full Professor at a young age at the Binghamton University.

S. N. Roy is presently being survived by his son Subir Roy who is an MD, Obstetrician & Gynecologist, Reproductive Endocrinologist at the LAC+USC Medical Center, LA,[13]and his daughter Sunanda R. McGarvey who works at the National Center of Public Health Informatics, CDC in Washington, DC.

Selected publications

Potthoff, R. F. and Roy, S. N. (1964), “A generalized multivariate analysis of variance model useful especially for growth curve problems”, Biometrika, vol. 51, pp. 313–326.
Roy, S. N. (1957), “Some Aspects of Multivariate Analysis”, New York: Wiley.
Roy, S. N. and Sarhan, A. E. (1956), “On inverting a class of patterned matrices”, Biometrika, 43, 227–231.

Selected Ph.D. dissertations under Roy's guidance

Olkin, Ingram (1951). "On distribution problems in multivariate analysis".
Pachares, James (1953). "On the distribution of quadratic forms".
Pillai, K.C. Sreedharan (1954). "On some distribution problems in multivariate analysis".
Mitra, Sujit K. (1956).[14] "Contributions to the statistical analysis of categorical data".
Gnanadesikan, Ramanathan (1957). "Contributions to multivariate analysis including univariate and multivariate variance components analysis and factor analysis".
Potthoff, Richard F. (1958). "Multi-dimensional incomplete block designs".
Diadmond, Earl L. (1958). "Asymptotic power and independence of certain classes of tests on categorical data".
Bargman, Rolf (1958). "A study of independence and dependence in multivariate normal analysis".
Cobb, Whitfield (1959). "Studies in univariate and multivariate variance components analysis connected with sampling from a finite population".
Bhapkar, Vasant P. (1959). "Contributions to the statistical analysis of experiments with one or more responses".
Sathe, Yashawande S. (1962). "Studies in certain types of nonparametric inference".
Das Gupta, Somesh (1963). "Some problems in classification".
Mudholkar, Govind S. (1963). "Some contributions to the theory of univariate and multivariate statistical analysis".

Sources:[7][9][15]

Recognition

Fellow of the Institute of Mathematical Statistics
Fellow of the International Statistical Institute in 1951
President of the Statistical Section of the Indian Science Congress in 1948
Fellow of The National Institute of Sciences (FNASc), India (now known as the Indian National Science Academy) in 1946

Damodar Dharmananda Kosambi

Damodar Dharmananda Kosambi (31 July 1907 – 29 June 1966) was an Indian mathematician, statistician, philologist, historian and polymath who contributed to genetics by introducing Kosambi map function.[1] He is well known for his work in numismaticsand for compiling critical editions of ancient Sanskrit texts. His father, Dharmananda Damodar Kosambi, had studied ancient Indian texts with a particular emphasis on Buddhism and its literature in the Pali language. Damodar Kosambi emulated him by developing a keen interest in his country's ancient history. Kosambi was also a Marxist historian specialising in ancient India who employed the historical materialist approach in his work.[2] He is particularly known for his classic work An Introduction to the Study of Indian History.

He is described as "the patriarch of the Marxist school of Indian historiography".[2] Kosambi was critical of the policies of then prime minister Jawaharlal Nehru, which, according to him, promoted capitalism in the guise of democratic socialism. He was an enthusiast of the Chinese revolution and its ideals, and, in addition, a leading activist in the World Peace Movement.

Early life

After a few years of schooling in India, in 1918, Damodar and his elder sister, Manik travelled to Cambridge, Massachusetts with their father, the eminent Buddhist and Palischolar, Dharmananda Damodar Kosambi. The latter was tasked by Professor Charles Rockwell Lanman of Harvard University to complete compiling a critical edition of Visuddhimagga, a book on Buddhist philosophy, which was originally started by Henry Clarke Warren. There, the young Damodar spent a year in a Grammar school and then was admitted to the Cambridge High and Latin School in 1920. He became a member of the Cambridge branch of American Boy Scouts.

It was in Cambridge that he befriended another prodigy of the time, Norbert Wiener, whose father Leo Wiener was the elder Kosambi's colleague at Harvard University.

Kosambi excelled in his final school examination and was one of the few candidates who was exempt on the basis of merit from necessarily passing an entrance examination essential at the time to gain admission to Harvard University. He enrolled in Harvard in 1924, but eventually postponed his studies, and returned to India. He stayed with his father who was now working in the Gujarat University, and was in the close circles of Mahatma Gandhi.

In January 1926, Kosambi returned to the US with his father, who once again studied at Harvard University for a year and half. Kosambi studied mathematics under George David Birkhoff, who wanted him to concentrate on mathematics, but the ambitious Kosambi instead took many diverse courses excelling in each of them. In 1929, Harvard awarded him the Bachelor of Arts degree with a summa cum laude. He was also granted membership to the esteemed Phi Beta Kappa Society, the oldest undergraduate honours organisation in the United States. He returned to India soon after.

Banaras and Aligarh

He obtained the post of professor at the Banaras Hindu University (BHU), teaching German alongside mathematics. He struggled to pursue his research on his own, and published his first research paper, "Precessions of an Elliptic Orbit" in the Indian Journal of Physics in 1930.

In 1931, Kosambi married Nalini from the wealthy Madgaonkar family. It was in this year that he was hired by mathematician André Weil, then Professor of Mathematics at Aligarh Muslim University, to the post of lecturership in mathematics at Aligarh.[3] His other colleagues at Aligarh included Vijayraghavan. During his two years stay in Aligarh, he produced eight research papers in the general area of Differential Geometry and Path Spaces. His fluency in several European languages allowed him to publish some of his early papers in French, Italian and German journals in their respective languages.

Fergusson College, Pune

Marxism cannot, even on the grounds of political expediency or party solidarity, be reduced to a rigid formalism like mathematics. Nor can it be treated as a standard technique such as work on an automatic lathe. The material, when it is present in human society, has endless variations; the observer is himself part of the observed population, with which he interacts strongly and reciprocally. This means that the successful application of the theory needs the development of analytical power, the ability to pick out the essential factors in a given situation. This cannot be learned from books alone. The one way to learn it is by constant contact with the major sections of the people. For an intellectual, this means at least a few months spent in manual labour, to earn his livelihood as a member of the working class; not as a superior being, nor as a reformist, nor as a sentimental "progressive" visitor to the slums. The experience gained from living with worker and peasant, as one of them, has then to be consistently refreshed and regularly evaluated in the light of one's reading. For those who are prepared to do this, these essays might provide some encouragement, and food for thought.

— From Exasperating Essays: Exercises in Dialectical Method(1957)

In 1933, he joined the Deccan Education Society's Fergusson College in Pune, where he taught mathematics for the next 12 years. In 1935, his eldest daughter, Maya was born, while in 1939, the youngest, Meera.

In Pune, while teaching mathematics and conducting research in the field, he started his interdisciplinary pursuit. In 1944 he published a small article of 4 pages titled 'The Estimation of Map Distance from Recombination Values' in Annals of Eugenics, in which he introduced what later came to be known as Kosambi map function.

One of the most important contributions of Kosambi to statistics is the widely known technique called proper orthogonal decomposition (POD). Although it was originally developed by Kosambi in 1943, it is now referred to as the Karhunen–Loève expansion. In the 1943 paper entitled 'Statistics in Function Space' presented in the Journal of the Indian Mathematical Society, Kosambi presented the Proper Orthogonal Decomposition some years before Karhunen (1945) and Loeve (1948). This tool has found application to such diverse fields as image processing, signal processing, data compression, oceanography, chemical engineering and fluid mechanics. Unfortunately this most important contribution of his is barely acknowledged in most papers that utilise the POD method. In recent years though, some authors have indeed referred to it as the Kosambi-Karhunen-Loeve decomposition.[4]

It was his studies in numismatics that initiated him into the field of historical research. He did extensive research in difficult science of numismatics. His evaluation of data was by modern statistical methods.[5] He made a thorough study of Sanskrit and ancient literature, and he started his classic work on the ancient poet Bhartṛhari. He published his critical editions of Bhartrihari's Śatakatraya and Subhashitas during 1945–1948.

It was during this period that he started his political activism, coming close to the radical streams in the ongoing Independence movement, especially the Communist Party of India. He became an outspoken Marxist and wrote some political articles.

Tata Institute of Fundamental Research

In 1945, Homi J. Bhabha invited Kosambi to join the Tata Institute of Fundamental Research (TIFR) as Professor of Mathematics, which he accepted. After independence, in 1948–49 he was sent to England and to the USA as a UNESCO Fellow to study the theoretical and technical aspects of computing machines. In London, he started his long-lasting friendship with Indologist and historian A.L. Basham. In the spring semester of 1949, he was a visiting professor of geometry in the Mathematics Department at the University of Chicago, where his colleague from his Harvard days, Marshall Harvey Stone, was the chair. In April–May 1949, he spent nearly two months at the Institute for Advanced Study in Princeton, New Jersey, discussing with such illustrious physicists and mathematicians as J. Robert Oppenheimer, Hermann Weyl, John von Neumann, Marston Morse, Oswald Veblen and Carl Ludwig Siegel amongst others.

After his return to India, in the Cold War circumstances, he was increasingly drawn into the World Peace Movement and served as a Member of the World Peace Council. He became a tireless crusader for peace, campaigning against the nuclearisation of the world. Kosambi's solution to India's energy needs was in sharp conflict with the ambitions of the Indian ruling class. He proposed alternative energy sources, like solar power. His activism in the peace movement took him to Beijing, Helsinki and Moscow. However, during this period he relentlessly pursued his diverse research interests, too. Most importantly, he worked on his Marxist rewriting of ancient Indian history, which culminated in his book, An Introduction to the Study of Indian History (1956).

He visited China many times during 1952–62 and was able to watch the Chinese revolution very closely, making him critical of the way modernisation and development were envisaged and pursued by the Indian ruling classes. All these contributed to straining his relationship with the Indian government and Bhabha, eventually leading to Kosambi's exit from the Tata Institute of Fundamental Research in 1962.

Post-TIFR days

His exit from the TIFR gave Kosambi the opportunity to concentrate on his research in ancient Indian history culminating in his book, The Culture and Civilisation of Ancient India, which was published in 1965 by Routledge, Kegan & Paul. The book was translated into German, French and Japanese and was widely acclaimed. He also utilised his time in archaeological studies, and contributed in the field of statistics and number theory. His article on numismatics was published in February 1965 in Scientific American.

Due to the efforts of his friends and colleagues, in June 1964, Kosambi was appointed as a Scientist Emeritus of the Council of Scientific and Industrial Research (CSIR) affiliated with the Maharashtra Vidnyanvardhini in Pune. He pursued many historical, scientific and archaeological projects (even writing stories for children). But most works he produced in this period could not be published during his lifetime. On 29 June 1966, he died in Pune. He was posthumously decorated with the Hari Om Ashram Award by the government of India's University Grant Commission in 1980.

His friend A.L. Basham, a well-known indologist, wrote in his obituary:

At first it seemed that he had only three interests, which filled his life to the exclusion of all others — ancient India, in all its aspects, mathematics and the preservation of peace. For the last, as well as for his two intellectual interests, he worked hard and with devotion, according to his deep convictions. Yet as one grew to know him better one realized that the range of his heart and mind was very wide...In the later years of his life, when his attention turned increasingly to anthropology as a means of reconstructing the past, it became more than ever clear that he had a very deep feeling for the lives of the simple people of Maharashtra.[6]

Kosambi's historiography

Cover of An Introduction to the Study of Indian History

Certain opponents of Marxism dismiss it as an outworn economic dogma based upon 19th century prejudices. Marxism never was a dogma. There is no reason why its formulation in the 19th century should make it obsolete and wrong, any more than the discoveries of Gauss, Faraday and Darwin, which have passed into the body of science... The defense generally given is that the Gita and the Upanishadsare Indian; that foreign ideas like Marxism are objectionable. This is generally argued in English the foreign language common to educated Indians; and by persons who live under a mode of production (the bourgeois system forcibly introduced by the foreigner into India.) The objection, therefore seems less to the foreign origin than to the ideas themselves which might endanger class privilege. Marxism is said to be based upon violence, upon the class-war in which the very best people do not believe nowadays. They might as well proclaim that meteorology encourages storms by predicting them. No Marxist work contains incitement to war and specious arguments for senseless killing remotely comparable to those in the divine Gita.

— From Exasperating Essays: Exercises in Dialectical Method (1957)

As a historian, Kosambi revolutionised Indian historiography with his realistic and scientific approach. He understood history in terms of the dynamics of socio-economic formations rather than just a chronological narration of "episodes" or the feats of a few great men – kings, warriors or saints. In the very first paragraph of his classic work, An Introduction to the Study of Indian History, he gives an insight into his methodology as a prelude to his life work on ancient Indian history:

"The light-hearted sneer “India has had some episodes, but no history“ is used to justify lack of study, grasp, intelligence on the part of foreign writers about India’s past. The considerations that follow will prove that it is precisely the episodes — lists of dynasties and kings, tales of war and battle spiced with anecdote, which fill school texts — that are missing from Indian records. Here, for the first time, we have to reconstruct a history without episodes, which means that it cannot be the same type of history as in the European tradition."[7]

According to A. L. Basham, "An Introduction to the Study of Indian History is in many respects an epoch making work, containing brilliantly original ideas on almost every page; if it contains errors and misrepresentations, if now and then its author attempts to force his data into a rather doctrinaire pattern, this does not appreciably lessen the significance of this very exciting book, which has stimulated the thought of thousands of students throughout the world."[6]

Professor Sumit Sarkar says: "Indian Historiography, starting with D.D. Kosambi in the 1950s, is acknowledged the world over – wherever South Asian history is taught or studied – as quite on a par with or even superior to all that is produced abroad. And that is why Irfan Habib or Romila Thapar or R.S. Sharma are figures respected even in the most diehard anti-Communist American universities. They cannot be ignored if you are studying South Asian history."[8]

In his obituary of Kosambi published in Nature, J. D. Bernal had summed up Kosambi's talent as follows: "Kosambi introduced a new method into historical scholarship, essentially by application of modern mathematics. By statistical study of the weights of the coins, Kosambi was able to establish the amount of time that had elapsed while they were in circulation and so set them in order to give some idea of their respective ages."

Legacy

Kosambi is an inspiration to many across the world, especially to Sanskrit philologists[9] and Marxist scholars. He is one of the few along with James Mill and Vincent Smith, who has so deeply influenced the writing of Indian history.[10] The Government of Goa has instituted the annual D.D. Kosambi Festival of Ideas since February 2008 to commemorate his birth centenary.[11]

Historian Irfan Habib said, "D. D. Kosambi and R.S. Sharma, together with Daniel Thorner, brought peasants into the study of Indian history for the first time."[12]

Kosambi was an atheist.[13]

India Post issued a commemorative postage stamp on 31 July 2008 to honour Kosambi.[14][15]

Books by D.D. Kosambi

Works on history and society

1956 An Introduction to the Study of Indian History (Popular Book Depot, Bombay)
1957 Exasperating Essays: Exercise in the Dialectical Method (People's Book House, Poona)
1962 Myth and Reality: Studies in the Formation of Indian Culture (Popular Prakashail, Bombay)
1965 The Culture and Civilisation of Ancient India in Historical Outline (Routledge & Kegan Paul, London)
1981 Indian Numismatics (Orient Blackswan, New Delhi)
2002 D.D. Kosambi: Combined Methods in Indology and Other Writings – Compiled, edited and introduced by Brajadulal Chattopadhyaya (Oxford University Press, New Delhi). Pdf on archive.org
2009 The Oxford India Kosambi – Compiled, edited and introduced by Brajadulal Chattopadhyaya (Oxford University Press, New Delhi)
2014 Unsettling The Past, edited by Meera Kosambi (Permanent Black, Ranikhet)
2016 Adventures into the Unknown: Essays, edited by Ram Ramaswamy (Three Essays Collective, New Delhi)

Edited works

1945 The Satakatrayam of Bhartrhari with the Comm. of Ramarsi, edited in collaboration with Pt. K. V. Krishnamoorthi Sharma (Anandasrama Sanskrit Series, No.127, Poona)
1946 The Southern Archetype of Epigrams Ascribed to Bhartrhari (Bharatiya Vidya Series 9, Bombay) (First critical edition of a Bhartrhari recension.)
1948 The Epigrams Attributed to Bhartrhari (Singhi Jain Series 23, Bombay) (Comprehensive edition of the poet's work remarkable for rigorous standards of text criticism.)
1952 The Cintamani-saranika of Dasabala; Supplement to Journal of Oriental Research, xix, pt, II (Madras) (A Sanskrit astronomical work which shows that King Bhoja of Dhara died in 1055–56.)
1957 The Subhasitaratnakosa of Vidyakara, edited in collaboration with V.V. Gokhale (Harvard Oriental Series 42)

Mathematical and scientific publications

In addition to the papers listed below, Kosambi wrote two books in mathematics, the manuscripts of which have not been traced. The first was a book on path geometry that was submitted to Marston Morse in the mid-1940s and the second was on prime numbers, submitted shortly before his death. Unfortunately, neither book was published. The list of articles below is complete but does not include his essays on science and scientists, some of which have appeared in the collection Science, Society, and Peace(People's Publishing House, 1995). Four articles (between 1962 and 1965) are written under the pseudonym S. Ducray.

1930 Precessions of an elliptical orbit, Indian Journal of Physics, 5, 359–364
1931 On a generalization of the second theorem of Bourbaki, Bulletin of the Academy of Sciences, U. P., 1, 145–147
1932 Modern differential geometries, Indian Journal of Physics, 7, 159–164
1932 On differential equations with the group property, Journal of the Indian Mathematical Society, 19, 215–219
1932 Geometrie differentielle et calcul des variations, Rendiconti della Reale Accademia Nazionale dei Lincei, 16, 410–415 (in French)
1932 On the existence of a metric and the inverse variational problem, Bulletin of the Academy of Sciences, U. P., 2, 17–28
1932 Affin-geometrische Grundlagen der Einheitlichen Feld–theorie, Sitzungsberichten der Preussische Akademie der Wissenschaften, Physikalisch-mathematische klasse, 28, 342–345 (in German)
1933 Parallelism and path-spaces, Mathematische Zeitschrift, 37, 608–618
1933 Observations sur le memoire precedent, Mathematische Zeitschrift, 37, 619–622 (in French)
1933 The problem of differential invariants, Journal of the Indian Mathematical Society, 20, 185–188
1933 The classification of integers, Journal of the University of Bombay, 2, 18–20
1934 Collineations in path-space, Journal of the Indian Mathematical Society, 1, 68–72
1934 Continuous groups and two theorems of Euler, The Mathematics Student, 2, 94–100
1934 The maximum modulus theorem, Journal of the University of Bombay, 3, 11–12
1935 Homogeneous metrics, Proceedings of the Indian Academy of Sciences, 1, 952–954
1935 An affine calculus of variations, Proceedings of the Indian Academy of Sciences, 2, 333–335
1935 Systems of differential equations of the second order, Quarterly Journal of Mathematics (Oxford), 6, 1–12
1936 Differential geometry of the Laplace equation, Journal of the Indian Mathematical Society, 2, 141–143
1936 Path-spaces of higher order, Quarterly Journal of Mathematics (Oxford), 7, 97–104
1936 Path-geometry and cosmogony, Quarterly Journal of Mathematics (Oxford), 7, 290–293
1938 Les metriques homogenes dans les espaces cosmogoniques, Comptes rendus de l’Acad ́emie des Sciences, 206, 1086–1088 (in French)
1938 Les espaces des paths generalises qu’on peut associer avec un espace de Finsler, Comptes rendus de l’Acad ́emie des Sciences, 206, 1538–1541 (in French)
1939 The tensor analysis of partial differential equations, Journal of the Indian Mathematical Society, 3, 249–253 (1939); Japanese version of this article in Tensor, 2, 36–39
1940 A statistical study of the weights of the old Indian punch-marked coins, Current Science, 9, 312–314
1940 On the weights of old Indian punch-marked coins, Current Science, 9, 410–411
1940 Path-equations admitting the Lorentz group, Journal of the London Mathematical Society, 15, 86–91
1940 The concept of isotropy in generalized path-spaces, Journal of the Indian Mathematical Society, 4, 80–88
1940 A note on frequency distribution in series, The Mathematics Student, 8, 151–155
1941 A bivariate extension of Fisher’s Z–test, Current Science, 10, 191–192
1941 Correlation and time series, Current Science, 10, 372–374
1941 Path-equations admitting the Lorentz group–II, Journal of the Indian Mathematical Society, 5, 62–72
1941 On the origin and development of silver coinage in India, Current Science, 10, 395–400
1942 On the zeros and closure of orthogonal functions, Journal of the Indian Mathematical Society, 6, 16–24
1942 The effect of circulation upon the weight of metallic currency, Current Science, 11, 227–231
1942 A test of significance for multiple observations, Current Science, 11, 271–274
1942 On valid tests of linguistic hypotheses, New Indian Antiquary, 5, 21–24
1943 Statistics in function space, Journal of the Indian Mathematical Society, 7, 76–88
1944 The estimation of map distance from recombination values, Annals of Eugenics, 12, 172–175
1944 Direct derivation of Balmer spectra, Current Science, 13, 71–72
1944 The geometric method in mathematical statistics, American Mathematical Monthly, 51, 382–389
1945 Parallelism in the tensor analysis of partial differential equations, Bulletin of the American Mathematical Society, 51, 293–296
1946 The law of large numbers, The Mathematics Student, 14, 14–19
1946 Sur la differentiation covariante, Comptes rendus de l’Acad ́emie des Sciences, 222, 211–213 (in French)
1947 An extension of the least–squares method for statistical estimation, Annals of Eugenics, 18, 257–261
1947 Possible Applications of the Functional Calculus, Proceedings of the 34th Indian Science Congress. Part II: Presidential Addresses, 1–13
1947 Les invariants differentiels d’un tenseur covariant a deux indices, Comptes rendus de l’Acad ́emie des Sciences, 225, 790–92
1948 Systems of partial differential equations of the second order, Quarterly Journal of Mathematics (Oxford), 19, 204–219
1949 Characteristic properties of series distributions, Proceedings of the National Institute of Science of India, 15, 109–113
1949 Lie rings in path-space, Proceedings of the National Academy of Sciences (USA), 35, 389–394
1949 The differential invariants of a two-index tensor, Bulletin of the American Mathematical Society, 55, 90–94
1951 Series expansions of continuous groups, Quarterly Journal of Mathematics (Oxford, Series 2), 2, 244–257
1951 Seasonal variations in the Indian birth–rate, Annals of Eugenics, 16, 165–192 (with S. Raghavachari)
1952 Path-spaces admitting collineations, Quarterly Journal of Mathematics (Oxford, Series 2), 3, 1–11
1952 Path-geometry and continuous groups, Quarterly Journal of Mathematics (Oxford, Series 2), 3, 307–320
1954 Seasonal variations in the Indian death–rate, Annals of Human Genetics, 19, 100–119 (with S. Raghavachari)
1954 The metric in path-space, Tensor (New Series), 3, 67–74
1957 The method of least–squares, Advancement in Mathematics, 3, 485–491 (in Chinese)
1958 Classical Tauberian theorems, Journal of the Indian Society of Agricultural Statistics, 10, 141–149
1958 The efficiency of randomization by card–shuffling, Journal of the Royal Statistics Society, 121, 223–233 (with U. V. R. Rao)
1959 The method of least–squares, Journal of the Indian Society of Agricultural Statistics, 11, 49–57
1959 An application of stochastic convergence, Journal of the Indian Society of Agricultural Statistics, 11, 58–72
1962 A note on prime numbers, Journal of the University of Bombay, 31, 1–4 (as S. Ducray)
1963 The sampling distribution of primes, Proceedings of the National Academy of Sciences (USA), 49, 20–23
1963 Normal Sequences, Journal of the University of Bombay, 32, 49–53 (as S. Ducray)
1964 Statistical methods in number theory, Journal of the Indian Society of Agricultural Statistics, 16, 126–135
1964 Probability and prime numbers, Proceedings of the Indian Academy of Sciences, 60, 159–164 (as S. Ducray)
1965 The sequence of primes, Proceedings of the Indian Academy of Sciences, 62, 145–149 (as S. Ducray)
1966 Numismatics as a Science, Scientific American, February 1966, pages 102–111
2016 Selected Works in Mathematics and Statistics, ed. Ramakrishna Ramaswamy, Springer. (Posthumous publication)

Sarvadaman Chowla

Sarvadaman D. S. Chowla (22 October 1907 – 10 December 1995) was a British-born Indian American mathematician, specializing in number theory.

Early life

He was born in London, since his father, Gopal Chowla, a professor of mathematics in Lahore, was then studying in Cambridge. His family returned to India, where he received his masters degree in 1928 from the Government College in Lahore. In 1931 he received his doctorate from the University of Cambridge, where he studied under J. E. Littlewood. [1]:594

Career and awards

Chowla then returned to India, where he taught at several universities, becoming head of mathematics at Government College in 1936.[1]:594 During the difficulties arising from the partition of India in 1947, he left for the United States.[2] There he visited the Institute for Advanced Study until the fall of 1949, then taught at the University of Kansas in Lawrence until moving to the University of Colorado in 1952.[1]:594 He moved to Penn State in 1963 as a research professor, where he remained until his retirement in 1976.[1]:594 He was a member of the Indian National Science Academy.[1]:595

Among his contributions are a number of results which bear his name. These include the Bruck–Ryser–Chowla theorem, the Ankeny–Artin–Chowla congruence, the Chowla–Mordell theorem, and the Chowla–Selberg formula, and the Mian–Chowla sequence.

Works

Chowla, Sarvadaman (2000). James G. Huard; Kenneth S. Williams, eds. The Collected Papers of Sarvadaman Chowla. Montréal: Centre de Recherches mathématiques, Université de Montréal. OCLC 43730416.
Chowla, S. (1965). Riemann Hypothesis and Hilbert's Tenth Problem. New York: Routledge. ISBN 978-0-677-00140-1. OCLC 15428640.

Lakkoju Sanjeevaraya Sharma

Lakkoju Sanjeevaraya Sharma (Telugu: లక్కోజు సంజీవరాయ శర్మ) (27 November 1907 – 1998) was an Indian mathematicianfrom Andhra Pradesh.[1] He was born blind[2] and gave many Mathematical Avadhanams (Ganitavadhanams) and surprised the elite and educated people.

He was born on 27 November 1907 at Kalluru village of Proddatur mandal in Cuddapah district. His parents are Lakkoju Pedda Pullaiah and Nagamma. He was born blind and had no formal education. He was married to Adilakshmamma at the age of 19 years.

He gave his first performance at Andhra Mahasabha at Nandyala in 1928 chaired by Sarvepalli Radhakrishnan. He was invited to New Delhi by Jawaharlal Nehru and performed before the President Rajendra Prasad and others dignitories. He traveled widely and gave about 7,000 performances in Andhra Pradesh, Tamil Nadu, Karnataka, Maharashtra and Delhi. One of the memorable performances was given on December 7, 1966 at Sri Krishna Devaraya Andhra Bhasha Nilayam, Hyderabad.[3] He received many gold medals, honors and felicitations. He has prepared Indian Calendar for 4000 years.

He was invited to United States in 1993 by the Telugu community. He could not attend due to a problem with the visa. Sri Venkateswara University honored him with a doctorate in 1996.

He spent his terminal phase of life at Srikalahasti. He played Violin every evening at Kalahasteswara temple and used to get some honourorium from the Devasthanam authorities.

Subrahmanyan Chandrasekhar

Subrahmanyan Chandrasekhar FRS PV [1] (/ˌtʃʌndrəˈʃeɪkər/ ( listen); 19 October 1910 – 21 August 1995),[3] was an Indian American astrophysicist who spent his professional life in the United States.[4] He was awarded the 1983 Nobel Prize for Physics with William A. Fowler for "...theoretical studies of the physical processes of importance to the structure and evolution of the stars". His mathematical treatment of stellar evolution yielded many of the best current theoretical models of the later evolutionary stages of massive stars and black holes.[5][6] The Chandrasekhar limit is named after him.

Chandrasekhar worked on a wide variety of physical problems in his lifetime, contributing to the contemporary understanding of stellar structure, white dwarfs, stellar dynamics, stochastic process, radiative transfer, the quantum theory of the hydrogen anion, hydrodynamic and hydromagnetic stability, turbulence, equilibrium and the stability of ellipsoidal figures of equilibrium, general relativity, mathematical theory of black holes and theory of colliding gravitational waves.[7] At the University of Cambridge, he developed a theoretical model explaining the structure of white dwarf stars that took into account the relativistic variation of mass with the velocities of electrons that comprise their degenerate matter. He showed that the mass of a white dwarf could not exceed 1.44 times that of the Sun – the Chandrasekhar limit. Chandrasekhar revised the models of stellar dynamics first outlined by Jan Oort and others by considering the effects of fluctuating gravitational fields within the Milky Way on stars rotating about the galactic centre. His solution to this complex dynamical problem involved a set of twenty partial differential equations, describing a new quantity he termed ‘dynamical friction’, which has the dual effects of decelerating the star and helping to stabilize clusters of stars. Chandrasekhar extended this analysis to the interstellar medium, showing that clouds of galactic gas and dust are distributed very unevenly.

Chandrasekhar studied at Presidency College, Madras (now Chennai) and the University of Cambridge. A long-time professor at the University of Chicago, he did some of his studies at the Yerkes Observatory, and served as editor of The Astrophysical Journal from 1952 to 1971. He was on the faculty at Chicago from 1937 until his death in 1995 at the age of 84, and was the Morton D. HullDistinguished Service Professor of Theoretical Astrophysics.[8]

Chandrasekhar married Lalitha Doraiswamy in September 1936. He had met her as a fellow student at Presidency College, Madras. Chandrasekhar was the nephew of C. V. Raman, who was awarded the Nobel Prize for Physics in 1930. He became a naturalized citizen of the U.S. in 1953. Others considered him as warm, positive, generous, unassuming, meticulous, and open to debate, as well as private, intimidating, impatient, and stubborn regarding non-scientific matters,[4] and unforgiving to those who ridiculed his work.[

Early life and education

Chandrasekhar was born on 19 October 1910 in Lahore, Punjab, British India (now Pakistan) in a Brahmin family, to Sitalakshmi (Divan Bahadur) Balakrishnan (1891–1931) and Chandrasekhara Subrahmanya Ayyar (1885–1960)[10] who was stationed in Lahore as Deputy Auditor General of the Northwestern Railways at the time of Chandrasekhar's birth. He had two elder sisters, Rajalakshmi and Balaparvathi, three younger brothers, Vishwanathan, Balakrishnan, and Ramanathan and four younger sisters, Sarada, Vidya, Savitri, and Sundari. His paternal uncle was the Indian physicist and Nobel laureate C. V. Raman. His mother was devoted to intellectual pursuits, had translated Henrik Ibsen's A Doll's House into Tamil and is credited with arousing Chandra's intellectual curiosity at an early age[11]. The family moved from Lahore to Allahabad in 1916, and finally settled in Madras in 1918.

Chandrasekhar was tutored at home until the age of 12[11]. In middle school his father would teach him Mathematics and Physics and his mother would teach him Tamil. He later attended the Hindu High School, Triplicane, Madras during the years 1922–25. Subsequently, he studied at Presidency College, Madras from 1925 to 1930, writing his first paper, "The Compton Scattering and the New Statistics", in 1929 after being inspired by a lecture by Arnold Sommerfeld. He obtained his bachelor's degree, B.Sc. (Hon.), in physics, in June 1930. In July 1930, Chandrasekhar was awarded a Government of India scholarship to pursue graduate studies at the University of Cambridge, where he was admitted to Trinity College, Cambridge, secured by R. H. Fowler with whom he communicated his first paper. During his travels to England, Chandrasekhar spent his time working out the statistical mechanics of the degenerate electron gas in white dwarf stars, providing relativistic corrections to Fowler's previous work (see Legacy below).

In his first year at Cambridge, as a research student of Fowler, Chandrasekhar spent his time calculating mean opacities and applying his results to the construction of an improved model for the limiting mass of the degenerate star. At the meetings of the Royal Astronomical Society, he met E. A. Milne. At the invitation of Max Born he spent the summer of 1931, his second year of post-graduate studies, at Born’s institute at Göttingen, working on opacities, atomic absorption coefficients, and model stellar photospheres. On the advice of P. A. M. Dirac, he spent his final year of graduate studies at the Institute for Theoretical Physics in Copenhagen, where he met Niels Bohr.

After receiving a bronze medal for his work on degenerate stars, in the summer of 1933, Chandrasekhar was awarded his PhD degree at Cambridge with a thesis among his four papers on rotating self-gravitating polytropes, and the following October, he was elected to a Prize Fellowship at Trinity College for the period 1933–1937.

During this time, Chandrasekhar made acquaintance with British physicist Sir Arthur Eddington. In an infamous encounter at the Royal Astronomical Society in London in 1935, Eddington publicly ridiculed the concept of the Chandrasekhar limit [11]. Although Eddington would later be proved wrong by computers and the first positive identification of a black hole in 1972, this encounter caused Chandrasekhar to contemplate employment outside the UK. Later in life, on multiple occasions, Chandrasekhar expressed the view that Eddington's behavior was in part racially motivated.[12]

Career and research

Early career

In January 1937, Chandrasekhar was recruited to the University of Chicago faculty as assistant professor by Otto Struve and President Robert Maynard Hutchins. He was to remain at the university for his entire career, becoming Morton D. Hull Distinguished Service Professor of Theoretical Astrophysics in 1952 and attaining emeritus status in 1985. In 1953, he and his wife took American citizenship.[13] Famously, Chandrasekhar declined many offers from other universities, including one to succeed Henry Norris Russell, the preeminent American astronomer, as director of the Princeton University Observatory.

Chandrasekhar did some work at Yerkes Observatory in Williams Bay, Wisconsin, which was run by the University of Chicago. After the Laboratory for Astrophysics and Space Research (LASR) was built by NASA in 1966 at the University, Chandrasekhar occupied one of the four corner offices on the second floor. (The other corners housed John A. Simpson, Peter Meyer, and Eugene N. Parker.) Chandrasekhar lived at 4800 Lake Shore Drive after the high-rise apartment complex was built in the late 1960s, and later at 5550 Dorchester Building.

World War II

During World War II, Chandrasekhar worked at the Ballistic Research Laboratory at the Aberdeen Proving Ground in Maryland. While there, he worked on problems of ballistics, resulting in reports such as 1943's On the decay of plane shock waves and The normal reflection of a blast wave.[7] Chandrasekhar's expertise in hydrodynamics led Robert Oppenheimer to invite him to join the Manhattan Project at Los Alamos, but delays in the processing of his security clearance prevented him from contributing to the project. It has been rumoured that he visited the Calutron project, where he suggested that young women be employed to operate the calutrons producing enriched radioactive materials for the atomic weapons.

Philosophy of systematization

He wrote that his scientific research was motivated by his desire to participate in the progress of different subjects in science to the best of his ability, and that the prime motive underlying his work was systematization. "What a scientist tries to do essentially is to select a certain domain, a certain aspect, or a certain detail, and see if that takes its appropriate place in a general scheme which has form and coherence; and, if not, to seek further information which would help him to do that." [14]

Chandrasekhar developed a unique style of mastering several fields of physics and astrophysics; consequently, his working life can be divided into distinct periods. He would exhaustively study a specific area, publish several papers in it and then write a book summarizing the major concepts in the field. He would then move on to another field for the next decade and repeat the pattern. Thus he studied stellar structure, including the theory of white dwarfs, during the years 1929 to 1939, and subsequently focused on stellar dynamics, theory of Brownian motion from 1939 to 1943. Next, he concentrated on the theory of radiative transfer and the quantum theory of the negative ion of hydrogen from 1943 to 1950. This was followed by sustained work on hydrodynamic and hydromagnetic stability from 1950 to 1961. In the 1960s, he studied the equilibrium and the stability of ellipsoidal figures of equilibrium, and also general relativity. During the period, 1971 to 1983 he studied the mathematical theory of black holes, and, finally, during the late 80s, he worked on the theory of colliding gravitational waves.[7]

Work with students

Chandra worked closely with his students and expressed pride in the fact that over a 50-year period (from roughly 1930 to 1980), the average age of his co-author collaborators had remained the same, at around 30. He insisted that students address him as "Chandrasekhar" until they received their Ph.D. degree, after which time they (as other colleagues) were encouraged to address him as "Chandra". When Chandrasekhar was working at the Yerkes Observatory in 1940s, he would drive 150 miles to and fro every weekend to teach a course at University of Chicago. Two of the students who took the course, Tsung-Dao Lee and Chen-Ning Yang, won the Nobel prize before he could get one for himself. Regarding classroom interactions during his lectures, noted astrophysicist Carl Sagan stated from firsthand experience that “frivolous questions” from unprepared students were “dealt with in the manner of a summary execution”, while questions of merit “were given serious attention and response”.[4]

Other activities

From 1952 to 1971 Chandrasekhar was editor of The Astrophysical Journal.[15] During the years 1990 to 1995, Chandrasekhar worked on a project devoted to explaining the detailed geometric arguments in Sir Isaac Newton's Philosophiae Naturalis Principia Mathematica using the language and methods of ordinary calculus. The effort resulted in the book Newton's Principia for the Common Reader, published in 1995. Chandrasekhar was an honorary member of the International Academy of Science.[citation needed]

Personal life

Chandrasekhar died of a sudden heart attack at the University of Chicago Hospital in 1995, having survived a prior heart attack in 1975.[4] He was survived by his wife, Lalitha Chandrasekhar, who died on 2 September 2013 at the age of 102.[16] In the Biographical Memoirs of the Fellows of the Royal Society of London, R. J. Tayler wrote: "Chandrasekhar was a classical applied mathematician whose research was primarily applied in astronomy and whose like will probably never be seen again."[1] He was a serious student of literature and western classical music.[9]

Religious view

Once when involved in a discussion about the Gita, Chandrashekhar said, "I should like to preface my remarks with a personal statement in order that my later remarks will not be misunderstood. I consider myself an atheist."[17] This was also confirmed many times in his other talks.[18] In an interview with Kevin Krisciunas at the University of Chicago, on 6 October 1987, Chandrasekhar commented: "Of course, he (Otto Struve) knew I was an atheist, and he never brought up the subject with me".[19]

Awards, honours and legacy

Nobel prize

Chandrasekhar was awarded the Nobel Prize in Physics in 1983 for his studies on the physical processes important to the structure and evolution of stars. Chandrasekhar accepted this honor, but was upset the citation mentioned only his earliest work, seeing it as a denigration of a lifetime's achievement. He shared it with William A. Fowler.

Other awards

An exhibition on life and works of Subrahmanyan Chandrasekhar was held at Science City, Kolkata, on January, 2011.

Elected a Fellow of the Royal Society (FRS) in 1944 [1]
Henry Norris Russell Lectureship (1949)[20]
Bruce Medal (1952)[21]
Gold Medal of the Royal Astronomical Society (1953)[22]
Rumford Prize of the American Academy of Arts and Sciences (1957)[23]
National Medal of Science, USA (1966)[24]
Padma Vibhushan (1968)
Henry Draper Medal of the National Academy of Sciences (1971)[25]
Copley Medal of the Royal Society (1984)
Honorary Fellow of the International Academy of Science (1988)[citation needed]
Gordon J. Laing Award (1989)
Jansky Lectureship before the National Radio Astronomy Observatory
Humboldt Prize[when?]

Legacy

Chandrasekhar's most notable work was the astrophysical Chandrasekhar limit. The limit describes the maximum mass of a white dwarf star, ~1.44 solar masses, or equivalently, the minimum mass which must be exceeded for a star to ultimately collapse into a neutron star or black hole (following a supernova). The limit was first calculated by Chandrasekhar in 1930 during his maiden voyage from India to Cambridge, England for his graduate studies. In 1979, NASA named the third of its four "Great Observatories" after Chandrasekhar. This followed a naming contest which attracted 6,000 entries from fifty states and sixty-one countries. The Chandra X-ray Observatory was launched and deployed by Space Shuttle Columbia on 23 July 1999. The Chandrasekhar number, an important dimensionless number of magnetohydrodynamics, is named after him. The asteroid 1958 Chandra is also named after Chandrasekhar.

Chandra Astrophysics Institute (CAI) is a program offered for high school students who are interested in astrophysics mentored by MIT scientists[26] sponsored by Chandra X-ray Observatory.[27] American astronomer Carl Sagan, who studied Mathematics under Chandrasekhar, at the University of Chicago, praised him in the book The Demon-Haunted World: "I discovered what true mathematical elegance is from Subrahmanyan Chandrasekhar." Chandrasekhar guided 50 students to their PhDs.[citation needed].

After his death, his widow Lalitha Chandrasekhar made a gift of his Nobel Prize money to the University of Chicago towards the establishment of the Subrahmanyan Chandrasekhar Memorial Fellowship. First awarded in the year 2000, each year, this fellowship is given to an outstanding applicant to graduate school in the Ph.D. programs of the Department of Physics or the Department of Astronomy and Astrophysics.[28]

On 19 October 2017, Google showed a Google Doodle in 28 countries honouring Chandrasekhar’s 107th birthday, and the Chandrasekhar limit.[29][30]

Publications

Books

Chandrasekhar, S. (1958) [1939]. An Introduction to the Study of Stellar Structure. New York: Dover. ISBN 0-486-60413-6.
Chandrasekhar, S. (2005) [1942]. Principles of Stellar Dynamics. New York: Dover. ISBN 0-486-44273-X.
Chandrasekhar, S. (1947). Heywood, Robert B., ed. The Works of the Mind:The Scientist. Chicago: University of Chicago Press. pp. 159–179. OCLC 752682744.
Chandrasekhar, S. (1960) [1950]. Radiative Transfer. New York: Dover. ISBN 0-486-60590-6.
Chandrasekhar, S. (1975) [1960]. Plasma Physics. Chicago: The University of Chicago Press. ISBN 0-226-10084-7.
Chandrasekhar, S. (1981) [1961]. Hydrodynamic and Hydromagnetic Stability. New York: Dover. ISBN 0-486-64071-X.
Chandrasekhar, S. (1987) [1969]. Ellipsoidal Figures of Equilibrium. New York: Dover. ISBN 0-486-65258-0.
Chandrasekhar, S. (1998) [1983]. The Mathematical Theory of Black Holes. New York: Oxford University Press. ISBN 0-19-850370-9.
Chandrasekhar, S. (1983) [1983]. Eddington: The Most Distinguished Astrophysicist of His Time. Cambridge University Press. ISBN 9780521257466.
Chandrasekhar, S. (1990) [1987]. Truth and Beauty. Aesthetics and Motivations in Science. Chicago: The University of Chicago Press. ISBN 0-226-10087-1.
Chandrasekhar, S. (1995). Newton's Principia for the Common Reader. Oxford: Clarendon Press. ISBN 0-19-851744-0.

Notes

Chandrasekhar, S. (1943). Stochastic Problems in Physics and Astronomy. Reviews of modern physics.
Chandrasekhar, S. (1993). Classical general relativity. Royal Society.
Chandrasekhar, S. (1979). The Role of General Relativity: Retrospect and Prospect. Proc. IAU Meeting.[31]
Spiegel, E.A. (2011) [1954]. The Theory of Turbulence : Subrahmanyan Chandrasekhar's 1954 Lectures. Netherlands: Springer. ISBN 978-94-007-0117-5.
Chandrasekhar, S. (1943). New methods in stellar dynamics. New York Academy of Sciences.
Chandrasekhar, S. (1954). The illumination and polarization of the sunlit sky on Rayleigh scattering. American Philosophical Society.
Chandrasekhar, S. (1983). On Stars, their evolution and their stability, Noble lecture. Stockholm: Noble Foundation.
Chandrasekhar, S. (1981). New horizons of human knowledge: a series of public talks given at Unesco. Unesco Press.
Chandrasekhar, S. (1975). Shakespeare, Newton, and Beethoven: Or, Patterns of Creativity. University of Chicago.
Chandrasekhar, S. (1973). P.A.M. Dirac on his seventieth birthday. Contemporary Physics.[32]
Chandrasekhar, S. (1995). Reminiscences and discoveries on Ramanujan's bust. Royal Society. ASIN B001B12NJ8.
Chandrasekhar, S. (1990). How one may explore the physical content of the general theory of relativity. American Mathematical Society. ASIN B001B10QTM.

Journals

Chandrasekhar had published around 380 papers[33][34] in his lifetime. He wrote his first paper in 1928 when he was still an undergraduate student about Compton effect [35] and last paper which was accepted for publication just two months before his death was in 1995 which was about non-radial oscillation of star.[36] The University of Chicago Press published selected papers of Chandrasekhar in seven volumes.

Chandrasekhar, S. (1989). Selected Papers, Vol 1, Stellar structure and stellar atmospheres. Chicago: University of Chicago Press. ISBN 9780226100890.
Chandrasekhar, S. (1989). Selected Papers, Vol 2, Radiative transfer and negative ion of hydrogen. Chicago: University of Chicago Press. ISBN 9780226100920.
Chandrasekhar, S. (1989). Selected Papers, Vol 3, Stochastic, statistical and hydromagnetic problems in Physics and Astronomy. Chicago: University of Chicago Press. ISBN 9780226100944.
Chandrasekhar, S. (1989). Selected Papers, Vol 4, Plasma Physics, Hydrodynamic and Hydromagnetic stability, and applications of the Tensor-Virial theorem. Chicago: University of Chicago Press. ISBN 9780226100975.
Chandrasekhar, S. (1990). Selected Papers, Vol 5, Relativistic Astrophysics. Chicago: University of Chicago Press. ISBN 9780226100982.
Chandrasekhar, S. (1991). Selected Papers, Vol 6, The Mathematical Theory of Black Holes and of Colliding Plane Waves. Chicago: University of Chicago Press. ISBN 9780226101019.
Chandrasekhar, S. (1997). Selected Papers, Vol 7, The non-radial oscillations of star in General Relativity and other writings. Chicago: University of Chicago Press. ISBN 9780226101040.

Books about Chandrasekhar

Miller, Arthur I. (2005). Empire of the Stars: Friendship, Obsession, and Betrayal in the Quest for Black Holes. Boston: Houghton Mifflin. ISBN 0-618-34151-X.
Srinivasan, G., ed. (1997). From White Dwarfs to Black Holes: The Legacy of S. Chandrasekhar. Chicago: The University of Chicago Press. ISBN 0-226-76996-8.
Penrose, R. (1996). Chandrasekhar, Black Holes and Singularities. J. Astrophys. Astr.[37]
Wali, Kameshwar C. (1991). Chandra: A Biography of S. Chandrasekhar. Chicago: The University of Chicago Press. ISBN 0-226-87054-5.
Wali, Kameshwar C., ed. (1997). Chandrasekhar: The Man Behind the Legend - Chandra Remembered. London: imperial College Press. ISBN 1-86094-038-2.
Wignesan, T., ed. (2004). The Man who Dwarfed the Stars. The Asianists' Asia. ISSN 1298-0358.
Venkataraman, G. (1992). Chandrasekhar and His Limit. Hyderabad, India: Universities Press. ISBN 81-7371-035-X.
Saikia, D J.; et al., eds. (2011). Fluid flows to Black Holes: A tribute to S Chandrasekhar on his birth centenary. Singapore: World Scientific Publishing Co. Ptd Ltd. ISBN 981-4299-57-X.
Kameshwar, C Wali, ed. (2001). A Quest For Perspectives. Singapore: World Scientific Publishing Co. Ptd Ltd. ISBN 1-86094-201-6.
Ramnath, Radhika, ed. (2012). S. Chandrasekhar: Man of Science. Harpercollins. ASIN B00C3EWIME.
Kameshwar, C Wali, ed. (2011). A Scientific Autobiography: S Chandrasekhar. Singapore: World Scientific Publishing Co. Ptd Ltd. ISBN 981-4299-57-X.
Salwi, Dilip, ed. (2004). S. Chandrasekhar: The scholar scientist. Rupa. ISBN 8129104911.
Pandey, Rakesh Kumar, ed. (2017). Chandrasekhar Limit: Size of White Dwarfs. Lap Lambert Academic Publishing. ISBN 3330317663.

Sharadchandra Shankar Shrikhande

Sharadchandra Shankar Shrikhande (born 19 October 1917) is an Indian mathematician with distinguished and well-recognized achievements in combinatorial mathematics. He is notable for his breakthrough work along with R. C. Bose and E. T. Parker in their disproof of the famous conjecture made by Leonhard Euler dated 1782 that there do not exist two mutually orthogonal latin squares of order 4n + 2 for every n.[1] Shrikhande's specialty was combinatorics, and statistical designs. Shrikhande graph [2] is used in statistical designs.

Shrikhande received a Ph.D. in the year 1950 from the University of North Carolina at Chapel Hill under the direction of R. C. Bose. Shrikhande taught at various universities in the USA and in India.[3] Shrikhande was a professor of mathematics at Banaras Hindu University, Banaras and the founding head of the department of mathematics, University of Mumbai and the founding director of the Center of Advanced Study in Mathematics, Mumbai until he retired in 1978. He is a fellow of the Indian National Science Academy, the Indian Academy of Sciences and the Institute of Mathematical Institute, USA.

Shrikhande's son Mohan Shrikhande[4] is a professor of combinatorial mathematics at Central Michigan University in Mt. Pleasant, Michigan.

Prahalad Chunnilal Vaidya

Prahalad Chunnilal Vaidya (P.C.Vaidya; 23 May 1918 – 12 March 2010), was an Indian physicist and mathematician, renowned for his instrumental work in the general theory of relativity. Apart from his scientific career, he was also an educationist and a follower of Gandhian philosophy in post-independence India, specifically in his domicile state Gujarat.

Biography

Early life

P. C. Vaidya was born in Shahpur of Junagadh district, Gujarat, India on 23 May 1918.

He completed most of his schooling in Bhavnagar, and went to Mumbai (formerly known as Bombay) for higher studies. There, after finishing high school at Ismail Yusuf College, he joined the Institute of Science (then known as Royal Institute of Science) in Mumbai. He received a BSc degree, majoring in Mathematics and Physics. He completed a MSc degree with Applied Mathematics major.

Vaidya's first stint at teaching was at the Dharmendra Singhji College in Rajkot, where he joined as a lecturer in 1940, soon after completing his MSc examinations. Vaidya taught trigonometry and arithmetics to undergraduate students. The college was then managed by the St Xavier's College, Bombay for half the term, after which the royal family of Rajkot under His Highness the Thakor Saheb of Rajkot took control of the college. Due to differences with the new management, Vaidya resigned in 1941 and subsequently started with freedom fighter Prithvi Singh Azad the Ahimsak Vyayam Sangh institute of physical education, where he was the principal for non-violent struggle training programme for youths.[1] Meanwhile, he continued teaching mathematics by conducting private tutions for school students.[2]

In 1942, P. C. Vaidya wrote to Professor V. V. Narlikar, father of renowned Indian physicist Jayant Narlikar, expressing his desire to study relativity. Narlikar approved this, and Vaidya immediately moved to Banaras Hindu University(BHU), Varanasi, where Narlikar was a faculty member at the school of relativity. Vaidya was at Banaras for about ten months.

At that time, India's freedom struggle was at full steam with Mahatma Gandhi leading the Quit India movement. The political situation was also chaotic due to World War II. Vaidya was living with his wife Vidya and six-month-old daughter Kumud and surviving solely on his earlier savings. Gandhi went on a prolonged fast then, which led to a period of great uncertainty as the fast had affected his health adversely. Desperate to know the developments, Vaidya would eagerly await the Hindi evening daily Aaj. Amid the tension, the idea of spacetime geometry sprouted in his mind. Within a week, Vaidya came up with Vaidya Metric.[3]

Professor Vaidya obtained his Doctoral degree (PhD) in mathematics in 1949.[citation needed]

Professional career

After his research stint at BHU, he went to a number of places to teach mathematics, including reputed Science institutions in Surat, Rajkot and Mumbai. During a small period of 1947 – 48, he went to Tata Institute of Fundamental Research as Research Associate. There he got associated with Homi Bhabha, father of India's nuclear program. Due to accommodation constraints, he left Mumbai, and continued the rest of his academic career in Gujarat. From 1948 to 1971, he taught at various colleges including: V. P. College, Vallabh Vidyanagar; Gujarat College, Ahmedabad; M.N. College, Visnagar; and University School of Sciences, Gujarat University.

Vaidya was a recipient of the Bombay University's Springer Research Scholarship. Vaidya's initial research under this scholarship resulted in a paper that he sent to Robert Oppenheimer, who appreciated the paper and sent it to the American Physical Society journal Physical Review. The journal published the paper in 1951.[4]

In 1971, he was appointed Chairman of Gujarat Public Service Commission. This was followed by Union Public Service Commission membership during 1977 – 78, during which he served Central Government. His final professional benchmark was the Vice-Chancellorship of Gujarat University during 1978 – 80.

Doctoral Students

Hasmukh M. Raval; Mathematics Department, University School of Sciences, Gujarat University, Ahmedabad, India

Thesis: Theoretical Studies on Rotating Universes in Relativistic Cosmology; 1972

International contributions

During 1964 to 1973, Vaidya served as visiting professor at number of international universities, including:

Visiting Professor of Mathematics at Washington State University, Pullman, Washington, US.
Visiting Professor of Mathematics at London University (Queen Elizabeth College), York University and Newcastle University in UK.
Visiting Scientist at Dublin Institute of Advanced Studies, Dublin, Ireland.
Visiting Scientist at International Centre for Theoretical Physics at Trieste, Italy.

In June 1971, he delivered a very informative course of lectures at the Institute Henrie Poincare, Paris in June 1971. In July 1971, he attended 6th International Conference on General Relativity and Gravitation at Copenhagen.

Death

For last several years, Vaidya had confined himself to his Shardanagar house in Ahmedabad due to deteriorating health. He was diagnosed with a kidney ailment in January 2010. He died on 12 March 2010 at Ahmedabad.[5]

He had four daughters, Kumud, Smita, Darshana and Hina.

Work

Einstein's theory of gravity is described by a set of complicated equations which use the mathematics of Riemannian geometry. Professor Vaidya took up on this mission, and accomplished pioneering work which led to conception of such a solution. The result was The Vaidya Metric.

Professor Vaidya's research on general theory of relativity was started when he went to Banaras Hindu University in 1942, where he joined the school of relativity started by Professor V. V. Narlikar. It was only ten months that he spent at BHU at that time, during which the revolutionary idea of developing a spacetime geometry was born within him, which would describe the gravitational potentials in the exterior of a radiating star.

There was pioneering work done around the same area, but it was helpful up to some extent. The well-known Schwarzschild Solution describes the geometry around a spherical star. However, it necessarily assumes the exterior of the star to be empty. Vaidya generalised this case to incorporate the radiation from the star, and the resulting solution was famous Vaidya metric. Till date, Vaidya is known to be one of the pioneers of the Golden age of general relativity.

His discovery of the Vaidya Metric gave him worldwide reputation at small age of 24, even before the beginning of his professional career.

The Vaidya Metric

Vaidya Metric applies to a set Einstein's equations that describes the gravitational field of a star which has a sizeable radiation. It pioneered the key idea of using the light rays as a co-ordinate frame. In other words, it was an idea of a null co-ordinate, which eventually played extremely significant role in subsequent research in gravitation theory during forthcoming decades. The Vaidya metric has by now found many applications in gravitation theory. It is widely used and internationally cited to study many problems in gravitationand general relativity.

Contributions to scientific community

In February 1969, in an occasion to felicitate Professor V. V. Narlikar on his 60th birthday, Professor Vaidya made a proposition to found a society of Indian relativists. The result was Indian Association for General Relativity and Gravitation (IAGRG), and Professor V. V. Narlikar assumed the position of founder President.

At his suggestion, Vikram Sarabhai laid foundation of mathematics laboratory in Ahmedabad, a pioneering institute of its kind in India. It is known today as the Community Science Center.

Professor Vaidya also established the Gujarat Mathematical Society.

Professional affiliations

Founder member of Indian Association for General Relativity and Gravitation (IAGRG).
Fellow of the Indian Academy of Sciences (F.A.Sc.).
Fellow of the Indian National Science Academy (F.N.A).
President of Calcutta Mathematical Society for 2 years.
Honorary Fellow of Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune.
President of the Indian Mathematical Society – 1976 and 1977.
A Founder Member of an International Journal "General Relativity and Gravitation", which is published from Berne, Switzerland.
Member of Indian National Committee on Astronomy- 1976 to 1979.
President of the UGC Committee on Relativity and Cosmology.
Participated in activities that led to revision of state-level syllabi of Mathematics textbooks.
Founded Gujarat Ganit Mandal in 1964 in Bhavnagar. He presided over it in 1964 and 1988. Since inception, it organises programs in rural areas periodically to spread mathematics across all layers of society.

Literary work

Professor Vaidya's profound scientific contributions had phenomenal impact in journals and publications. During his life, he authored or co-authored more than thirty research papers in General Relativity and Gravitation. They are cited quite frequently in the field research since their publication.

In his efforts to popularise mathematics among Indian students, Professor Vaidya started 'Suganitam' mathematics magazine in 1960s. Since its inception, it has been continually read in numerous schools and colleges, and has inspired generations of mathematics teachers and students alike.

He authored several popular science books in Gujarati:

Akhil Brahamandman ("In the entire Universe")
Dashansh Paddhati Sha Mate? ("Why Decimal System?")
Dadaji Ni Vato ("Grandpa's Tales") – a collection of science stories for children
What is Modern Mathematics?
Ganit Darshan ("Discourses in Mathematics"). This book won the 1970–71 prize for Gujarati Scientific Literature from the Gujarat State Government.

Besides, Vaidya published several memoirs from his days as a teacher, such as 'Chalk ane Duster' (Chalk and Duster), and 'America ane apne' (America and Us), which are from his days as a visiting professor at the Washington State University.

He also wrote mathematical and scientific articles in the leading Gujarati cultural magazine, Kumar, founded by Kalaguru Ravishankar Raval, the leading painter, art teacher, art critic, journalist and essayist from Gujarat.

Thought

Professor Vaidya was known among his colleagues and friends as staunch follower of Gandhian principles – simplicity and honesty. Even in his old age, he used to ride a bicycle. He strongly believed that for a mathematician, his brain was the best tool in itself, and research had very less dependency over resources or money. His lectures, always delivered using a chalk and black board, never failed to captivate the student. His memoirs of his teaching and research are titled 'Chalk and Duster' – his tools of learning and teaching mathematics. Rather than being limited to opinions, he was quite pragmatic in living out his principles.

Honesty in public and personal life was his another remarkable trait. According to his close aides, even during peak years of his scientific career, he exercised extreme prudence and wisdom in using his influence for personal gains of his family, or people related to him.

Spending his senior years in Gujarat University, he initiated statewide efforts to revolutionise mathematics and science education – his motivation being "I am the highest paid mathematics teacher in Gujarat. It cannot be (limited) for teaching MSc classes."

As a visionary educationist, he felt a top-down need to change the way mathematics training was imparted to students, and began programs to educate mathematics teachers on "How to teach mathematics." He frequently interacted with primary students, and tried to awaken their curiosity in mathematics. His foundation of Gujarat Mathematical Society in Bhavnagar, 1964 was aimed at this objective. He tried to reach to farthest rural areas, and aimed the society's efforts to empower teachers and eradicate fear of the subject from students' minds.

Anil Kumar Gain

Anil Kumar Gain (Bengali: অনীল কুমার গায়েন), (1 February 1919 – 7 February 1978) (also spelt Anil Kumar Gayen) was an Indianmathematician and statistician best known for his works on the Pearson product-moment correlation coefficient in the field of applied statistics, with his colleague Ronald Fisher. He received his Ph.D. from the University of Cambridge under the supervision of Henry Ellis Daniels, who was the then President of the Royal Statistical Society. He was honoured as a Fellow of the Royal Statistical Society and the famous Cambridge Philosophical Society.[1]

Gain was the president of the statistics section of the Indian Science Congress Association, as well as the head of the Department of Mathematics at the Indian Institute of Technology Kharagpur. He later went on to found Vidyasagar University, naming it after the famous social reformer of the Bengali renaissance, Ishwar Chandra Vidyasagar.

Early life

Anil Kumar Gain was born in a poor Bengali family of a village named Lakkhi in Purba Medinipur, West Bengal, to Jibankrishna Gain and Panchami Devi. His father having died in his childhood, he and his siblings were brought up by his widowed mother under economic hardship. He started his education in an informal local school and was admitted to a formal school when he was eight. In his schooldays, he showed particular interest in English and mathematics, subjects he was primarily taught by his mother. Upon finishing school, he travelled to Kolkata to study mathematics from Surendranath College, followed by a master's degree in applied mathematicsfrom the University of Calcutta. He was declared the University Gold Medalist for the year 1943.[3]

Career

Sir Ronald Fisher

After briefly teaching at Presidency College and Bengal College of Engineering & Technology, Gain got married to Krishna Chongdar, the daughter of a famous and wealthy Bengali businessman. He travelled to England in 1947, to pursue his Ph.D. from the University of Cambridge in mathematical statistics – only to complete it in the year 1950. It was there that he met the famous Henry Ellis Daniels, under whose supervision he wrote most of his papers. He also befriended Sir Ronald Fisher there, and spent much of his time working with him in the field of applied statistics.

After returning to India, he started teaching at the Indian Statistical Institute as well as the University of Calcutta, and finally joined the Indian Institute of Technology Kharagpur, where he spent most of his remaining career. During his years at Kharagpur, he began to work on educational projects such as the National Council for Educational Research and Training (NCERT) to reform the education sector in Bengal. This interest in revolutionizing education eventually led to the inception of Vidyasagar University, which he founded with the vision of having a non-traditional teaching and learning environment at the University level. The University was finally established by the University Grants Commission (India) under the Vidyasagar University Act of 1981.[4]

Legacy and death

Due to his efforts to revolutionise education in Bengal, he became a key figure in the latter half of the Bengali renaissance, as well as a renowned scholar and academic. In 2012, Vidyasagar University announced the establishment of the Anil Kumar Gain Memorial Lecture, in honour of his contributions to the university, and to Bengal as a whole.[5]

He died a week after his birthday, on 7 February 1978, of natural causes at his residence in Kolkata, India. His descendants still live in Kolkata, as well as abroad.

C. R. Rao

Calyampudi Radhakrishna Rao, FRS known as C R Rao (born 10 September 1920) is an Indian-born, naturalised American, mathematician and statistician. He is currently professor emeritus at Penn State University and Research Professor at the University at Buffalo. Rao has been honoured by numerous colloquia, honorary degrees, and festschrifts and was awarded the US National Medal of Science in 2002.[2] The American Statistical Association has described him as "a living legend whose work has influenced not just statistics, but has had far reaching implications for fields as varied as economics, genetics, anthropology, geology, national planning, demography, biometry, and medicine."[2] The Times of India listed Rao as one of the top 10 Indian scientists of all time.[3] Rao is also a Senior Policy and Statistics advisor for the Indian Heart Association non-profit focused on raising South Asian cardiovascular disease awareness.

Early life

R. Rao was born in Hadagali, Bellary, Karnataka, India. He received an MScin mathematicsfrom Andhra University and an MA in statistics from Calcutta University in 1943.[2] He obtained a PhD degree at King's College in Cambridge University under R. A. Fisher in 1948, to which he added a Sc.D. degree, also from Cambridge, in 1965.

Academic career

Rao worked at the Indian Statistical Institute and the Anthropological Museum in Cambridge.

He held several important positions, as the Director of the Indian Statistical Institute, Jawaharlal Nehru Professor and National Professor in India, University Professor at the University of Pittsburgh and Eberly Professor and Chair of Statistics and Director of the Center for Multivariate Analysis at the Pennsylvania State University. As Head and later Director of the Research and Training School at the Indian Statistical Institute for a period of over 40 years, Rao developed research and training programs and produced several leaders in the field of Mathematics. On the basis of Rao's recommendation, the ASI (The Asian Statistical Institute) now known as Statistical Institute for Asia and Pacific was established in Tokyo to provide training to statisticians working in government and industrial organisations.[5]

Among his best-known discoveries are the Cramér–Rao bound and the Rao–Blackwell theorem [6] both related to the quality of estimators. Other areas he worked in include multivariate analysis, estimation theory, and differential geometry. His other contributions include the Fisher–Rao Theorem, Rao distance, and orthogonal arrays. He is the author of 14 books and has published over 400 journal publications.

Rao has received 38 honorary doctoral degrees from universities in 19 countries around the world and numerous awards and medals for his contributions to statistics and science. He is a member of eight National Academies in India, the United Kingdom, the United States, and Italy. Rao was awarded the United States National Medal of Science, that nation's highest award for lifetime achievement in fields of scientific research, in June 2002. He was given the India Science Award in 2010, the highest honour conferred by the government of India in scientific domain. In 2013, he was nominated for the Nobel Peace Prize, along with Miodrag Lovric[7] (Editor) and Shlomo Sawilowsky, for their contribution to the International Encyclopedia of Statistical Science [8]. He was most recently honoured with his 38th honorary doctorate by the Indian Institute of Technology, Kharagpur, on 26 July 2014 for "his contributions to the foundations of modern statistics through the introduction of concepts such as Cramér–Rao inequality, Rao–Blackwellization, Rao distance, Rao measure, and for introducing the idea of orthogonal arrays for the industry to design high-quality products."

He was the President of the International Statistical Institute, Institute of Mathematical Statistics (USA), and the International Biometric Society. He was inducted into the Hall of Fame of India's National Institution for Quality and Reliability (Chennai Branch) for his contribution to industrial statistics and the promotion of quality control programs in industries.

The Journal of Quantitative Economics published a special issue in Rao's honour in 1991. "Dr Rao is a very distinguished scientist and a highly eminent statistician of our time. His contributions to statistical theory and applications are well known, and many of his results, which bear his name, are included in the curriculum of courses in statistics at bachelor's and master's level all over the world. He is an inspiring teacher and has guided the research work of numerous students in all areas of statistics. His early work had greatly influenced the course of statistical research during the last four decades. One of the purposes of this special issue is to recognise Dr Rao's own contributions to econometrics and acknowledge his major role in the development of econometric research in India."

Areas of research contributions

Awards and medals

Sardar PatelLifetime Achievement Award (Sardar Ratna) of the Sardar Patel Foundation India (2015)
Guy Medalin Gold (2011) of the Royal Statistical Society [9]
India Science Award2010 (the highest award in a scientific field presented by government of India)[10]
International Mahalanobis Prize (2003) of the International Statistical Institute [11]
Srinivasa Ramanujan Medal (2003) of the Indian National Science Academy
President George W. Bush, on 12 June 2002, honoured him with the National Medal of Science, the highest award in US in the scientific field, as a "prophet of new age" with the citation, "for his pioneering contributions to the foundations of statistical theory and multivariate statistical methodology and their applications, enriching the physical, biological, mathematical, economic and engineering sciences."
Padma Vibhushan(2001) by the Government of India [12]
Mahalanobis Centenary Gold Medal (1993?) of the Indian Science Congress
Wilks Memorial Award(1989) of the American Statistical Association
Padma Bhushan(1968)[12]
Megnadh Saha Medal (1969) of the Indian National Science Academy
Guy Medalin Silver (1965) of the Royal Statistical Society
S. Bhatnagar Award(1963) of Council of Scientific and Industrial Research
JC Bose Gold Medal of the Bose Institute
Gold Medal of the University of Calcutta
He was also awarded an honorary Doctor of Scienceby the University of Calcutta in 2003.[13] Also honorary doctorates from a number of universities and institutes around the world.

In his honour

The Pennsylvania State Universityhas established C. R. and Bhargavi Rao Prize in Statistics,
CR Rao Advanced Institute of Mathematics, Statistics and Computer Science
National Award in Statisticsestablished by Ministry of Statistics and Programme Implementation (MoSPI), Government of India.
The road from IIIT Hyderabadpassing along Central University of Hyderabadcrossroads to Alind Factory, Lingampally is named as "Prof. C.R. Rao Road".

Harish-Chandra

Harish-Chandra FRS [2] (11 October 1923 – 16 October 1983) was an Indian American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups.

Early life

Harish-Chandra was born in Kanpur [6]. He was educated at B.N.S.D. College, Kanpur and at the University of Allahabad [7]. After receiving his master's degree in Physics in 1943, he moved to the Indian Institute of Science, Bangalore for further studies in theoretical physics and worked with Homi J. Bhabha.

In 1945, he moved to University of Cambridge, Cambridge and worked as a research student under Paul Dirac[8]. While at Cambridge, he attended lectures by Wolfgang Pauli, and during one of them pointed out a mistake in Pauli's work. The two were to become lifelong friends. During this time he became increasingly interested in mathematics. At Cambridge he obtained his PhD in 1947.

Honors and awards

He was a member of the(NAS) National Academy of Sciences of the U.S. and a Fellow of the Royal Society.[2] He was the recipient of the Cole Prize of the American Mathematical Society, in 1954. The Indian National Science Academy honoured him with the Srinivasa Ramanujan Medal in 1974. In 1981, he received an honorary degree from Yale University.

The mathematics department of V.S.S.D. College, Kanpur celebrates his birthday every year in different forms, which includes lectures from students and professors from various colleges, institutes and students' visit to Harish-Chandra Research Institute.

The Indian Government named the Harish-Chandra Research Institute, an institute dedicated to Theoretical Physics and Mathematics, after him.

Robert Langlands wrote in a biographical article of Harish-Chandra:

“

He was considered for the Fields Medal in 1958, but a forceful member of the selection committee in whose eyes Thomwas a Bourbakist was determined not to have two. So Harish-Chandra, whom he also placed on the Bourbaki camp, was set aside.

”

He was also a recipient of the Indian civilian honour of the Padma Bhushan (1977).

P. K. Srinivasan

P.K. Srinivasan (PKS) (November 4, 1924 – June 20, 2005) was a well known mathematics teacher in India. He taught mathematics at the Muthialpet High School in Chennai, Indiauntil his retirement. His singular dedication to education of mathematics would bring him to the United States, where he worked for a year, and then to Nigeria, where he would work for six years. He is known in India for his dedication to teaching mathematics and in creating pioneering awareness of the Indian mathematician Ramanujan. He has authored several books in English, Telugu and Tamil that introduce mathematics to children in novel and interesting ways. He was also a prominent reviewer of math books in the weekly Book Reviewcolumn of the Indian newspaper The Hindu in Chennai.

Experience

PKS, as he was known to the world-at-large, among his colleagues, students and friends, travelled to the United States as a Fulbright exchange teacher and worked in Liverpool Central School, New York, in 1965-'66. Later he served as a Senior education officer and a Senior lecturer in Mathematics in Nigeria for seven years. He served as a lecturer in the National Council of Educational Research and Training (NCERT) in India. He organized over sixty math expositions and fairs in India, Nigeria and the United States, and participated actively in four International Congress on Mathematical Education (ICME) conferences.

He inspired many a creative idea and gave them shape through demonstrative displays by his students. Conduct of Bharat Dharshan, World Exhibition, three-day three-tier Math Expositions to make all his math students walk across the curriculum[clarification needed] which was highly talked about in his days and research papers on subjects from English to Social Studies by students studying from 7 to 12 Standards was an everlasting contribution made by him to the students and to The Muthialpet High School, Chennai where he was a teacher par excellence.

He could pick the brightest of students and discuss esoteric topics such as Boolean Algebra, Ramanujan's theorems and at same time deploy easy-to-understand teaching tools for teaching mathematics and English. As a class teacher, he could reach across to the poorest performers and made them cross average levels in English and mathematics.

Although he hailed from a traditional family, he was always clad in a white kurta and dhoti spun out of khadi - rough and homespun cotton cloth that symbolized the Swadeshiconcept of Gandhiji. He sported a Gandhian cap as well.

His vision and unquenchable thirst for knowledge transcended the narrow barriers of caste, language and religion. Personal and family interests always took a backseat in his mission for spreading knowledge and awareness and imparting a sense of purpose in his students to go beyond the narrow frontiers of a syllabus-oriented formal education to exploration of the unfathomable depths of knowledge. He displayed the same missionary zeal in making classroom education and teaching of English and mathematics in particular a matter of fun and curiosity among the low-mark scoring students as well.

In addition, he was the founder and the curator-director of Ramanujan Museum & Math Education Centre [1] which he helped establish in 1993. He was also one of the founders of the Association of Mathematics Teachers of India (AMTI) in 1965.

Honours

He was awarded the National Science Award by the Government of India in 1991 and a State Award by the Tamil Nadu Science and Technology Centre for popularisation of mathematics education to enable children to learn mathematics with interest and enthusiasm.

Philosophy

He advocated the use of no cost teaching aids and no cost, no material teaching aids, improvising them to illustrate mathematical concepts whenever they are introduced and almost lead a crusade against the mere rote learning techniques and drill. He also indicated the efficacy of introducing non-routine thinking when the child can grasp, using the concepts introduced, leading to problem solving techniques and innovative strategies.

Memorial Meeting

A meeting to pay homage to his memory was organised on 8 July 2005 by AMTI in the Dakshinamoorthy auditorium of the P.S. Higher Secondary School in Mylapore, Chennai. Many speakers who were his associates, admirers, friends and relatives spoke of the momentum he brought to the movement of mathematics education in the country. It was mentioned by more than one speaker at that time that the best tribute to his memory would be for mathematics teachers to make mathematics learning interesting for the children.

Memorial

A commemorative website was created, for a short while (no longer actively maintained, as of 2012) by his immediate family. P.K. Srinivasan is survived by his wife, five sons, and five daughters. P.K. Srinivasan's obituary was published note in the newspaper The Hindu.

Books and Articles

Maths Club Activities
A Mathematical Delight
Number Fun with a Calendar
How to Promote Creativity in Learning Mathematics- published by Lakshmi Ganapathy Educational and Charitable Trust, Chennai, India.
Ramanujam Memorial Number Vol 1 - Letter and Reminiscences - a compilation(Ed.)
Ramanujam Memorial Number Vol 2 - An Inspiration(Ed.)
Introduction to the Creativity of Ramanujan - Instruction Guides to Primary, Middle and High School Teachers
Game Way Math- published by Lakshmi Ganapathy Educational and Charitable Trust, Chennai, India.
Mathematics and Magic Squares.

Book Reviews

Srinivasan was a prominent reviewer of math books in the weekly Book Review column of the Indian newspaper The Hindu in Chennai:

Personal

P.K. Srinivasan was the strict father of ten children, but most of his professional colleagues did not know he had such a large family. His wife, Alamelu Srinivasan, has been credited with bringing up their children almost single-handedly. She died in Chennai on November 10, 2012.

Raghu Raj Bahadur

Raghu Raj Bahadur (30 April 1924 – 7 July 1997) was an Indian statistician considered by peers to be "one of the architects of the modern theory of mathematical statistics".

Biography

Bahadur was born in Delhi, India, and received his BA (1943)[2] and MA (1945)[2] in mathematics from University of Delhi. He received his doctorate from the University of North Carolina under Herbert Robbins in 1950 after which he joined University of Chicago. He worked as a research statistician at the Indian Statistical Institute in Calcutta from 1956 to 1961. He spent the remainder of his academic career in the University of Chicago.[3]

Contributions

He published numerous papers[4] and is best known for the concepts of "Bahadur Efficiency"[5] and the Bahadur-Ghosh-Kiefer representation (with J. K. Ghosh and Jack Kiefer)[6]

He also framed the Anderson–Bahadur algorithm [7] along with Theodore Wilbur Anderson which is used in statistics and engineering for solving binary classification problems when the underlying data have multivariate normal distributions with different covariance matrices.

Legacy

He held the John Simon Guggenheim Fellowship (1968-69)[8] and was the 1974 Wald Lecturer of the IMS.[2] He was the President of the Institute of Mathematical Statistics from 1974-75[8] and was elected a Fellow of the American Academy of Arts and Sciences in 1986.

Gopinath Kallianpur (1925–2015)[1] was an Indian American mathematician and statistician who became the first director of the Indian Statistical Institute (1976–79) under its new Memorandum of Association. During his tenure as the director the new centre of ISI at Bangalore, Karnataka was founded.

Early life

He received his Bachelor's degree in 1945 and Master's degree in 1946 from the University of Madras.[2] In Bombay Kallianpur came in contact with D. D. Kosambi, a well-known mathematician and versatile scholar and with Subbaramiah Minakshisundaram at Andhra University which was then in Guntur. Under the supervision of Herbert Robbins, of the Statistics Department of the University of North Carolina, Chapel Hill he obtained his doctoral degree in 1951 in the then developing field of stochastic processes.

Academic life

In the start of his career he held the position of lecturer at the University of California, Berkeley, during 1951–52, and was a member of the Institute for Advanced Study, Princeton, from 1952 to 1953. In 1953 Kallianpur joined ISI and stayed in Calcutta until the summer of 1956. During the period distinguished scientists, like Ronald Fisher, Norbert Wiener and Yuri Linnik visited the Institute. Norbert Wiener collaborated with Kallianpur on topics including ergodic theory, prediction theory and generalized harmonic analysis [3] Between 1956 and 1976 Kallianpur hold a number of faculty positions at the Michigan State University (1956–59, 1961–63); Indiana University (1959–61); University of Minnesota. At University of Minnesota Kallianpur collaborated with Charlotte Striebel, a member of the Research Group of Lockheed Corporation working on filtering and control problems.[4] From 1976 to 1979 he became the first director of the Indian Statistical Institute. In 1979 he was appointed Alumni Distinguished Professor at the University of North Carolina, Chapel Hill (1979–2001). Six years after his retirement in 2001 Professor Kallianpur moved to Nashville, Tennessee where he continued to work on mathematics, until moving to Cleveland, Ohio in 2014.

Shreeram Shankar Abhyankar

Shreeram Shankar Abhyankar (22 July 1930 – 2 November 2012)[1][2] was an Indian American mathematician known for his contributions to algebraic geometry. He, at the time of his death, held the Marshall distinguished professor of mathematics chair at Purdue University, and was also a professor of computer science and industrial engineering. He is known for Abhyankar's conjecture of finite group theory.

His latest research was in the area of computational and algorithmic algebraic geometry.

Career

Abhyankar was born in Ujjain, Madhya Pradesh, India. He earned his B.Sc. from Royal Institute of Science of University of Mumbai in 1951, his A.M. at Harvard University in 1952, and his Ph.D. at Harvard in 1955. His thesis, written under the direction of Oscar Zariski, was titled Local uniformization on algebraic surfaces over modular ground fields.[3][4] Before going to Purdue, he was an associate professor of mathematics at Cornell University and Johns Hopkins University.

Abhyankar was appointed the Marshall Distinguished Professor of Mathematics at Purdue in 1967. His research topics include algebraic geometry (particularly resolution of singularities, a field in which he made significant progress over fields of finite characteristic), commutative algebra, local algebra, valuation theory, theory of functions of several complex variables, quantum electrodynamics, circuit theory, invariant theory, combinatorics, computer-aided design, and robotics. He popularized the Jacobian conjecture.

Death

Abhyankar died of a heart condition on 2 November 2012 at his residence near Purdue University.[5]

Selected publications

Algebraic surfaces, Oscar Zariski, SS Abhyankar, J Lipman, David Mumford1995
Lectures on expansion techniques in algebraic geometry, SS Abhyankar, B Singh – 1977 – Tata Institute of Fundamental Research
Local rings of high embedding dimension, SS Abhyankar – American Journal of Mathematics, 1967

Honours

Abhyankar has won numerous awards and honours.

Abhyankar received the Herbert Newby McCoy Award from Purdue University in 1973 .
Fellow of the Indian Academy of Sciences
Editorial board member of the Indian Journal of Pure and Applied Mathematics
Chauvenet Prizefrom the Mathematical Association of America (1978)[6]
Honorary Doctorate Degree (Docteur Honoris Causa) by the University of Angersin France (29 October 1998)
Fellow of the American Mathematical Society(2012)

M. S. Narasimhan

Mudumbai Seshachalu Narasimhan FRS (born 1932) is an eminent Indian mathematician. He is well known along with C S Seshadrifor their proof of the Narasimhan–Seshadri theorem, and both were elected as Fellows of the Royal Society

Education

Narasimhan did his undergraduate studies at Loyola College, Chennai, where he was taught by Fr Racine. Fr Racine had studied with the famous French mathematicians Élie Cartan and Jacques Hadamard, and connected his students with the latest developments in modern mathematics. Among Racine's other students who achieved eminence, we may count Subbaramiah Minakshisundaram, K. G. Ramanathan, C S Seshadri, Raghavan Narasimhan, and C. P. Ramanujam.

Narasimhan went to the Tata Institute of Fundamental Research (TIFR), Bombay, for his graduate studies. He obtained his Ph.D. from University of Mumbai in 1960; his advisor was K. Chandrasekharan. Among Narasimhan's distinguished students is M. S. Raghunathanwho followed in this footsteps to bag the Shanti Swarup Bhatnagar Prize as well as become FRS. Two other students who made a mark as top-notch mathematicians are S. Ramanan and V. K. Patodi.

Career

Degrees and posts held

Visiting Scholar, Institute for Advanced Study (1968-1969)[1]

Fellow of the Royal Society, London
Head, Mathematics Group of the Abdus Salam International Centre for Theoretical Physics (1992–1999)
Honorary Fellow, Tata Institute of Fundamental Research, Bangalore Centre.

Awards and felicitations

Shanti Swarup Bhatnagar Prize(1975)
TWAS Prizefor Mathematics (1987)[2]
Padma Bhushan(1990)[3]
King Faisal International Prize for Science, 2006 (jointly with Simon Donaldson, Imperial College).

C. S. Seshadri

C.S. Seshadri FRS (born 29 February 1932[1]) is an eminent Indian mathematician. He is Director-Emeritus of the Chennai Mathematical Institute,[2] and is known for his work in algebraic geometry. The Seshadri constant is named after him.

He is a recipient of the Padma Bhushan in 2009,[3] the third highest civilian honor in the country.

Degrees and posts

He received his B.A. (Hons) degree in Mathematics from Madras University in 1953 and was mentored by Fr. Racine and S Naryanan there.[5] He completed his Ph.D. from Bombay University in 1958 under the supervision of K. Chandrasekharan.[6] He was elected Fellow of the Indian Academy of Sciences in 1971.[7]

Seshadri worked in the School of Mathematics at TIFR in Bombay from 1953 to 1984 starting as a Research Scholar and rising to a senior professor. From 1984 to 1989, he worked in IMSc in Chennai. From 1989 to 2010, he worked as the founding director of the Chennai Mathematical Institute. He stepped down from his Directorship of the Chennai Mathematical Institute (CMI) in December 2010. He continues to be a part of CMI as "Director-Emeritus" from 1 January 2011.

Visiting professorships

University of Paris, France
Harvard University, Cambridge
UCLA
Brandeis University
University of Bonn, Bonn
Kyoto University, Kyoto, Japan

He has given talks at the ICM.

Awards

Honorary degree, Université Pierre et Marie Curie(UPMC), Paris, 2013.[8]
Honoris Causa, University of Hyderabad, India
Padma Bhushan
Shanti Swarup Bhatnagar Award
Srinivasa Ramanujan Medalfrom the Indian Academy of Sciences
Honorary D.Sc. from Banaras Hindu University
TWAS Science Award
Fellow of IAS, INSA and a Fellow of the Royal Society.
Membership of the United States National Academy of Sciences [9]
Fellow of the American Mathematical Society, 2012.[10]

Research work

Seshadri's main work is in algebraic geometry. His work with M S Narasimhan on unitary vector bundles and the Narasimhan–Seshadri theorem has influenced the field. His work on Geometric Invariant Theory and on Schubert varieties, in particular his introduction of standard monomial theory, is widely recognized. Seshadri's contributions include the creation of the Chennai Mathematical Institute, an institute for the study of mathematics in India.

Publications

Narasimhan, M. S.; Seshadri, C. S. (1965). "Stable and unitary vector bundles on a compact Riemann surface". Annals of Mathematics. The Annals of Mathematics, Vol. 82, No. 3. 82(3): 540–567. doi:2307/1970710. JSTOR 1970710. MR 0184252.
Seshadri, C. S. (2007), Introduction to the theory of standard monomials, Texts and Readings in Mathematics, 46, New Delhi: Hindustan Book Agency, ISBN 9788185931784, MR 2347272
Seshadri, C. S. (2012), Collected papers of C. S. Seshadri. Volume 1. Vector bundles and invariant theory, New Delhi: Hindustan Book Agency, ISBN 9789380250175, MR 2905897
Seshadri, C. S. (2012), Collected papers of C. S. Seshadri. Volume 2. Schubert geometry and representation theory., New Delhi: Hindustan Book Agency, ISBN 9789380250175, MR 2905898.

K. S. S. Nambooripad

S. S. Nambooripad(born 6 April 1935) is an Indian mathematician who has made fundamental contributions to the structuretheory of regular semigroups. Nambooripad was also instrumental in popularising the TeX software in India and also in introducing and championing the cause of the free software movement in India.

He was with the Department of Mathematics, University of Kerala, since 1976. He served the Department as its Head from 1983 until his retirement from University service in 1995. After retirement, he is associating with the academic and research activities of the Center for Mathematical Sciences, Thiruvananthapuram in various capacities.

Early years

Nambooripad was born on 6 April 1935 in Puttumanoor near Cochin in central Kerala. He received traditional vedic education up to the age of fifteen after which he joined a modern school offering formal education.[1] He obtained the BSc(Hons) degree of University of Kerala from Maharaja's College, Ernakulam, in 1956. He spent a few years teaching mathematics in some privately managed colleges[1] before joining the newly started Department of Mathematics, University of Kerala, as a research scholar in mathematics in 1965. He was initially under the supervision of Prof. M. R. Parameswaran. A year later he came under the guidance of Prof. B. R. Srinivasan. About two years later, consequent on the departure of Prof. B. R. Srinivasan from University of Kerala, Nambooripad became a student of Prof. Y. Sitaraman. He was awarded the PhD degree in 1974.

Major contributions

Nambooripad's basic contributions are in the structure theory of regular semigroups. A semigroup is a set S together with an associative binary operation in S. A semigroup S is said to be regular if for every a in S there is an element b in S such that aba = a. Nambooripad axiomatically characterised the structure of the set of idempotents in a regular semigroup. He called a set having this structure a biordered set. "The axioms defining a biordered set are quite complicated. However, considering the general nature of semigroups, it is rather surprising that such a finite axiomatization is even possible."[2] A full treatment of the theory was published as a single paper number of the Memoirs of American mathematical Society in 1979. "In the mid 70s A. H. Clifford became very much excited by the work of Nambooripad on the structure of regular semigroups in terms of idempotent ordering and sandwich matrices and wrote several expository papers on Nambooripad structure theorem for regular semigroups".[3]

He later developed an alternative approach to describe the structure of regular semigroups. This particular work utilizes the abstract theory of cross-connections to provide a useful framework for studying various classes of regular semigroups.[4][5]

As a TeX populariser

TeX was introduced into Kerala by Nambooripad. After a visit to the United States in early 1990s, he brought the TeX programme to Kerala in fourteen floppy disks. Nambooripad encouraged his students to learn and use TeX, especially for preparing their theses. One of his students was E. Krishnan, one of the authors of the LaTeX primer2 published as an electronic book by the Indian TeX User Group. Krishnan also played an important role in establishing the Free Software Foundation of India. Another person inspired by Nambooripad was C.V.Radhakrishnan who is running a company called River Valley Technologies since 1995 for typesetting of scientific journals and books.[6] Nambooripad was the prime catalyst for the formation of Indian TeX Users Group in 1998. He was the inaugural Chairman of the Group.

Ramaiyengar Sridharan

Ramaiyengar Sridharan is a mathematician at Chennai Mathematical Institute, formerly at the Tata Institute of Fundamental Research(TIFR). He was born in Cuddalore in the year 1935 [1] and he obtained his Ph.D. from Columbia under the guidance of Samuel Eilenberg in Filtered algebras and Representations of Lie Algebras.

Sridharan was awarded the Shanti Swarup Bhatnagar Prize for Science and Technology (SSB prize) in Mathematical Science in 1980. Another SSB Prize awardee, Raman Parimala was his Ph.D. student at TIFR.

Selected publications[edit]

Filtered algebras and representations of Lie algebras, R Sridharan - Transactions of the American Mathematical Society, 1961 - jstor.org
On the global dimension of some algebras, MP Murthy, R Sridharan - Mathematische Zeitschrift, 1963 - Springer [1]

Notes

Jump up^Google scholar

Mathematics in ancient India

Sridharan, the man and his work

Combinatorial methods in Indian music

Combinatorial methods in ancient India

Mathematics before S. Ramanujan

Hermann Weyl

Balagangadharan

Marginal note of Fermat

A theorem of Artin

Homi Bhabha

50 years of Indian mathematics

Mathematical Semantics

On the division by 5 of periods of elliptic functions

Otto Holder

Pratyayas and Fibonacci numbers

Proof of the Riemann Roch theorem.

Vinod Johri

Vinod Johri (10 June 1935 – 10 May 2014) was an Indian astrophysicist. He was an eminent cosmologist, a retired professor of astrophysics at Indian Institute of Technology, Madras and an emeritus professor at Lucknow University since 1995. Johri had over 75 research publications and articles published in pioneering journals. His major contributions in cosmological research included 'power law inflation, genesis of quintessence fields of dark energy and phantom cosmologies'. He was the co-author of the first model of power law inflation in Brans–Dicke theory along with C. Mathiazhagan. He was honored by Uttar Pradesh Government by Research Award of the Council of Science & Technology (CSIR).[1]

Johri spent over 45 years researching in cosmology, acting as a research guide and principal investigator of various research projects of Council of Scientific and Industrial Research,[2] Department of Science & Technology[3] and University Grants Commission of India.[4]Johri was a Commonwealth Fellow, a senior visitor at Cambridge University (UK) and a Fellow of Royal Astronomical Society[5] of London. He worked as consultant for UNESCO at United Nations Development Program[6] in Iran and as a DAAD Fellow[7] at University of Mainz (Germany), as a visiting scientist at Hansen Lab[8] (Gravity Probe B Group) Stanford University (USA) and as an International Scholar at Fine Theoretical Physics Institute[9] at University of Minnesota at Minneapolis (USA). He died in Dallas, USA at the age of 78 due to complications arising from Kidney failure.

Career

Johri was born in Etah (Uttar Pradesh), India on 10 June 1935. His father Dr. Bhairon Prasad Johri graduated from Veterinary College, Patna and worked as Livestock Officer, Allahabad, India. Johri's mother, Sarojini Johri was a home maker.

Johri completed his High School at Narain College,[10] Shikohabad, India, with 12th rank in the state merit list. He was awarded First Prize in Chiranjeevi Dhiri Singh Provincial English Debate. He completed his Bachelor's in 1953 and Master's in Applied Mathematics in 1956 from Allahabad University scoring high ranks in the merit list. In 1957 he was appointed as Assistant Professor in the Department of Mathematics Allahabad University,[11] Allahabad. In 1960, Johri was appointed Assistant Professor in Mathematics at Gorakhpur University,[12] Gorakhpur India, where he was conferred PhD degree in 1966 on his thesis "Gravitational Waves in Bondi Space-Time".

In 1967 Johri was awarded Commonwealth Fellowship for Post-Doctorate work at Department of Mathematics and Theoretical Physics,[13] Cambridge University (UK), where he worked in close collaboration on cosmological problems with Dr.Dennis Sciama and research scholars Friedrich Hehl (Cologne), Fernando de Felice[14] (Padova, Italy), George Ellis (Cape Town) and Stephen Hawking(UK) for his Post Doctorate program. In 1968 Johri was appointed Reader in Mathematics at Gorakhpur University. Between 1970 and 1972, Johri worked as UNESCO consultant to Government of Iran under United Nations Development Program at Teheran. Johri was elected a Fellow of Royal Astronomical Society of London in 1978.

Johri accepted the position of Professor of Cosmology at the Mathematics Department of Indian Institute of Technology,[15] Madras (Chennai) in 1980. In addition to teaching and advising PhD students, he introduced new courses on General Relativity and Cosmology. In 1984, he discovered the first model of 'power law inflation' under Brans–Dicke theory(later on called as 'extended inflation') along with his student C.Mathizhagan. Johri worked on various visiting assignments at National University[16] (Kuwait), International Centre for Theoretical Physics, Trieste (Italy), Copernicus Astronomical Institute, Polish Academy of Sciences (Warsaw), National University of Singapore, South West Technical University of Sydney; Australian National University, Canberra, Hansen Lab (Gravity Probe B Group)[17]), Stanford University, Cologne University, International Institute of Physics and Chemistry Brussels University (hosted by Nobel Laureate Prof. Ilya Prigogine).

Johri organized the International Symposium on Cosmology at Indian Institute of Technology,[18] Madras in 1995. The proceedings of the symposium were published by Hadronic Press,[19] Florida (USA) in 1997.

In 1995 Professor Johri retired from Indian Institute of Technology[20] and joined as CSIR[21] Professor Emeritus in the Department of Mathematics & Astronomy at Lucknow University,[22] Lucknow (India). In 2001, Johri got interested in dark energy during his visiting assignment at Fine Theoretical Physics Institute,[23] University of Minnesota (USA), he gave the theory of 'integrated tracking' of Quintessence Fields. His research work on 'Genesis of Quintessence'[24] and 'Phantom Cosmologies'[25] was published in reputed journals Physical Review D & Classical and Quantum Gravity.

Honors

Johri was a Commonwealth Fellow and a senior visitor to Cambridge University (UK). He was also a resident of Clare College, Cambridge. He was elected a Fellow of Royal Astronomical Society of London in 1978. Johri was awarded the Royal Society Visiting Fellowship in 1989 for visiting Department of Applied Mathematics and Theoretical Physics (DAMTP) Cambridge,[26] Southampton University (UK), Cardiff University and Queen Mary College, London under the Distinguished Scientists Exchange Program of British Council.[27] In 1993, Johri was awarded DAAD Fellowship for his visiting assignment at Theoretical Physics Institute, Johannes Gutenberg University, Mainz(Germany).

Johri was honored with a 'Silver Plaque' by Council of Science & Technology,[28] India for his distinguished work on Dark Energy in the year 2006. He was Life Member of Ganita Parishad[29] and a founding member of the Indian Association of General Relativity and Gravitation.[30]

Writings and hobbies

Besides scientific research papers and popular science articles, Johri wrote several books on Mathematical Physics and Astrophysics. His research monograph Early Universe was published by Hadronic Press (USA) in 1996. Vinod Johri loved to compose verses in Hindi, Urdu and English during his leisure time.

Personal life

Johri was married to Aruna (maiden name Kodesia) for 54 years. The couple had 2 sons Manoj (Software Architect, HP), Vivek (IT Manager, UnitedHealth Group) and 2 daughters Manisha (Teacher at Shishya,[31] Chennai) and Anvita (Software Development, UnitedHealth Group).

K. R. Parthasarathy (probabilist)

Kalyanapuram Rangachari Parthasarathy (born 25 June 1936) is professor emeritus at the Indian Statistical Institute and a pioneer of quantum stochastic calculus.

Biography

He was born in 1936 [2][3] at Chennai. He studied at the Ramakrishna Mission Vivekananda College, where he completed the B.A. (Honours) course in Mathematics, and moved to the Indian Statistical Institute, Kolkata, where he completed his Ph.D., under the supervision of C. R. Rao in 1962. He was one of the "famous four"[4] (the others were R. Ranga Rao, Veeravalli S. Varadarajan, and S. R. Srinivasa Varadhan ) in ISI during 1956-1963. He was awarded the first Ph.D. degree of ISI.[2] He received the Shanti Swarup Bhatnagar Prize for Science and Technology in Mathematical Science in 1977 and the TWAS Prize in 1996.[5]

Research

He worked at the Steklov Mathematical Institute, USSR Academy of Sciences (1962–63), as Lecturer where he collaborated with Andrey Kolmogorov.[6] Later he came in United Kingdom as Professor of Statistics in University of Sheffield (1964–68), University of Manchester (1968-70) and later at University of Nottingham where he collaborated with Robin Lyth Hudson on their pioneering work in quantum stochastic calculus.[3][7][8][9] Then he returned to India, and after a few years in Bombay University and the Indian Institute of Technology, Delhi, he came back in 1976 to the new Indian Statistical Institute, Delhi Centre and he stayed there till he retired in 1996.[2]

He is the namesake of Kostant–Parthasarathy–Ranga Rao–Varadarajan determinants along with Bertram Kostant, R. Ranga Rao and Veeravalli S. Varadarajan which they introduced in 1967.[10]

Books authored

Among the books he has authored are:

R. Parthasarathy. Probability measureson metric spaces. Vol. 352. American Mathematical Society, 1967.
Robin Lyth Hudsonand K. R. Parthasarathy. "Quantum Ito's formula and stochastic evolutions." Communications in Mathematical Physics3 (1984): 301-323.
R. Parthasarathy. An introduction to quantum stochastic calculus. Vol. 85. Springer, 1992.
R. Parthasarathy and Klaus Schmidt. "Positive definite kernels, continuous tensor products, and central limit theoremsof probability theory" (series: Lecture Notes in Mathematics). (1972).

Ramdas L. Bhirud

Ramdas L. Bhirud (Ramdas Lotu Bhirud or R. L. Bhirud) (April 7, 1937 – January 4, 1997), was an Indian mathematician from Maharashtra region of India worked in the field of numerical analysis, special functions and number theory.

Early life and education

Ramdas was born on April 7, 1937, in Chinawal, Maharashtra, India. He was one of six children. He did his elementary school education in Chinawal. After that moved to Pune for middle school and high school education. He attended Elphinstone College in Mumbai for his bachelor's degree in mathematics.

Career

After his bachelor's degree from University of Mumbai Ramdas set out for his Masters and PhD from University of Michigan in United States. For his PhD he contributed to the theory and construction of the Padé table. After his studies for a brief period he worked as an Assistant Professor at Purdue University in Indianapolis before returning to India where he worked as a professor of mathematics at MPKV Rahuri.

S. Ramanan

S (Sundararaman) Ramanan (born 20 July 1937) is an Indian mathematician who works in the area of algebraic geometry, moduli spaces and Lie groups. He is one of India's leading mathematicians and internationally recognised as an outstanding expert in algebraic geometry, especially in the area of modulii problems. He has also done some very beautiful work in differential geometry: his joint paper with MS Narasimhan on universal connections has been very influential. It enabled, among other things, SS Chern and B Simons to introduce what is known as the Chern-Simons invariant, which has proved useful in theoretical physics.[1]

The honours awarded to Professor Ramanan include the Shanti Swarup Bhatnagar Prize, India's highest science prize,[2] in 1979; the TWAS Prize for Mathematics in 2001[3] and the Ramanujan Medal in 2010.

He is the nephew of the Sanskrit scholar and Vedanta expert, the late Ramachandra Dikshitar, who was a professor at the Banaras Hindu University. Professor Ramanan is also a great aficionado and an amateur singer of Carnatic music.

He is an alumnus of the Ramakrishna Mission School in Chennai and the Vivekananda College in Chennai, where he completed a BA Honours in mathematics, standing second in mathematics and first in English among students of the science stream in the final exams in what was then Madras Presidency. He completed his PhD at the Tata Institute of Fundamental Research, under the direction of MS Narasimhan, with whom he collaborated for decades. He did his post-doctoral studies at Oxford University, Harvard University and ETH Zurich.

He later pursued a lengthy career at TIFR, with many international visits. He picked up the methods of modern differential geometry from the French mathematician Jean-Louis Koszul,[4] and later successfully applied it for his research centred on algebraic geometry. He has also made important contributions to the topics of abelian varieties and also vector bundles.

He was a senior colleague of M S Raghunathan and influenced him considerably.[5] Vijay Kumar Patodi who proved part of the Atiyah-Singer index theorem, was found and encouraged by Ramanan, and Patodi's PhD was done under the combined direction of Narasimhan and Ramanan.[6] He has a considerable number of students. Mathematicians influenced by Ramanan include N Mohan Kumar,[7] Shrawan Kumar,[8] D S Nagaraj,[9] Kapil H Paranjape,[10] Jaya Iyer,[11] Annamalai Ramanathan and several others.[12]

He was very close to, and has closely collaborated with, many Western mathematicians of note, like the late Raoul Bott, who was at Harvard University. While in TIFR as distinguished professor, he was one of the important figures in the school of mathematics in India. He now continues his contributions via teaching and mentoring at the Chennai Mathematical Institute,[13] where he is adjunct professor, and the Institute of Mathematical Sciences, Chennai.

He is a great lecturer[14] and expositor. He has written the book Modulii of Abelian Varieties with Allan Adler, published by Springer-Verlag, and a graduate-level book on algebraic geometry called Global Calculus, published by the American Mathematical Society.[15]

He has been a visiting professor at many of the world's leading universities, including Harvard University, University of California at Berkeley, the Institute of Advanced Study in Princeton, UCLA, Oxford University, Cambridge University, the Max Planck Institute and University of Paris. In 1978 he gave one of the prestigious 50 minute invited talks at the International Congress of Mathematicians in Helsinki, and has also been a speaker at many major international conferences. In 1999, he was extended the privilege of speaking on some aspects of the work of André Weil, one of the greatest mathematicians of 20th century, on the occasion of his being awarded the prestigious Inamouri Prize.

He is married to Anuradha Ramanan, a translator and former librarian, and has two daughters.

The first daughter is Sumana Ramanan, a senior journalist. A graduate of Brandeis University and the University of Chicago, she is now a freelance journalist and columnist. Earlier, she was the managing editor of Scroll.in, an award-winning pioneering digital newspaper, which she helped start; a senior editor and readers' editor at the Hindustan Times; an editor with Reuters; and a foreign correspondent based in Jerusalem, Israel. She was part of the team that won the Ramnath Goenka Award for spot reporting of the 2008 Mumbai terror attacks and is the recipient of the Red Ink journalism award in 2016 for her writing on culture. She is married to Jaikumar Radhakrishnan, a theoretical computer scientist.

Professor Ramanan's second daughter is Kavita Ramanan,[16] a noted mathematician who is now a professor of applied mathematics at Brown University in Providence, Rhode Island, USA. She was previously at Bell Labs and Carnegie Mellon University. A graduate of the Indian Institute of Technology, Mumbai, she got her Phd from Brown University, Rhode Island, USA. She is the recipient of the prestigious international Erlang Prize for outstanding contributions to applied probability, in 2006.kr

Selected publications

Biswas& S. Ramanan (1994). "An infinitesimal study of the moduli of Hitchin pairs". Journal of the London Mathematical Society. 49 (2): 219–231. doi:10.1112/jlms/49.2.219.

Pranab K. Sen

Pranab Kumar Sen (born November 7, 1937 in Calcutta, India)[1] is a statistician, a professor of statistics and the Cary C. Boshamer Professor of Biostatistics at the University of North Carolina at Chapel Hill.

Academic biography

Sen was the second of seven siblings; his father, a railway officer, died of leukemia when Sen was ten, and he was raised by his mother, the daughter of a physician.[3] He began his undergraduate studies at Presidency College, Kolkata, initially intending to study medicine but shifting to statistics when it was discovered that he was too young for medical college.[3] He received a B.S. from the University of Calcutta in 1955, an M.Sc. in 1957, and a Ph.D. in 1962;[1][2][4] his doctoral advisor was Hari Kinkar Nandi.[3] He taught for three years at the University of Calcutta and one more year at the University of California, Berkeley before joining the UNC faculty in 1965; although he has held visiting positions at other universities, he has remained at Chapel Hill for the rest of his career.[1][2] He was the founding co-editor of two journals, Sequential Analysis and Statistics and Decisions,[3] and was joint editor-in-chief of the Journal of Statistical Planning and Inference from 1980 to 1983.[1]

Research and graduate advising

Sen is the author or co-author of multiple books on non-parametric statistics, the advisor of over 80 Ph.D. students, and the author of over 600 research publications.[1][5] He is known for inventing the Hodges–Lehmann estimator independently of and contemporaneously with Hodges and Lehmann[3][6] and for the Theil–Sen estimator, a form of robust regression that fits a line to two-dimensional sample points by choosing the slope of the fit line to be the median of the slopes of the lines through pairs of samples.[7][8]

Awards and honors

Sen is a fellow of the Institute of Mathematical Statistics [9] and of the American Statistical Association.[10] He became the Cary C. Boshamer Professor in 1982.[1] He was the Lukacs Distinguished Visiting Professor at Bowling Green State University in 1996–1997.[11] In 2002 he won the Gottfried E. Noether Senior Scholar Award of the American Statistical Association,[12] and he was the 2010 winner of the Wilks Memorial Award of the ASA "for outstanding contributions to statistical research, especially in nonparametric statistics and biostatistics; and for exceptional service in mentoring doctoral students."[13] The Government of India awarded him the civilian honour of Padma Shri in 2011.[14] In 2012, the University of Calcutta awarded him an honorary Doctor of Science degree.[15]

In 2007, a festschrift was dedicated to him on the occasion of his 70th birthday.

Veeravalli S. Varadarajan

Biography

Varadarajan received his undergraduate degree in 1957 from Presidency College, Madras and his doctorate in 1960 from the Indian Statistical Institute in Calcutta, under the supervision of C. R. Rao. He was one of the "famous four"[1] (the others were R. Ranga Rao, K. R. Parthasarathy, and S. R. Srinivasa Varadhan ) in ISI during 1956-1963. After short periods at the Institute for Advanced Study and the University of Washington, Seattle he joined the Department of Mathematics at UCLA in 1965.

Contributions

Varadarajan's early work, including his doctoral thesis, was in the area of probability theory. He then moved into representation theorywhere he has done some of his best known work. In the 1980s, he wrote a series of papers with Donald Babbitt on the theory of differential equations with irregular singularities. His latest work has been in supersymmetry.

He introduced Kostant–Parthasarathy–Ranga Rao–Varadarajan determinants along with Bertram Kostant, K. R. Parthasarathy and R. Ranga Rao in 1967,[2] the Trombi–Varadarajan theorem [3] in 1972 and the Enright–Varadarajan modules [4] in 1975.

Recognition

He was awarded the Onsager Medal in 1998 for his work. He was recognized along with 23 Indian and Indian American members "who have made outstanding contributions to the creation, exposition advancement, communication, and utilization of mathematics" by the Fellows of the American Mathematical Society program in November 1, 2012.

Jayanta Kumar Ghosh

Jayanta Kumar Ghosh or Jaẏanta Kumāra Ghosha (Bengali: জয়ন্ত কুমার ঘোষ, born May 23, 1937, died September 30, 2017) was an Indian statistician, an emeritus professor at Indian Statistical Institute and a professor of statistics at Purdue University.

Education

He obtained a B.S. from Presidency College, then affiliated with the University of Calcutta, and subsequently a M.A. and a Ph.D. from the University of Calcutta. He started his research career in the early 1960s, studying sequential analysis as a graduate student in the department of statistics at the University of Calcutta.[1]

Research

Among his best-known discoveries are the Bahadur–Ghosh–Kiefer representation (with R. R. Bahadur and Jack Kiefer)[2] and the Ghosh–Pratt identity along with John W. Pratt.[3]

His research contributions fall within the fields of:

Awards/Honors

Elected Member of the International Statistical Institute
Advisory Editor, Journal of Statistical Planning and Inference
Fellow of the Institute of Mathematical Statistics
Fellow of the Indian National Science Academy
Life Member and Director of the Calcutta Statistical Association
Fellow of the Indian Academy of Sciences
Japanese Society for Promotion of Sciences Fellowship, 1978
Shanti Swarup Bhatnagar Prize for Science and Technology, 1981
President, Statistics Section of the Indian Science Congress Association, 1991
President, International Statistical Institute, 1993
Mahalanobis Gold Medal of Indian Science Congress Association, 1998
P. V. Sukhatme Prize for Statistics, 2000
Mahalanobis Memorial Lecture, State Science and Technology Congress, W. Bengal, 2003
D.Sc. (h.c.), B.C. Roy Agricultural University, W. Bengal, India, 2006
International Indian Statistical Association (IISA) Lifetime Achievement Award, 2010 [4]
Padma Shree (2014) by the Government of India

Bibliography

He has published over 50 research papers. He has also published four books, which are:

Invariance in Testing and Estimation (Lecture Notes), 1967, published by Indian Statistical Institute, Calcutta.
Higher Order Asymptotics (based on CBMS-NSF lecture), published jointly by Institute of Mathematical Statistics and American Statistical Association, 1994.
(with R.V. Ramamoorthi) Bayesian Nonparametrics (Springer 2003).
(with Mohan Delampady and Tapas Samanta) An Introduction to Bayesian Analysis - Theory and Methods, Springer 2006.

C. P. Ramanujam

Chakravarthi Padmanabhan Ramanujam (9 January 1938 – 27 October 1974) was an Indian mathematician who worked in the fields of number theory and algebraic geometry. He was elected a fellow of the Indian Academy of Sciences in 1973.

Like his namesake Srinivasa Ramanujan, Ramanujam also had a very short life.[1]

As David Mumford put it, Ramanujam felt that the spirit of mathematics demanded of him not merely routine developments but the right theorem on any given topic. "He wanted mathematics to be beautiful and to be clear and simple. He was sometimes tormented by the difficulty of these high standards, but in retrospect, it is clear to us how often he succeeded in adding to our knowledge, results both new, beautiful and with a genuinely original stamp".

Early life and education

Ramanujam was born to a Tamil family on 9 January 1938 in Madras (now Chennai), India, as the eldest of seven, to Chakravarthi Srinivasa Padmanabhan. He finished his schooling and joined Loyola College in Madras in 1952. He wanted to specialise in mathematics and he set out to master it with vigour and passion. He also enjoyed music and his favourite musician was Dr. M. D. Ramanathan, a maverick concert musician. His teacher and friend at this time was Father Racine, a missionary who had obtained his doctorate under the supervision of Élie Cartan. With Father Racine's encouragement and recommendation, Ramanujam applied and was admitted to the graduate school at the Tata Institute of Fundamental Research in Bombay. His father had wanted him to join the Indian Statistical Institute in Calcutta as he had passed the entrance exam meritoriously.

Career

Ramanujam set out for Mumbai at the age of eighteen to pursue his interest in mathematics. He and his friend and schoolmate Raghavan Narasimhan, and S. Ramanan joined TIFRtogether in 1957. At the Tata Institute there was a stream of first-rate visiting mathematicians from all over the world. It was a tradition for some graduate student to write up the notes of each course of lectures. Accordingly, Ramanujam wrote up in his first year, the notes of Max Deuring's lectures on Algebraic functions of one variable. It was a nontrivial effort and the notes were written clearly and were well received. The analytical mind was much in evidence in this effort as he could simplify and extend the notes within a short time period. "He could reduce difficult solutions to be simple and elegant due to his deep knowledge of the subject matter" states Ramanan. "Max Deuring's lectures gave him a taste for algebraic number theory. He studied not only algebraic geometry and analytic number theory of which he displayed a deep knowledge but he became an expert in several other allied subjects as well".

On the suggestion of his doctoral advisor, K. G. Ramanathan, he began working on a problem relating to the work of the German number theorist Carl Ludwig Siegel. In the course of proving the main result to the effect that every cubic form in 54 variables over any algebraic number field K had a non-trivial zero over that field, he had also simplified the earlier method of Siegel. He took up Waring's problem in algebraic number fields and got interesting results. In recognition of his work and his contribution to Number Theory, the Institute promoted him to associate professor. He protested against this promotion as 'undeserved', and had to be persuaded to accept the position. He proceeded to write his thesis in 1966 and took his doctoral examination in 1967. Dr. Siegel, who was one of the examiners, was highly impressed with the young man's depth of knowledge and his great mathematical abilities.

Ramanujam was a scribe for Igor Shafarevich's course of lectures in 1965 on minimal models and birational transformation of two-dimensional schemes. Professor Shafarevich subsequently wrote to say that Ramanujam not only corrected his mistakes but complemented the proofs of many results. The same was the case with Mumford's lectures on abelian varieties, which were delivered at TIFR around 1967. Mumford wrote in the preface to his book that the notes improved upon his work and that his current work on abelian varietieswas a joint effort between him and Ramanujam. A little-known fact is that during this time he started teaching himself German, Italian, Russian and French so that he could study mathematical works in their original form. His personal library contained quite a few non-English mathematical works.

Illness and death

Between 1964 and 1968, he was making great strides in number theory and his contacts with Shafarevich and Mumford led him on to algebraic geometry. According to Ramanathan and other colleagues, his progress and deep understanding of algebraic geometry was phenomenal. In 1964, based on his participation in the International Colloquium on Differential Analysis, he earned the respect of Alexander Grothendieck and of David Mumford, who invited him to Paris and Harvard. He accepted the invitation and was in Paris, but for a brief period. He was diagnosed in 1964 with schizophrenia with severe depression and left Paris for Chennai. He later decided to quit his position at TIFR.

He quit his post at Mumbai in 1965 after a bout of illness and secured a tenured position as a professor in Chandigarh, Punjab. There he met the young student Chitikila Musili, who later went on to prove interesting results in the geometry connected with the theory of Lie groups and wrote good expository books. Ramanujam stayed in Chandigarh only eight months and he had to return to Chennai again for treatment. TIFR was his real home and he was back there again in June 1965. Around this time he accepted an invitation from Institut des Hautes Études Scientifiques, near Paris. He was barely there before he was flown back to Chennai. Unfortunately schizophrenia, a highly treatable condition today, was not properly diagnosed and treated at that time. Thus he continued until the end of his life to be highly creative for short periods before the recurrent illness overtook him. Again, in 1970, he sent his resignation letter to TIFR but the institute would not take it seriously. Around this time, Mumford invited him to Warwick as a visiting professor during the algebraic geometry year. Mumford writes that he spent many delightful evenings with Ramanujam and that his presence contributed importantly to the success of the algebraic geometry year. A famous paper written during this time, by Michael Artin and David Mumford acknowledges Ramanujam's suggestions and help. He also had a short tenure at Turin where he was widely appreciated and accepted. Just after his death a commemorative hall was named after him in the former Istituto di Matematica (Institute of Mathematics) of the university of Genoa.

Back in India after his year at the University of Warwick, Ramanujam requested for a professorship at the Tata Institute but to be made tenable in their Bangalore campus. The Tata Institute had an applied mathematics wing in Bangalore. Although Ramanjuam had nothing to do with this area, the Institute, wishing him to continue his research, made a special arrangement by which he could stay and work there. By this time, he was deeply affected and depressed by his illness. He was put in charge of a new branch dealing with applied mathematics. He settled down in Bangalore, but again in the depths of depression caused by his illness, he tried to leave the Institute and obtain a university teaching post. During one of the attacks, he tried to take his life, but was rescued in time. However, late one evening on 27 October 1974, after a lively discussion with a visiting foreign professor he took his life with an overdose of barbiturates. He was barely thirty-seven.

Vasanti N. Bhat-Nayak

Vasanti N. Bhat-Nayak (10 June 1938 – 12 February 2009) was a professor of combinatorics and head of the department of mathematics, University of Mumbai. Vasanti Nayak was known for her work in BIBD designs, bivariegated graphs, graceful graphs, graph equations and frequency partitions.

Vasanti Bhat a Goud Saraswat Brahmin was raised in Pune. She earned a M.Sc. in mathematics from Pune University and a Ph.D. from the University of Mumbai in 1970 on her work "Some New Results in PBIBD Designs and Combinatorics". S. S. Shrikhande was her advisor. Bhat had guided Ph.D. students in graph theory and combinatorial mathematics. She died on 12 February 2009 in her residence in Chembur.

S. R. Srinivasa Varadhan

Sathamangalam Ranga Iyengar Srinivasa Varadhan FRS (born 2 January 1940) is an Indian American mathematician who is known for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviations.

Early life and education

Srinivasa was born in Chennai (then Madras) in 1940.[2] Varadhan received his under-graduate degree in 1959 from Presidency College, Madras, and then moved to the Indian Statistical Institute in Kolkata. He was one of the "famous four" (the others were R Ranga Rao, K R Parthasarathy, and Veeravalli S Varadarajan) in ISI during 1956–1963.[3] He received his doctorate from ISI in 1963 under C R Rao,[4][5] who arranged for Andrey Kolmogorov to be present at Varadhan's thesis defence.[6] Since 1963, he has worked at the Courant Institute of Mathematical Sciences at New York University, where he was at first a postdoctoral fellow (1963–66), strongly recommended by Monroe D Donsker. Here he met Daniel Stroock, who became a close colleague and co-author. In an article in the Notices of the American Mathematical Society, Stroock recalls these early years:

Varadhan, whom everyone calls Raghu, came to these shores from his native India in the fall of 1963. He arrived by plane at Idlewild Airport and proceeded to Manhattan by bus. His destination was that famous institution with the modest name, The Courant Institute of Mathematical Sciences, where he had been given a postdoctoral fellowship. Varadhan was assigned to one of the many windowless offices in the Courant building, which used to be a hat factory. Yet despite the somewhat humble surroundings, from these offices flowed a remarkably large fraction of the post-war mathematics of which America is justly proud.

Varadhan is currently a professor at the Courant Institute.[7][8] He is known for his work with Daniel W Stroock on diffusion processes, and for his work on large deviations with Monroe D Donsker.

Varadhan is married to Vasundra Varadhan who is also an academic (in media studies in the Gallatin School of Individualised Study). They had two sons, one of whom died in the September 11 attacks in 2001. The other, Ashok Varadhan, is an executive at Goldman Sachs in New York City.[9][10]

Awards and honours

Varadhan's awards and honours include the National Medal of Science (2010) from President Barack Obama, "the highest honour bestowed by the United States government on scientists, engineers and inventors".[11] He received also the Birkhoff Prize (1994), the Margaret and Herman Sokol Award of the Faculty of Arts and Sciences, New York University (1995), and the Leroy P Steele Prize for Seminal Contribution to Research (1996) from the American Mathematical Society, awarded for his work with Daniel W Stroock on diffusion processes.[12] He was awarded the Abel Prize in 2007 for his work on large deviations with Monroe D Donsker.[7][13] In 2008, the Government of India awarded him the Padma Bhushan.[14] He also has two honorary degrees from Université Pierre et Marie Curie in Paris (2003) and from Indian Statistical Institute in Kolkata, India (2004).

Varadhan is a member of the US National Academy of Sciences (1995),[15] and the Norwegian Academy of Science and Letters (2009).[16] He was elected to Fellow of the American Academy of Arts and Sciences (1988),[17] the Third World Academy of Sciences (1988), the Institute of Mathematical Statistics (1991), the Royal Society (1998),[18] the Indian Academy of Sciences (2004), the Society for Industrial and Applied Mathematics (2009),[19] and the American Mathematical Society (2012).

M. S. Raghunathan

Madabusi Santanam "M. S." Raghunathan is an Indian mathematician. He is currently Head of the National Centre for Mathematics, Indian Institute of Technology, Mumbai. Formerly Professor of eminence at TIFR in Homi Bhabha Chair.[1]Raghunathan received his PhD in Mathematics from (TIFR), University of Mumbai; his advisor was M. S. Narasimhan. Raghunathan is a Fellow of the Royal Society, of the Third World Academy of Sciences, and of the American Mathematical Society [2] and a recipient of the civilian honour of Padma Bhushan.

Early life and education

Madabusi Santanam Raghunathan was born on 11 August 1941 at Anantapur, Andhra Pradesh, his maternal grandparents' place. The family lived in Chennai. His father Santanam continued the family's timber business and expanded it through exports to Europe and Japan. He had earlier joined the Indian Institute of Science, Bangalore, after a BSc in Physics, but had to leave his studies mid-way to take care of the family business. Raghunathan fondly recalls that his father had a feeling for science and used to talk about it, making it very interesting to the children. Raghunathan's mother came from a family with an academic tradition. Her father was an esteemed Professor of English, who had contributed articles to the Cornhill Magazine. He also wrote, and published on his own, a book on William Makepeace Thackerey, which was later found to have been reprinted in the United States, without his knowledge, indeed in violation of the copyright he held.[4]

Raghunathan had his schooling in Chennai, in P.S. High School, Mylapore and the Madras Christian College High School. He passed his SSLC (Secondary School Leaving Certificate) examination in 1955. There is a rather interesting story about it: after the Sanskrit paper he absent-mindedly left the examination hall along with his answer paper, and was intercepted on his way home by a fellow student, following commotion at the examination hall on account of the missing answer paper. He narrowly escaped having to reappear for the entire examination, thanks to the headmaster vouching for his integrity.[5]

The University of Madras had the curious restriction of not admitting anyone under the age of 14 years and six months, though after attaining that age it was possible to be admitted even in higher classes. Raghunathan therefore pursued his Intermediate at the St. Joseph's College, Bangalore during 1955–57. He then returned to Chennai and joined B.A.(Hons.) in mathematics, in Vivekananda College, which had a very good reputation.[6]

Research

After initial training during 1960–62, he worked on a research problem suggested by Prof. M.S. Narasimhan, on "Deformations of linear connections and Riemannian metrics", and solved it by the summer of 1963.

He wrote his PhD thesis under the guidance of Professor Narasimhan and was awarded the degree by the University of Bombay in 1966. After completing his PhD, Raghunathan spent a year at the Institute for Advanced Study, Princeton, US. Through the years, he has held visiting position in many academic institutions in the US, Europe and Japan, for durations ranging from a few weeks to a year, and has spoken at several international conferences.

Discrete subgroups of Lie groups have been the central objects of his researches. He has made contributions to rigidity and arithmeticity problems.

Contribution

Raghunathan has also played an important role in the promotion of mathematics through various scientific bodies, in both advisory and administrative capacities. He organised the Ramanujan centenary celebrations in Chennai in 1987, with an international conference attended by the foremost number theorists.

His most important and comprehensive contribution in this sphere has been his role in the National Board for Higher Mathematics (NBHM). Raghunathan was a member of the Board since it was formed in 1983 and became its chairman in 1987. He continues to serve in that capacity. The Board has undertaken a variety of activities through the years: apart from providing financial support to mathematics libraries around the country and grants for research projects, organising conferences, travel to both national and international events and so on, the Board has taken a pro-active role in tapping mathematical talent through various activities, such as Olympiad activity, Mathematics Training and Talent Search, Scholarships/Fellowships at the M.Sc., Ph.D. and post- doctoral levels, and the rather innovative Nurture Programme conceived by Raghunathan to support learning of mathematics by students even while pursuing other career options. He organized the International Congress of Mathematicians (ICM 2010) at Hyderabad in India. This was the first ever ICM held in India.

Book Written

Raghunathan's book Discrete Subgroups of Lie Groups, published by Springer Verlag, Germany, in 1972 is now a classic in the area. It is unique in its coverage of various results which in recent decades have been put to considerable use, and as such it is much appreciated and widely referred to.[according to whom?] The book has been translated into the Russian and published with a foreword by Grigory Margulis.

Vashishtha Narayan Singh

Vashishtha Narayan Singh is an Indian mathematician [1] from Basantpur, Bhojpur District, Bihar, India.

Early life and education

He was born on 2 April 1942 in Basantpur village of Bhojpur district in Bihar, India[2] to Lal Bahadur Singh and Lahaso Devi. He received his primary and secondary education from Netarhat Residential School and college education from Patna Science College. Vashishtha Narayan Singh became a legend[3] as a student when he was allowed by Patna University to appear in the two-year course of B.Sc. (Hons.) in Mathematics in its very first year. His achievements are still mentioned with a sense of pride by Netarhat Vidyalaya[4]He received Ph.D. in Reproducing Kernels and Operators with a Cyclic Vector from University of California, Berkeley, in 1969.[5] His doctoral advisor was John L. Kelley.

Personal life

He married in the year 1974. He has suffered from schizophrenia. After a few years of marriage he and his wife separated due to his illness. He currently stays in his village without any attention from government.[6]

Career

After receiving his Ph.D.(on Cycle Vector Space Theory), he worked at NASA[citation needed] and then returned to India in 1971 to teach at I.I.T.Kanpur.[Indian Institute of Technology Kanpur], after eight months, he joined T.I.F.R. (Tata Institute of Fundamental Research)Bombay. In 1973, he was appointed as a permanent faculty in I.S.I.(Indian Statistical Institution) Kolkata.In 2014, he was appointed as guest faculty in Bhupendra Narayan Mandal University (BNMU) in Madhepura as a visiting professor.[1][7]

Works

Singh, Vashishtha N. (1974). Reproducing kernels and operators with a cyclic vector. I. Pacific J. Math. 52 (1974), no. 2, 567–584.

S. B. Rao

Siddani Bhaskara Rao a Graph theorist was a professor and director of the Indian Statistical Institute (ISI), Calcutta. Rao is the first director of the CR Rao Advanced Institute of Mathematics, Statistics and Computer Science.[2] S. B. Rao is known for his work on line graphs, frequency partitions and degree sequences.

Rao hails from Andhra Pradesh and completed his M.A. (1965) in mathematics from Andhra University.[citation needed] He received his Ph.D. (1971) from the Indian Statistical Institute, Calcutta under the supervision of renowned Statistician CR Rao. After completing his Ph.D., he moved to the University of Mumbai to work with S. S. Shrikhande. At the same time, he visited King's College, Aberdeen to work with Crispin St. J. A. Nash-Williams. From the University of Mumbai, Rao went back to the Indian Statistical Institute (ISI). While at ISI, he visited Ohio State University. Rao has guided students for their Ph.D.s in graph theory. He was the director of ISI Calcutta from 1995 to 2000. After retirement from ISI, he went to Andhra Pradesh to work as the first director of the C. R. Rao Advanced Institute of Mathematics, Statistics and Computer Science, Hyderabad.

References

Gopal Prasad

Gopal Prasad (born 31 July 1945 in Ghazipur, India) is an Indian-American mathematician. His research interests span the fields of Lie groups, their discrete subgroups, algebraic groups, arithmetic groups, geometry of locally symmetric spaces, and representation theory of reductive p-adic groups.

He is the Raoul Bott Professor of Mathematics [1] at the University of Michigan in Ann Arbor.

Education

He earned his bachelor's degree with honors in Mathematics from Magadh University in 1963. Two years later, in 1965, he received his masters in Mathematics from Patna University. After a brief stay at the Indian Institute of Technology Kanpur in their Ph.D. program for Mathematics, Prasad joined TIFR for his PhD program in 1966. There Prasad began a long and extensive collaboration with his advisor M. S. Raghunathan on several topics including the study of lattices in semi-simple Lie groups. In 1976, Prasad received his Ph.D. from University of Mumbai. Prasad became an Associate Professor at TIFR in 1979, and a Professor in 1984. He left TIFR to join the faculty at the University of Michigan in Ann Arbor in 1992, where he is the Raoul Bott Professor of Mathematics.

Family

In 1969, he married Indu Devi of Deoria. Gopal Prasad and Indu Devi have a son and a daughter and five grand-children. Shrawan Kumar, a professor of mathematics at the University of North Carolina, and Dipendra Prasad, a professor of mathematics at the Tata Institute of Fundamental Research, are his younger brothers.

Some contributions to mathematics

Prasad's early work was on discrete subgroups of real and p-adic semi-simple groups. He proved the "strong rigidity" of lattices in real semi-simple groups of rank 1 and also of lattices in p-adic groups, see [1] and [2]. He then tackled group-theoretic and arithmetic questions on semi-simple algebraic groups. He proved the "strong approximation" property for simply connected semi-simple groups over global function fields [3]. In collaboration with M. S. Raghunathan, Prasad determined the topological central extensions of these groups, and computed the "metaplectic kernel" for isotropic groups, see [11], [12] and [10]. Later, together with Andrei Rapinchuk, Prasad gave a precise computation of the metaplectic kernel for all simply connected semi-simple groups, see [14]. Prasad and Raghunathan have also obtained results on the Kneser-Tits problem, [13].

In 1987, Prasad found a formula for the volume of S-arithmetic quotients of semi-simple groups, [4]. Using this formula and certain number theoretic and Galois-cohomological estimates, Armand Borel and Gopal Prasad proved several finiteness theorems about arithmetic groups, [6]. The volume formula, together with number-theoretic and Bruhat-Tits theoretic considerations led to a classification, by Gopal Prasad and Sai-Kee Yeung, of fake projective planes (in the theory of smooth projective complex surfaces) into 28 non-empty classes [21] (see also [22] and [23]). This classification, together with computations by Donald Cartwright and Tim Steger, has led to a complete list of fake projective planes. This list consists of exactly 50 fake projective planes, up to isometry (distributed among the 28 classes). This work was the subject of a talk in the Bourbaki seminar.

Prasad has worked on the representation theory of reductive p-adic groups with Allen Moy. The filtrations of parahoric subgroups, referred to as the "Moy-Prasad filtration", is widely used in representation theory and harmonic analysis. Moy and Prasad used these filtrations and Bruhat-Tits theory to prove the existence of "unrefined minimal K-types", to define the notion of "depth" of an irreducible admissible representation and to give a classification of representations of depth zero, see [8] and [9].

In collaboration with Andrei Rapinchuk, Prasad has studied Zariski-dense subgroups of semi-simple groups and proved the existence in such a subgroup of regular semi-simple elements with many desirable properties, [15], [16]. These elements have been used in the investigation of geometric and ergodic theoretic questions. Prasad and Rapinchuk introduced a new notion of "weak-commensurability" of arithmetic subgroups and determined "weak- commensurability classes" of arithmetic groups in a given semi-simple group. They used their results on weak-commensurability to obtain results on length-commensurable and isospectral arithmetic locally symmetric spaces, see [17], [18] and [19].

Together with Jiu-Kang Yu, Prasad has studied the fixed point set under the action of a finite group of automorphisms of a reductive p-adic group G on the Bruhat-Building of G, [24]. In another joint work, Prasad and Yu determined all the quasi-reductive group schemes over a discrete valuation ring (DVR), [25].

In collaboration with Brian Conrad and Ofer Gabber, Prasad has studied the structure of pseudo-reductive groups, and also provided proofs of the conjugacy theorems for general smooth connected linear algebraic groups, announced without detailed proofs by Armand Borel and Jacques Tits; their research monograph [26] contains all this. The monograph [27] contains a complete classification of pseudo-reductive groups, including a Tits-style classification and also many interesting examples. The classification of pseudo-reductive groups already has many applications. There was a Bourbaki seminar in March 2010 on the work of Tits, Conrad-Gabber-Prasad on pseudo-reductive groups.

Honors

Prasad has received the Guggenheim Fellowship, the Humboldt Senior Research Award, and the Raoul Bott Professorship at the University of Michigan. He was awarded the Shanti Swarup Bhatnagar prize (by the Council of Scientific and Industrial Research of the Government of India). He has received Fellowships in the Indian National Science Academy, the Indian Academy of Sciences and the American Mathematical Society. Prasad gave an invited talk in the International Congress of Mathematicians held in Kyoto in 1990. In 2012 he became a fellow of the American Mathematical Society.[2]

Prasad was the Managing Editor of the Michigan Mathematical Journal for over a decade, an Associate Editor of the Annals of Mathematics for six years, and is an editor of the Asian Journal of Mathematics since its inception.

Vijay Kumar Patodi

Vijay Kumar Patodi (12 March 1945 – 21 December 1976) was an Indian mathematician who made fundamental contributions to differential geometry and topology. He was the first mathematician to apply heat equation methods to the proof of the Index Theoremfor elliptic operators.[citation needed] He was a professor at Tata Institute of Fundamental Research, Mumbai (Bombay).

Education

Patodi was a graduate of Government High School, Guna, Madhya Pradesh. He received his bachelor's degree from Vikram University, Ujjain, his master's degree from the Benaras Hindu University, and his Ph.D. from the University of Bombay under the guidance of M. S. Narasimhan and S. Ramanan at the Tata Institute of Fundamental Research.[1]

In the two papers based on his Ph.D. thesis, "Curvature and Eigenforms of the Laplace Operator" (Journal of Differential Geometry), and "An Analytical Proof of the Riemann-Roch-Hirzebruch Formula for Kaehler Manifolds" (also Journal of Differential Geometry), Patodi made his fundamental breakthroughs.[2]

Research career

He was invited to spend 1971–1973 at the Institute for Advanced Study in Princeton, New Jersey, where he collaborated with Michael Atiyah, Isadore Singer, and Raoul Bott. The joint work led to a series of papers, "Spectral Asymmetry and Riemannian Geometry" (Math. Proc. Cambridge. Phil. Soc.) with Atiyah and Singer, in which the η-invariant was defined. This invariant was to play a major role in subsequent advances in the area in the 1980s.[3]

Patodi was promoted to full professor at Tata Institute at age 30, however, he died at age 31, as a result of complications prior to surgery for a kidney transplant.

S. G. Dani

Shrikrishna Gopalrao Dani is a Professor of mathematics at the Indian Institute of Technology, Bombay, Mumbai who works in the broad area of ergodic theory.

Education

He did a Masters from the University of Mumbai in 1969. He then joined the Tata Institute of Fundamental Research (TIFR), Mumbai for a PhD and was awarded the same in 1975. After that, he joined TIFR as the faculty. He was a visiting scholar at the Institute for Advanced Study during 1976-77 and 1983-84.[2]

Administration

He has been a member of the NBHM since 1996 and was the Chairman of the NBHM. He is also the Chairman, Commission for Development and Exchange (CDE) of International Mathematical Union, for the period 2007‐10. He has served as Editor of Proceedings (Math. Sci.) of the Indian Academy of Sciences, Bangalore for many years since 1987.[3]

Awards and recognition

Dani was awarded the Shanti Swarup Bhatnagar Prize in 1990. He received the Third World Academy of Sciences prize in 2007. He was invited to give a talk at the International Congress of Mathematicians held in Zurich, Switzerland in 1994. He is also a recipient of the 2007 TWAS Prize.

Raman Parimala

Raman Parimala (born 21 November 1948)[1] is an Indian mathematician known for her contributions to algebra. She is the Arts & Sciences Distinguished Professor of mathematics at Emory University.[2] For many years, she was a professor at Tata Institute of Fundamental Research (TIFR), Mumbai.

Background

Parimala was born and raised in Tamil Nadu, India.[3] She studied in Saradha Vidyalaya Girls' High School and Stella Maris College at Chennai. She received her M.Sc. from Madras University (1970) and Ph.D. from the University of Mumbai (1976); her advisor was R. Sridharan from TIFR.[4]

Selected publications

Failure of a quadratic analogue of Serre's conjecture, Bulletin AMS, vol. 82, 1976, pp. 962–964 MR 0419427
Quadratic spaces over polynomial extensions of regular rings of dimension 2, Mathematische Annalen, vol. 261, 1982, pp. 287–292 doi:10.1007/BF01455449
Galois cohomology of the Classical groups over fields of cohomological dimension≦2, E Bayer-Fluckiger, R Parimala - Inventiones mathematicae, 1995 - Springer doi:10.1007/BF01231443
Hermitian analogue of a theorem of Springer, R Parimala, R. Sridharan, V Suresh - Journal of Algebra, 2001 - Elsevier doi:10.1006/jabr.2001.8830
Classical groups and the Hasse principle, E Bayer-Fluckiger, R Parimala - Annals of Mathematics, 1998 - jstor.org[5]doi:10.2307/120961

Honors

Parimala was an invited speaker at the International Congress of Mathematicians in Zurich in 1994 and gave a talk Study of quadratic forms — some connections with geometry. She gave a plenary address Arithmetic of linear algebraic groups over two dimensional fields at the Congress in Hyderabad in 2010.

Fellow of the Indian Academy of Sciences [1]
Fellow of Indian National Science Academy [1]
Bhatnagar Award in 1987[1]
Honorary doctorate from the University of Lausanne in 1999[1]
Srinivasa Ramanujan Birth Centenary Award in 2003.[1]
TWAS Prize for Mathematics (2005).[1][6]
Fellow of the American Mathematical Society (2012)

Navin M. Singhi

Navin Madhavprasad Singhi (born 1949) is an Indian mathematician and a professor at Tata Institute of Fundamental Research [1]Mumbai, specializing in combinatorics and graph theory. He is the recipient of the prestigious Shanti Swarup Bhatnagar Prize for Science and Technology. Singhi is known for his research in block designs, projective planes, line graphs, and coding theory.

Early life

Singhi was born in Indore and raised in Goregaon, Mumbai and earned a M.A. in mathematics from the University of Mumbai.

Career

Singhi earned a Ph.D. (1974) from the University of Mumbai, his advisor was S. S. Shrikhande. Professor Singhi is a Fellow of the Indian National Science Academy.

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Sujatha Ramdorai

Sujatha Ramdorai is a professor of mathematics and Canada Research Chair at University of British Columbia, Canada [1][2]. Previously a professor at Tata Institute of Fundamental Research,, Ramdorai is an algebraic number theorist known for her work on Iwasawa theory. She is the first Indian to win the prestigious ICTP Ramanujan Prize in 2006 and also a winner of the Shanti Swarup Bhatnagar Award, the highest honour in scientific fields by Indian Government in 2004. She was a member of the National Knowledge Commission from 2007 to 2009. She is at present a member of the Prime Minister’s Scientific Advisory Council from 2009 onwards and also a member of the National Innovation Council.[3] She is also on the advisory board of Gonit Sora.[4] She holds an adjunct professorship position at Indian Institute of Science Education and Research, Pune.

Education

She completed her B.Sc in 1982 at St. Joseph's college, Bangalore and then got her M.Sc. through correspondence from Annamalai University in 1985. After that she went for PhD at Tata Institute of Fundamental Research and was awarded her PhD under supervision of Raman Parimala in 1992.[6] Her dissertation was "Witt Groups of Real Surfaces and Real Geometry".

Contribution to mathematics

Together with Coates, Fukaya, Kato, and Venjakob she formulated a non-commutative version of the main conjecture of Iwasawa theory, on which much foundation of this important subject is based.[7] Iwasawa theory has its origins in the work of a great Japanese mathematician, Kenkichi Iwasawa.[8]

Editorial position

Managing Editor, International Journal of Number Theory (IJNT)
Editor, Journal of Ramanujan Mathematical Society (JRMS)[9]
Associate Editor, Expositiones Mathematicae.

Ramachandran Balasubramanian

Ramachandran Balasubramanian (born 15 March 1951) is an Indian mathematician and was Director of the Institute of Mathematical Sciences in Chennai, India.[1] He is known for his work in number theory, which includes settling the final g(4) case of Waring's problem in 1986.[2][3]

His works on moments of Riemann zeta function is highly appreciated and he was a plenary speaker from India at ICM in 2010. He was a visiting scholar at the Institute for Advanced Study in 1980-81.

Awards and honours

He has received the following awards:

The Shanti Swarup Bhatnagar Prize for Science and Technology in 1990.
The French government's Ordre National du Mérite for "furthering Indo-French cooperation in the field of mathematics" in 2003.[5]
The Padma Shri, the fourth highest civilian award in India, in 2006.[6]
Fellow of the American Mathematical Society, 2012.[7]
The Lifetime Achievement Award, 2013 awarded by Manmohan Singh, the Prime Minister of India.[8]
Fellow of the Indian National Science Academy (1988)

M. Ram Murty

Maruti Ram Pedaprolu Murty, FRSC (born 16 October 1953 in Guntur, India)[1] is an Indo-Canadian mathematician, currently head of the Department of Mathematics and Statistics at Queen's University, where he holds a Queen's Research Chair [2] in mathematics.

Career

Specialising in number theory, Murty is a researcher in the areas of modular forms, elliptic curves, and sieve theory. He was elected a Fellow of the Royal Society of Canada in 1990,[3] was elected to the Indian National Science Academy (INSA) in 2008,[4] and has won numerous prestigious awards in mathematics, including the Coxeter–James Prize. A highly learned Indian scholar, Murty is also cross-appointed as a professor of philosophy at Queen's, specialising in Indian philosophy.

Murty graduated with a B.Sc. from Carleton University in 1976. He received his Ph.D. in 1980 from the Massachusetts Institute of Technology, supervised by Harold Stark.[5] He was on the faculty of McGill University from 1982 until 1996, when he joined Queen's.

Murty has Erdős number 1, and has collaborated with dozens of other researchers, including frequent joint work with his brother, V. Kumar Murty. In 2012 he became a fellow of the American Mathematical Society.[6]

Selected publications

Cojocaru, Alina Carmen; Murty, M. Ram (2006). An introduction to sieve methods and their applications. London Mathematical Society Student Texts. 66. Cambridge University Press. ISBN 0-521-84816-4. MR 2200366..

Murty, M. Ram; Murty, V. Kumar (1991). "Mean values of derivatives of modular L-series". Annals of Mathematics. Second Series. Annals of Mathematics. 133 (3): 447–475. doi:10.2307/2944316. JSTOR 2944316. MR 1109350..

Alok Bhargava

Alok Bhargava (born 13 July 1954) is an Indian-American econometrician. He studied mathematics at Delhi University and economicsand econometrics at the London School of Economics. He is currently a full professor at the University of Maryland School of Public Policy.[1]

Career and research

Bhargava received his Ph.D. in econometrics from the London School of Economics under the supervision of John Denis Sargan in 1982. His thesis (The Theory of the Durbin–Watson Statistic with special reference to the Specification of Models in Levels as against in Differences) led to many tests for unit roots that were used in co-integration analyses. Bhargava also worked on econometric methods for longitudinal ("panel") data.[2]

Since 1991, Bhargava has been publishing on important aspects of nutrition, food policy, population health, child development, demography, epidemiology, AIDS, and finance in developing and developed countries.[3] His academic publications demonstrate the usefulness of rigorous econometric and statistical methods in addressing issues of under-nutrition and poor child health in developing countries, as well as obesity in developed countries.

Bhargava is an associate editor of the multi-disciplinary journal Economics and Human Biology.[4] He has held teaching positions at the University of Pennsylvania, Harvard University and University of Houston,and has published over 50 articles in academic journals.[5]

Books and reviews

A collection of his works has been reprinted in a separate volume in 2006 entitled "Econometrics, statistics and computational approaches in food and health sciences".[

Rattan Chand

Dr. Rattan Chand (Hindi : डॉ रतन चन्द ) is a former senior bureaucrat who has served Government of India at various positions for more than 35 years. Apart from working at senior positions in Government of India he has also been a resource person to USAID, World Bank, International Monetary Fund (IMF), United Nations, World Health Organisation and many other organisations. He has represented Government of India at many International Meetings, Conferences, Working Sessions, Seminars worldwide mainly in the Health Sector. He is currently member of the working group of National Statistical Commission, Government of India and senior expert for World Bank.

Early life and education

Rattan Chand was born to a humble brahmin family. He was born on 10 May 1955 in a small village in Bilaspur district, Himachal Pradesh, India. His father Pandit Harsukh, had been a freedom fighter and then lived as a farmer for rest of his life . He received elementary education from a small govt school in Bilaspur (HP) and went on to get his matriculation from Government Senior Secondary School, Kiratpur Sahib. He did his Bachelors of Science (B.Sc.) from Punjabi University, Punjab. He did his Masters in Mathematics from Punjabi University, Punjab and then completed his Masters of Philosophy (M.Phil.) from Jawaharlal Nehru University, Delhi, New Delhi.

Dr. Rattan Chand at US Bureau of Census, Washington DC, USA on 20th May 1992

He later completed his Ph.D. in Demography from International Institute of Population Sciences. He is also an alumnus from US Bureau of Census. Dr Rattan Chand joined the Indian Civil Service at Group A level through Union Public Service Commission (UPSC) in 1982.

Career

Dr. Chand has reorganised the ‘Bulletin on Rural Health Statistics’ and updated it regularly. As a senior official in National Sample Survey Organisation, he handled important work of survey of unorganised sector enterprises and drafted the first ever survey report on ‘Informal Sector in India’. For better dissemination of the results, system of organising the National Seminars on survey results was introduced by him. As National Director & Convenor (PNDT) in the Ministry of Health and Family Welfare, he strengthened the programme implementation by involving all major stakeholders. For the first time, an annual report on implementation of Prenatal Diagnosis Techniques (PNDT) Act was prepared and released. A website giving information on the programme and activities undertaken was launched. While working in National Accounts Division, he developed a ‘Guide to Government Budget Analysis and Preparation of Accounts’ for the general government sector. As Chief Director, in the Ministry of Health and Family Welfare, under him, a Web based Health Management Information System (HMIS) was developed to increase the flow and use of information at all levels. Also, Annual Health Surveys were introduced under him to track changes in program and impact indicators in the poor performing districts. He contributed significantly to the launch of the fourth round of National Family Health Survey.

Dr. Rattan Chand, Chief Guest of 56th Convocation of International Institute of Population Sciences (IIPS), delivering convocation lecture on May 2014.

Dr Rattan Chand has been one of the key people involved in the National Family Health Surveys conducted by Government of India. While talking to New York Times, Dr. Chand said, “This round will provide both district and national level data because of a revamped format. The plan is to conduct such a survey every three years.” Dr. Chand also said that he wasn’t aware why the national survey had been delayed but explained that consultations had been under way for a while to come up with a new redesigned survey.[1] Dr. Rattan Chand had also said that the annual healthy survey (AHS) was being scrapped to avoid duplication. “There was no reason to carry multiple surveys to map the same indicators. It was thought to be better to have one comprehensive survey instead of four fragmented ones,the new version of NFHS will not compromise of any indicators that were mapped in other surveys. We have asked the census commissioner to come out with district-level infant, under-5 and maternal mortality once in three years.”[2]

Efforts Against Female Foeticide

Given the dismal Child Sex Ratio in the country, and the Supreme Court directive of 2003 to State governments to enforce the law banning the use of sex determination technologies, the Ministry set up a National Inspection and Monitoring Committee (NIMC) in October last. Dr. Rattan Chand, Director (PNDT) was made the convenor of the NIMC. The NIMC under the guidance of Dr. Rattan Chand conducted raids in some of the districts in Maharashtra, Punjab, Haryana, Himachal Pradesh, Delhi and Gujarat. In April, it conducted raids on three clinics in Delhi. In its reports sent to the Chief Secretaries of the respective States, the committee observed that the Authorities had failed to monitor or supervise the registered clinics.[3]

As Director (PNDT) & Convenor of PNDT Act, 1994, Dr. Chand raided many hospitals and clinic all over India and shut them down for running illegal Sex determination tests. Dr. Rattan Chand said that even if the NIMC conducted raids, it could only send a report to the highest levels in the State Government. It was up to the State authorities to act on it.[3]

Membership and Affiliations

Member, Executive Council, Indian Association for Study of Population[4]
Member, Working Group, National Statistical Commission, Government of India
Member, Executive & General Council, International Institute for Population Sciences (IIPS), Mumbai, INDIA
Member, Technical Advisory Committee, Annual Health Survey (AHS), Government of India
Member, Technical Advisory Committee, District Level Household Survey (DLHS), Government of India
Member, Technical Advisory Committee, National Family Health Survey (NFHS), Government of India
Member, Technical Advisory Committee, Sample Registration System (SRS), Government of India
Member, Technical Coordination Committee, Health Management information System (HMIS)
Member, Technical Group on population projections, Government of India
Member, Working Group on Survey of Health and Morbidity, National Sample Survey Organisation (NSSO), Government of India
Member, Indian Association for Research in National Income and Wealth
Former Convenor, National Inspection and Monitoring Committee, Ministry of Health and Family Welfare, Government of India
Former Member, Governing Body, National Institute of Health & Family Welfare (NIHFW)
Visiting Faculty, National Academy of Statistical Administration, Greater Noida, India
Visiting Faculty, National Judicial Academy (India)

Important Publications

‘Quality of Life approach for identification of the poor’: Journal of Rural development, Vol.19, No.1, January – March 2004. ISSN 0970-3357
‘Socio-economic Dimensions of Unemployment in India’, The Indian Journal of Social Development, Vol.1.no.2.December-2004. ISSN 0972-3692
‘Inequality in Household Consumption in Indian States, 1973-2000’: Indian Journal of Millennium Development Studies, Vol.1.No.1.March-2006; ISSN 0973-3981
‘Incidence of Hunger in India, 1983-2002’. Paper Presented at the National Seminar on the Results of NSS 57th Round Survey Results, March 2004.
‘Health Implications of Poverty and Hunger’, INDIAN DEVELOPMENT REVIEW Vol.3.No.1,June 2005, ISSN 0972-9437
‘An Integrated Summary of the NSS 55th Round Consumer Expenditure Survey Results’, Published in ‘Sarvekshana’- Journal of National Sample Survey Organisation, 86thIssue,Vol.XIV, no.4 and XV, No.1.ISSN 2249-197X
‘An Integrated Summary of the NSS 55th Round Employment-Unemployment Survey Results’, Published in ‘Sarvekshana’ – Journal of National Sample Survey Organisation, 87th Issue, Vol. XV, no.2&3. ISSN 2249-197X
‘Population Change and Demand for Cereals – An Inter-State Analysis 1981-2001’, Jawaharlal Nehru University.
‘Fertility and Infant Mortality in Developing Countries – A study of the Relative role of social and economic dimensions of development’,
‘Socio-economic Dimensions of Tuberculosis in India’

References

Jump up^ India’s Battle Against Nutrition Data Deficiency The New York Times, October 24, 2013
Jump up^ http://www.livemint.com/Politics/zjD4pm80nNrUgpvbpcBRKK/Govt-discontinues-annual-health-survey.html
^ Jump up to:a b Small gain for the girl child Front Line)
Jump up^ http://www.iasp.ac.in/Executive_council2008_2010.html

Narendra Karmarkar

Narendra Krishna Karmarkar (born 1957) is an Indian mathematician, who developed Karmarkar's algorithm. He is listed as an ISI highly cited researcher.

Biography

Karmarkar received his B.Tech in Electrical Engineering from IIT Bombay in 1978, M.S. from the California Institute of Technology in 1979,[3] and Ph.D. in Computer Science from the University of California, Berkeley in 1983 under the supervision of Richard M. Karp.[4]

He invented the first provably polynomial time algorithm for linear programming also known as the interior point method. The algorithm is a cornerstone in the field of Linear Programming. He published his famous result in 1984 while he was working for Bell Laboratories in New Jersey. Karmarkar was a professor at the Tata Institute of Fundamental Research in Mumbai from 1998 to 2005. He was briefly the scientific advisor to the chairman of the TATA group. He is currently working on a new architecture for supercomputing.

Karmarkar has received a number of awards:

Paris Kanellakis Award, 2000 given by The Association for Computing Machinery for "specific theoretical accomplishments that have had a significant and demonstrable effect on the practice of computing".
Srinivasa Ramanujan Birth Centenary Award for 1999, presented by the Prime Minister of India.
Distinguished Alumnus Award, Indian Institute of Technology, Bombay, 1996
Distinguished Alumnus Award, Computer Science and Engineering, University of California, Berkeley (1993)
Fulkerson Prize in Discrete Mathematics given jointly by the American Mathematical Society & Mathematical Programming Society (1988)
Fellow of Bell Laboratories (1987– )
Texas Instruments Founders' Prize (1986)
Marconi International Young Scientist Award (1985)
American Academy of Achievement award, presented by former U.S. president (1985)
Frederick W. Lanchester Prize of the Operations Research Society of America for the Best Published Contributions to Operations Research (1984)
President of India gold medal, I.I.T. Bombay (1978)

Work

Karmarkar's algorithm

Karmarkar's algorithm solves linear programming problems in polynomial time. These problems are represented by "n" variables and "m" constraints. The previous method of solving these problems consisted of problem representation by an "x" sided solid with "y" vertices, where the solution was approached by traversing from vertex to vertex. Karmarkar's novel method approaches the solution by cutting through the above solid in its traversal. Consequently, complex optimization problems are solved much faster using the Karmarkar algorithm. A practical example of this efficiency is the solution to a complex problem in communications network optimization where the solution time was reduced from weeks to days. His algorithm thus enables faster business and policy decisions. Karmarkar's algorithm has stimulated the development of several interior point methods, some of which are used in current codes for solving linear programs.

Paris Kanellakis Award

The Association for Computing Machinery awarded him the prestigious Paris Kanellakis Award in 2000 for his work on polynomial time interior point methods for linear programming.

Galois geometry

After working on the Interior Point Method, Karmarkar worked on a new architecture for supercomputing, based on concepts from finite geometry, especially projective geometry over finite fields.[5][6][7][8]

Current investigations

Currently, he is synthesizing these concepts with some new ideas he calls sculpturing free space (a non-linear analogue of what has popularly been described as folding the perfect corner).[9] This approach allows him to extend this work to the physical design of machines. He is now publishing updates on his recent work,[10] including an extended abstract.[11]This new paradigm was presented at IVNC, Poland on 16 July 2008,[12] and at MIT on 25 July 2008.[13] Some of his recent work is published at ieeexplore.[14] He delivered a lecture on his on going work at IIT Bombay in September 2013.[15] He gave a four-part series of lectures at FOCM 2014 (Foundations of Computational Mathematics)[16] titled "Towards a Broader View of Theory of Computing". First part of this lecture series is available at Cornell archive.

Dinesh Thakur (mathematician)

Dinesh S. Thakur is a mathematician and a professor of mathematics at University of Rochester.[1] Before moving to Rochester, Thakur was a professor at University of Arizona.[2] His main research interest is number theory.

Early life

Thakur was born in Mumbai, India. He attended Balmohan Vidyamandir School in Bombay. He completed his undergraduate studies at Ruia College, University of Bombay. He got his Ph.D in 1987 at Harvard University [3] under the guidance of Professor John Tate.[4]

Career

Thakur has spent three and half years at Institute for Advanced Study, Princeton and three years at Tata Institute of Fundamental Research, Bombay. He held positions at University of Minnesota and University of Michigan. He moved to University of Arizona in 1993. He joined University of Rochester in July 2013. Thakur wrote a research monograph Function Field Arithmetic. Thakur has been serving on the editorial boards of Journal of Number Theory, International Journal of Number Theory, and P-adic Numbers,[5] Ultrametric Analysis and Applications. Thakur is a founding member of-and for 15 years a participant in-the NSF-funded Southwest Center for Arithmetic Geometry and the Arizona Winter School.[6]

He was elected as a member of the 2017 class of Fellows of the American Mathematical Society "for contributions to the arithmetic of function fields, exposition, and service to the mathematical community".[7]

Work

His main work has been in number theory, where he has been instrumental in developing various aspects of function field arithmetic and arithmetic geometry.[8][9]

References

Manindra Agrawal (born 20 May 1966) is a professor at the Department of Computer Science and Engineering and the Deputy Director at the Indian Institute of Technology, Kanpur.[1] He was also the recipient of the first Infosys Prize for Mathematics[2] and the Shanti Swarup Bhatnagar Award in Mathematical Sciences in 2003. He has been honored with Padma Shri in 2013.

Early life

Manindra Agrawal obtained a B.Tech. from IIT Kanpur. (1986 batch) and a Ph.D. from the same institute.

Career

He co-created the AKS primality test with Neeraj Kayal and Nitin Saxena, for which he and his co-authors won the 2006 Fulkerson Prize, and the 2006 Gödel Prize. He was also awarded with 2002 Clay Research Award for this work.The test is the first unconditional deterministic algorithm to test an n-digit number for primality in a time that has been proven to be polynomial in n.[4]

In September 2008, Agrawal was chosen for the first Infosys Mathematics Prize for outstanding contributions in the broad field of mathematics.[5] He was a visiting scholar at the Institute for Advanced Study in 2003-04.[6]

Awards and honors

Clay Research Award (2002)
Shanti Swarup Bhatnagar Prize for Science and Technology (2003)
ICTP Prize (2003)
IIT Kanpur Distinguished Alumnus Award (2003)
Fulkerson Prize (2006)
Gödel Prize (2006)
G D Birla Award (2009)
TWAS Prize (2010)[7]
Padma Shri (2013)
ACCS-CDAC Foundation Award (2014).

Madhu Sudan

(born September 12, 1966)[1] is an Indian-American computer scientist. He has been a Gordon McKay Professor of Computer Science at the Harvard John A. Paulson School of Engineering and Applied Sciences since 2015.

Career

He received his bachelor's degree in computer science from IIT Delhi in 1987[1] and his doctoral degree in computer science at the University of California, Berkeley in 1992.[1][2] He was a research staff member at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York from 1992 to 1997 and moved to MIT after that.[1] From 2009 to 2015 he was a permanent researcher at Microsoft Research New England before joining Harvard University in 2015.

Research contribution and awards

He was awarded the Rolf Nevanlinna Prize at the 24th International Congress of Mathematicians in 2002. The prize recognizes outstanding work in the mathematical aspects of computer science. Sudan was honored for his work in advancing the theory of probabilistically checkable proofs—a way to recast a mathematical proof in computer language for additional checks on its validity—and developing error-correcting codes.[1] For the same work, he received the ACM's Distinguished Doctoral Dissertation Award in 1993 and the Gödel Prize in 2001. He is a Fellow of the ACM (2008).[3] In 2012 he became a fellow of the American Mathematical Society.[4] In 2014 he won the Infosys Prize in the mathematical sciences.[5] In 2017 he was elected to the National Academy of Sciences.[6]

Sudan has made important contributions to several areas of theoretical computer science, including probabilistically checkable proofs, non-approximability of optimization problems, list decoding, and error-correcting codes.

Chandrashekhar Khare

Chandrashekhar B. Khare (born 1968) is a professor of mathematics at the University of California Los Angeles. In 2005, he made a major advance in the field of Galois representations and number theory by proving the level 1 Serre conjecture,[1] and later a proof of the full conjecture with Jean-Pierre Wintenberger.

Professional career

Resident of Mumbai, India and completed his undergraduate education at Trinity College, Cambridge University. He finished his thesis in 1995 under the supervision of Haruzo Hida at California Institute of Technology. His Ph.D. thesis was published in the Duke Mathematical Journal. He proved Serre's conjecture with Jean-Pierre Wintenberger, published in Inventiones Mathematicae.[2]

He started his career as a Fellow at Tata Institute of Fundamental Research. Currently, he is a professor at University of California, Los Angeles.

Awards and honors

Khare is the winner of the INSA Young Scientist Award (1999),[3] Fermat Prize (2007), the Infosys Prize (2010),[4] and the Cole Prize (2011).

He gave an invited talk at the International Congress of Mathematicians in 2010, on the topic of "Number Theory".[5]

In 2012 he became a fellow of the American Mathematical Society [6] and was elected as a Fellow of the Royal Society.[7]

U. S. R. Murty

Uppaluri Siva Ramachandra Murty,[1][2] or U. S. R. Murty (as he prefers to write his name), is a Professor Emeritus of the Department of Combinatorics and Optimization, University of Waterloo.[3][4]

U. S. R. Murty received his Ph.D. in 1967 from the Indian Statistical Institute, Calcutta, with a thesis on extremal graph theory;[5] his advisor was C. R. Rao.[6] Murty is well known for his work in matroid theory and graph theory, and mainly for being a co-author with J. A. Bondy of a textbook on graph theory. Murty has served as a managing editor and co-editor-in-chief of the Journal of Combinatorial Theory, Series B.[7]

Selected publications

John Adrian Bondy and U. S. R. Murty (1976), Graph Theory with Applications. North-Holland. Book's page at the University of Paris VI.
John Adrian Bondy and U. A. R. Murty (1979), "Graph Theory and Related Topics." Academic Press Inc. ISBN 978-0121143503.
U. S. R. Murty (1971) How Many Magic Configurations are There? The American Mathematical Monthly.
U. S. R. Murty (1971) Equicardinal matroids. Journal of Combinatorial Theory, Series B
U. S. R. Murty (1970) Matroids with Sylvester property. Aequationes Mathematicae.
Murty, U. S. R. (1968), "On some extremal graphs", Acta Mathematica Academiae Scientiarum Hungaricae, 19: 69–74, doi:10.1007/BF01894681, MR 0224509
de Carvalho, Marcelo H.; Lucchesi, Cláudio L.; Murty, U. S. R. (2002), "On a conjecture of Lovász concerning bricks. II. Bricks of finite characteristic", Journal of Combinatorial Theory, Series B, 85 (1): 137–180, doi:10.1006/jctb.2001.2092, MR 1900684.

Lakshminarayanan Mahadevan

Lakshminarayanan Mahadevan FRS is a mathematician and scientist of Indian origin, and is currently the Lola England de Valpine Professor of Applied Mathematics, Organismic and Evolutionary Biology and Physics at Harvard University. His work centers around using mathematics to understand the organization of matter in space and time, i.e. how it is shaped and how it flows, particularly at the scale observable by the unaided senses.

Education

Mahadevan graduated from the Indian Institute of Technology, Madras, and then received an M.S from the University of Texas at Austin, and an M.S. and Ph.D. from Stanford University in 1995.

Career and research

He started his independent career on the faculty at the Massachusetts Institute of Technology in 1996. In 2000, he was elected the inaugural Schlumberger Professor of Complex Physical Systems in the Department of Applied Mathematics and Theoretical Physics, and a Professorial Fellow of Trinity College, Cambridge, University of Cambridge, the first Indian to be appointed Professor to the Faculty of Mathematics there. He moved to Harvard in 2003.

Awards

2016 Elected Fellow of the Royal Society [1]
2009 MacArthur Fellows Program [2]
2006 Guggenheim Fellowship [3]
2007 Ig Nobel Prize for physics[4][5]
2006 George Ledlie Prize

Kapil Hari Paranjape

Kapil Hari Paranjape (कपिल हरी परांजपे) is an Indian mathematician specializing in algebraic geometry. He is a Professor of Mathematics at the Indian Institute of Science Education and Research, Mohali.

Biography

He was born in Mumbai, Maharashtra near the Kabootar Khana in Dadar but grew up in New Delhi. He completed his schooling from Sardar Patel Vidyalaya in 1977. He joined Indian Institute of Technology Kanpur where he pursued a five-year integrated master’s programme in Mathematics and graduate with first class distinction in 1982.

He joined School of Mathematics, Tata Institute of Fundamental Research as was awarded his PhD Mathematics in 1992.

Paranjape is also involved in the promotion of Linux and GNU [1] and writes a blog Mast Kalandar.[2]

Career

He worked as a Reader at TIFR from 1993-1998. During this he also held various visiting positions at University of Chicago, University of Paris-Sud and University of Warwick. He was appointed as Professor at the Theoretical Statistics and Mathematics Unit of the Indian Statistical Institute, Bangalore. He moved to Institute of Mathematical Sciences, Chennai in 1996. Between 2001 and 2009 he has held visiting positions at California Institute of Technology. Since 2009 he is a professor of Mathematics at Indian Institute of Science Education and Research, Mohali.

Awards/Honors

He was awarded the Shanti Swarup Bhatnagar Prize for Science and Technology in 2005, the highest science award in India, in the mathematical sciences category. His citation read "Dr Paranjape has made outstanding contributions in the field of algebraic geometry, especially the theory of algebraic cycles. He has made highly significant contributions in connecting Hodge Theory to the study of Chow Groups. He has also established deep relations between Calabi-Yau varieties and modular forms.".[3][4] He is also a recipient of various other honors, among them are

Fellowship of Indian Academy of Sciences Bangalore, 1997
Fellowship of National Academy of Sciences Allahabad, 1999
Associate of the ICTP, Trieste, 1999-2001
B. M. Birla award for young scientists, 1999
DST Swarnajayanti Grant, 2001
NBHM National Lecturer, India, 2004-2005
Debian Developer, 2007
Fellowship of Indian National Science Academy, New Delhi, 2009
J C Bose Fellowship, 2010

Vijay Vazirani

Vijay Virkumar Vazirani (Hindi: विजय वीरकुमार वज़ीरानी; b. 1957[1]) is an Indian American distinguished professor of computer science in the Donald Bren School of Information and Computer Sciences at the University of California, Irvine.

Education and career

Vazirani received his Bachelor's degree from MIT in 1979 and his Ph.D. from the University of California, Berkeley in 1983.His dissertation, Maximum Matchings without Blossoms, was supervised by Manuel Blum.[2] After postdoctoral research with Michael O. Rabin and Leslie Valiant at Harvard University, he joined the faculty at Cornell University in 1984. He moved to the Indian Institute of Technology, Delhi as a full professor in 1990, and moved again to the Georgia Institute of Technology in 1995. He was also a McKay Visiting Professor at the University of California, Berkeley, and a Distinguished SISL Visitor at the Social and Information Sciences Laboratory at the California Institute of Technology. In 2017 he moved to the University of California, Irvine as a distinguished professor.

Research

Vazirani's research career has been centered around the design of algorithms, together with work on computational complexity theory, cryptography, and algorithmic game theory.

During the 1980s, he made seminal contributions to the classical maximum matching problem,[3] and some key contributions to computational complexity theory, e.g., the isolation lemma and the Valiant-Vazirani theorem. During the 1990s he worked mostly on approximation algorithms, championing the primal-dual schema, which he applied to problems arising in network design, facility location and web caching, and clustering. In July 2001 he published what is widely regarded as the definitive book on approximation algorithms (Springer-Verlag, Berlin). Since 2002, he has been at the forefront of the effort to understand the computability of market equilibria, with an extensive body of work on the topic.

Two of his most significant research results were proving, along with Leslie Valiant, that if UNIQUE-SAT is in P, then NP = RP (Valiant–Vazirani theorem), and obtaining in 1980, along with Silvio Micali, an algorithm for finding maximum matchings in general graphs; the latter is still the most efficient known algorithm for the problem.

Awards and honors

In 2005 both Vazirani and his brother Umesh Vazirani (also a theoretical computer scientist, at the University of California, Berkeley) were inducted as Fellows of the Association for Computing Machinery.[4][5] In 2011, he was awarded a Guggenheim Fellowship.

Umesh Vazirani

Umesh Virkumar Vazirani (Hindi: उमेश वीरकुमार वज़ीरानी) is the Roger A. Strauch Professor of Electrical Engineering and Computer Science at the University of California, Berkeley, and the director of the Berkeley Quantum Computation Center. His research interests lie primarily in quantum computing. He is also the author of a textbook on algorithms.

Biography

Vazirani received a BS from MIT in 1981[1] and received his Ph.D. in 1986 from UC Berkeley under the supervision of Manuel Blum.[2]

He is the brother of Georgia Tech College of Computing professor Vijay Vazirani.

Research

Vazirani is one of the founders of the field of quantum computing. His 1993 paper with his student Ethan Bernstein on quantum complexity theory [3] defined a model of quantum Turing machines which was amenable to complexity based analysis. This paper also gave an algorithm for the quantum Fourier transform, which was then used by Peter Shor within a year in his celebrated quantum algorithm for factoring integers.

Awards and honors

In 2005 both Vazirani and his brother were inducted as Fellows of the Association for Computing Machinery, Umesh for “contributions to theoretical computer science and quantum computation”[4] and his brother Vijay for his work on approximation algorithms.[5] Vazirani was awarded the Fulkerson Prize for 2012 for his work on improving the approximation ratio for graph separators and related problems (jointly with Satish Rao and Sanjeev Arora).

Mahan Maharaj

Mahan Mj (Bengali: মহান মহারাজ)), (born Mahan Mitra (Bengali: মহান মিত্র), 5 April 1968 [1]), also known as Mahan Maharaj and Swami Vidyanathananda (Bengali: স্বামী বিদ্যানাথানন্দ), is an Indian mathematician and monk of the Ramakrishna Order. He is a recipient of the 2011 Shanti Swarup Bhatnagar Award in Mathematical Sciences.[2][3] and the Infosys Prize 2015 for Mathematical Sciences.[4] He is best known for his work in hyperbolic geometry, geometric group theory, low-dimensional topology and complex geometry.

Early education

Mahan Mitra studied at St. Xavier's Collegiate School, Calcutta, till Class XII. He then entered the Indian Institute of Technology Kanpur, with an AIR (All India Rank) rank of 67 in the Joint Entrance Examination, where he initially chose to study electrical engineering but later switched to mathematics. He graduated with a Masters in mathematics from IIT Kanpur in 1992.[5]

Career

Mahan Mitra joined the PhD program in mathematics at University of California, Berkeley with Andrew Casson as his advisor.[6] He received the Earle C. Anthony Fellowship, U.C. Berkeley in 1992–1993 and the prestigious Sloan Fellowship for 1996–1997.[5] After earning a doctorate from the University of California at Berkeley in 1997, he worked briefly at the Institute of Mathematical Sciences in 1998. Spiritually inclined, he joined the Ramakrishna Math as a renunciate upon being impressed by the life and work of the Vedantic philosopher Ramakrishna Paramahansa.[5] His initial name was Brahmachari BrahmaChaitanya. He was renamed as Swami Vidyanathananda after receiving his saffron robe in 2008.[7] Swami Vidyanathananda is a monk at the order's headquarters at Belur Math. He was Professor of Mathematics and Dean of Research at the Ramakrishna Mission Vivekananda University till 2015. He is currently Professor of Mathematics at Tata Institute of Fundamental Research Mumbai .[8]

He has widely published and presented his research in the area of hyperbolic manifolds and ending lamination spaces. His most notable work is the proof of existence of Cannon–Thurston maps,.[9][10] This led to the resolution of the conjecture that connected limit sets of finitely generated Kleinian groups are locally connected.[5] He is also the author of a book titled Maps on boundaries of hyperbolic metric spaces.[11]

Personality

Mahan Maharaj, as he is known to his students and colleagues, is fluent in English, Hindi and Bengali. He also knows a bit of Tamil, learnt from his stay in southern part of India at IMSc. He has been quoted as saying "I am enjoying being a monk as much as I enjoy my mathematics".

Santosh Vempala

Santosh Vempala (born 18 October 1971) is a prominent computer scientist. He is a Distinguished Professor of Computer Science at the Georgia Institute of Technology. His main work has been in the area of Theoretical Computer Science.

Biography

Vempala attended Carnegie Mellon University, where he received his Ph.D. in 1997 under professor Avrim Blum.[3]

In 1997, he was awarded a Miller Fellowship at Berkeley. Subsequently, he was a Professor at MIT in the Mathematics Department, until he moved to Georgia Tech in 2006.

Work

His main work has been in the area of theoretical computer science, with particular activity in the fields of algorithms, randomized algorithms, computational geometry, and computational learning theory, including the authorship of books on random projection[1] and spectral methods.[2]

In 2008, he co-founded the Computing for Good (C4G)[4] program at Georgia Tech.

Honors and awards

Vempala has received numerous awards, including a Guggenheim Fellowship, Sloan Fellowship, and being listed in Georgia Trend's 40 under 40.[5] He was named Fellow of ACM"For contributions to algorithms for convex sets and probability distributions" in 2015.

Kannan Soundararajan

Kannan Soundararajan is a mathematician and a professor of mathematics at Stanford University. Before moving to Stanford in 2006, he was a faculty member at University of Michigan where he pursued his undergraduate studies. His main research interest is in analytic number theory, particularly in the subfields of automorphic L-functions, and multiplicative number theory.

Early life

Soundararajan grew up in Chennai and was a student at Padma Seshadri High School in Nungambakkam in Madras (now Chennai), India. In 1989, he attended the prestigious Research Science Institute. He represented India at the International Mathematical Olympiadin 1991 and won a Silver Medal.

Education

Soundararajan joined the University of Michigan, Ann Arbor, in 1991 for undergraduate studies, and graduated with highest honours in 1995. Soundararajan won the inaugural Morgan Prize in 1995 for his work in analytic number theory whilst an undergraduate at the University of Michigan,[1] where he later served as professor. He joined Princeton University in 1995 and did his Ph.D under the guidance of Professor Peter Sarnak.

Career

After his Ph.D. he received the first five-year fellowship from the American Institute of Mathematics, and held positions at Princeton University, the Institute for Advanced Study, and the University of Michigan. He moved to Stanford University in 2006 where he is currently a Professor of Mathematics and the Director of the Mathematics Research Center (MRC) at Stanford.

Work

He proved a conjecture of Ron Graham in combinatorial number theory jointly with Ramachandran Balasubramanian. He made important contributions in settling the arithmetic Quantum Unique Ergodicity conjecture for Maass wave forms and modular forms.

Awards

He received the Salem Prize in 2003 "for contributions to the area of Dirichlet L-functions and related character sums". In 2005, he won the $10,000 SASTRA Ramanujan Prize, shared with Manjul Bhargava, awarded by SASTRA in Thanjavur, India, for his outstanding contributions to number theory.[2] In 2011, he was awarded the Infosys science foundation prize 2011.[3] He was awarded the Ostrowski prize[4] in 2011, shared with lb Madsen and David Preiss, for a cornucopia of fundamental results in the last five years to go along with his brilliant earlier work.

He gave an invited talk at the International Congress of Mathematicians in 2010, on the topic of "Number Theory".[5] He was elected to the 2018 class of fellows of the American Mathematical Society.[6]

Personal life

Soundararajan resides in Palo Alto, California with his wife and one son.

Kiran Sridhara Kedlaya

(/ˈkɪrən ˈʃriːdər kɛdˈlɑːjə/;[2] born July 1974) is an Indian American mathematician. He currently is a Professor of Mathematics and the Stefan E. Warschawski Chair in Mathematics[3] at the University of California, San Diego.

At age 16, Kedlaya won a gold medal at the International Mathematics Olympiad,[4] and would later win a silver and another gold medal. While an undergraduate student at Harvard, he was a three-time Putnam Fellow. A 1996 article by The Harvard Crimsondescribed him as "the best college-age student in math in the United States".[5]

Kedlaya was runner-up for the 1995 Morgan Prize, for a paper[6] in which he substantially improved on results of Babai and Sós(1985)[7] on the size of the largest product-free subset of a finite group of order n.

He gave an invited talk at the International Congress of Mathematicians in 2010, on the topic of "Number Theory".[8]

In 2012 he became a fellow of the American Mathematical Society.[9]

He was also a contestant on the game show Jeopardy! in 2011, winning one episode.[10]

Selected works

p-adic Differential Equations, Cambridge Studies in Advanced Mathematics, Band 125, Cambridge University Press 2010[11]
with David Savitt, Dinesh Thakur, Matt Baker, Brian Conrad, Samit Dasgupta, Jeremy Teitelbaum p-adic Geometry, Lectures from the 2007 Arizona Winter School, American Mathematical Society 2008
with Bjorn Poonen, Ravi Vakil The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions and Commentary, Mathematical Association of America, 2002

Ritabrata Munshi

Ritabrata Munshi (born 14 September 1976) is an Indian mathematician specialising in number theory. He was awarded the Shanti Swarup Bhatnagar Prize for Science and Technology, the highest science award in India, for the year 2015 in mathematical science category.[1] He is affiliated to Tata Institute of Fundamental Research, Mumbai, and the Indian Statistical Institute, Kolkata. Munshi obtained PhD degree from Princeton University in 2006 under the guidance of Andrew John Wiles.[2]

Munshi was awarded the Swarna-Jayanti fellowship by the Department of Science and Technology, Government of India in 2012. He also received the B.M. Birla Science prize in 2013, and was elected a fellow of the Indian Academy of Sciences in 2016. For his outstanding contributions to analytic aspects of number theory, he was awarded the Infosys Prize 2017 in Mathematical Sciences[3].

He serves in the editorial board of The Journal of the Ramanujan Mathematical Society and the Hardy-Ramanujan journal.

Subhash Khot

Subhash Khot FRS (born June 10, 1978) is an Indian-American mathematician and theoretical computer scientist who is the Julius Silver Professor of Computer Science in the Courant Institute of Mathematical Sciences at New York University. Khot's unexpected and original contributions are providing critical insight into unresolved problems in the field of computational complexity. He is best known for his unique games conjecture.[1]

Khot was awarded the 2014 Rolf Nevanlinna Prize by the International Mathematical Union. He received the MacArthur Fellowship in 2016 [2] and was elected a Fellow of the Royal Society in 2017.[3]

Education

Khot obtained his bachelor's degree in computer science from the Indian Institute of Technology Bombay in 1999.

He received his doctorate degree in computer science from Princeton University in 2003 under the supervision of Sanjeev Arora. He also received an honorable mention in the ACM doctoral dissertation award in 2003 for his dissertation, "New Techniques for Probabilistically Checkable Proofs and Inapproximability Results."[4]

Honours and awards

Khot is a two time silver medalist representing India at the International Mathematical Olympiad in the years 1994 and 1995.[5][6]

In 1995, Khot topped the prestigious Indian Institute of Technology Joint Entrance Examination.[7][8]

In 2005, he received the Microsoft Research New Faculty Fellowship Award.[9] The fellowship recognizes innovative, promising new faculty members who are exploring breakthrough, high-impact research that has the potential to help solve some of today’s most challenging societal problems.[10]

In 2010, Khot received the prestigious Alan T. Waterman Award, which recognizes an early career scientist for their outstanding contributions in their respective field.[11] The National Science Foundation citation for the Waterman award states: "For unexpected and original contributions to computational complexity, notably the Unique Games Conjecture, and the resulting rich connections and consequences in optimization, computer science and mathematics".[12]

Khot gave an invited talk at the International Congress of Mathematicians in 2010, on the topic of "Mathematical Aspects of Computer Science".[13]

Khot was awarded the 2014 Rolf Nevanlinna Prize by the International Mathematical Union, for his work related to the Unique Games Conjecture, as well as for posing the conjecture itself. According to the International Mathematical Union citation,[14] "he is awarded the Nevanlinna Prize for his prescient definition of the “Unique Games” problem, and leading the effort to understand its complexity and its pivotal role in the study of efficient approximation of optimization problems; his work has led to breakthroughs in algorithmic design and approximation hardness, and to new exciting interactions between computational complexity, analysis and geometry".

Khot received the MacArthur Fellowship (or "Genius Grant") in 2016. The MacArthur foundation states that these are "unrestricted fellowships to talented individuals who have shown extraordinary originality and dedication in their creative pursuits and a marked capacity for self-direction".[15]

He was elected a Fellow of the Royal Society in 2017.[16] Fellows are elected based on having made "a substantial contribution to the improvement of natural knowledge, including mathematics, engineering science and medical science".

Sourav Chatterjee

Sourav Chatterjee (born November 1979)[1] is a mathematician, specializing in mathematical statistics and probability theory. Chatterjee is credited with work on Stein's method on spin glasses and also the Universality of Lindeberg principle. For these achievements, he was awarded a Sloan Fellowship in 2007 from the Alfred P. Sloan Foundation [2] and the Tweedie New Researcher Award in 2008 from the Institute of Mathematical Statistics.

Career

Chatterjee received a Bachelor and Master of Statistics from Indian Statistical Institute, Kolkata, and a Ph.D from Stanford University in 2005, where he worked under the supervision of Persi Diaconis.[4] Chatterjee joined University of California, Berkeley, as a Visiting Assistant Professor, then received a tenure-track Assistant Professor position in 2006. In July 2009 he became an Associate Professorof Statistics and Mathematics at University of California, Berkeley. Then in September 2009, Chatterjee became an Associate Professor of Mathematics at the Courant Institute of Mathematical Sciences, New York University.[5] He spent the academic year 2012-2013 as a Visiting Associate Professor of Mathematics and Statistics at Stanford University. Since autumn 2013 he has joined the faculty of Stanford University as a full professor with joint appointments in the departments of Mathematics and Statistics.[6]

He is also an associate editor of Annals of Probability, since January 2009, and Annales de l'Institut Henri Poincaré (B) Probabilities et Statistiques, since January 2008.[5][7][8]

Achievements

2008 Tweedie New Researcher Award, from the Institute of Mathematical Statistics.
Sloan Research Fellowship in Mathematics, 2007-2009.
Rollo Davidson Prize 2010.
IMS Medallion Lecture, 2012.[9]
Inaugural Wolfgang Doeblin Prize in Probability, 2012.
Loeve Prize 2013.
ICM Invited talk, 2014.

Neena Gupta (mathematician)

Neena Gupta is an Assistant Professor at the Statistics and Mathematics Unit of the Indian Statistical Institute (ISI), Kolkata. Her primary fields of interest are commutative algebra and affine algebraic geometry. Gupta was previously a visiting scientist at the ISI and a visiting fellow at the Tata Institute of Fundamental Research (TIFR).

Gupta graduated with honours in Mathematics from Bethune College in 2006. She earned her Post Graduation in Mathematics from the ISI in 2008[1] and subsequently, her Ph.D. degree in 2011 with algebraic geometry as her specialization, under the guidance of Professor Amartya Kumar Dutta.[2] Gupta's Ph.D. dissertation was "Some Results on Laurent Polynomial Fibrations and Quasi A* Algebras"[3]

Gupta received the Indian National Science Academy Young Scientist award in 2014[4] for the solution she proposed to the ZariskiCancellation Problem[5] in positive characteristic. Her work on the conjecture had also earned her the inaugural Saraswathi Cowsik Medal in 2013, awarded by the TIFR Alumni Association.

Positions held

Assistant Professor at Statistical and Mathematics Unit (SMU), ISI Kolkata (Jun 2014 -)
INSPIRE Faculty at ISI Kolkata (Dec 2012 - Jun 2014)[7]
Visiting Fellow at TIFR Mumbai (May 2012 - Dec 2012)
Visiting Scientist at ISI Kolkata (Feb 2012 - Apr 2012)
Shyama Prasad MukherjeeResearch Fellow at ISI Kolkata (Sep 2008 - Feb 2012)

Awards and honours

M. Birla Science Prize in Mathematics (2017)[8]
The Swarna Jayanti Fellowship Award, Department of Science and Technology (India)(2015)[9]
The inaugural Professor A. K. Agarwal Award for best research publication by the Indian Mathematical Society(2014)[10]
The INSA Young Scientist Award (2014)[4]
The Ramanujan Prize from the University of Madras(2014)[11]
Associateship of the Indian Academy of Sciences (2013)[12]
The Saraswathi Cowsik Medal by the TIFR Alumni Association for her work on the Zariski Cancellation Problem in positive characteristic (2013)[13]
INSPIRE Faculty Fellowship Award (2012)[7]
Shyama Prasad Mukherjee fellowship, the Council of Scientific and Industrial Research(2008)
PC Panesar Gold Medal for Outstanding Performance in the Masters program in Mathematics, ISI (2008)
The distinction of standing first also earned her six awards from Bethune College in 2006 (PC Chandra Award for overall excellence, Sushama Basu Memorial Medal for the highest marks amongst all science graduates, Shantilata Basu Medal for standing first amongst mathematics graduates, the GC and Pratima Das Medal for standing first in B.Sc. (Hons) and the Nalini Das Medal for Most Promising Student)
First position in B.Sc. (Hons) in Mathematics in the University of Calcutta (2006)[1]

Publications

With S.M. Bhatwadekar and S. Lokhande, Some K-theoretic properties of the kernel of a locally nilpotent derivation on k[X1, . . . , X4], Transactions of American Mathematical Society, 2017[14]
With A. K. Dutta, The Epimorphism Theorem and its generalisations, Journal of Algebra and its Applications, (special issue in honour of Prof. Shreeram S. Abhyankar), 14(9) (2015) 15400101–30[15]
With S.M. Bhatwadekar, A Note on the Cancellation property of k[X, Y ], Journal of Algebra and its Applications, (special issue in honour of Prof. Shreeram S. Abhyankar)14(9) (2015)15400071–5[16]
A Survey on Zariski Cancellation Problem, Indian Journal of Pure and Applied Mathematics, December 2015[17]
On Faithfully Flat Fibrations by a Punctured Line, Journal of Algebra, October 2014[18]
On the family of affine threefolds x^m y= F(x, z, t), Compositio Mathematica, June 2014[19]
With Dhvanita R. Rao, On the non-injectivity of the Vaserstein symbol in dimension three, Journal of Algebra, February 2014
On Zariski's Cancellation Problem in Positive Characteristic, Advances in Mathematics, September 2013
A Counter-example to the Cancellation Problem for the Affine Space A^3 in Characteristic p, Inventiones Mathematicae, August 2012
With Amartya K. Dutta and Nobuharu Onoda, Some Patching Results on Algebras over Two-dimensional Factorial Domains, Journal of Pure and Applied Algebra, July 2012
With S.M. Bhatwadekar, The Structure of a Laurent Polynomial Fibration in n Variables, Journal of Algebra, March 2012
With S.M. Bhatwadekar, On Locally Quasi A ∗ Algebras in Codimension-one over a Noetherian Normal Domain, Journal of Pure and Applied Algebra, September 2011

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Baudhayana

Baudhāyana Sulbasūtra

Pythagorean theorem

Circling the square

Square root of 2

Kātyāyana(c. 3rd century BC)

was a Sanskrit grammarian, mathematician and Vedic priest who lived in ancient India.

Works

Views

Pingala

Combinatorics

Panini:

Father of linguistics.The history of linguistics begins not with Plato or Aristotle, but with the Indian grammarian Panini.

Date and context

Biography and location

Legends and later reception

Aṣṭādhyāyī

Rules

List of IT markers

Auxiliary texts

Shiva Sutras

Dhatupatha

Ganapatha

Commentary

Editions

Bhaṭṭikāvya

Modern linguistics

De Saussure

Leonard Bloomfield

Comparison with modern formal systems

Other works

Āryabhatta

Academic work

Biography

Name

Time and place of birth

Other hypothesis

Education

Works

Aryabhatiya

Mathematics

Place value system and zero

Approximation of π

Trigonometry

Indeterminate equations

Algebra

Astronomy

Motions of the solar system

Eclipses

Sidereal periods

Heliocentrism

Legacy

Varāhamihira

Works:

Pancha-Siddhantika

Brihat-Samhita

On Astrology

Influences

Contributions

Trigonometry

Combinatorics

Optics

Yativṛṣabha

Brahmagupta

Life and career

Controversy

Reception

Mathematics

Algebra

Arithmetic

Series

Zero

Diophantine analysis

Pythagorean triples

Pell's equation

Geometry

Brahmagupta's formula

Triangles

Brahmagupta's theorem

Pi

Father of linguistics.
The history of linguistics begins not with Plato or Aristotle, but with the Indian grammarian Panini.