To most people, the words “Indian Mathematics” probably suggest, first, the romantic figure of Srinivasa Ramanujan and then, perhaps, the contributions of the ancients (Aryabhata, Bhaskara, Brahmagupta and so on). To a professional mathematician of today, they might also suggest the Tata Institute of Fundamental Research in Bombay and the many excellent mathematicians it has produced. However, during the period 1900–1950, there were several centres of mathematical activity in India about which people outside India, and even most Indians, know very little.

In what follows, I shall try to describe some of the figures who, often under very difficult circumstances, kept mathematical tradition alive in India. Their work may not be of great interest to a present day mathematician, but it was good work for the time and was very influential in attracting talented people to the subject. I am well aware that many more people merit inclusion than the ones I shall mention, but I hope that despite the shortcomings of my discussion, the general picture that emerges will reflect, at least with some accuracy, the state of affairs during the period 1900–1950.

India’s position in the world of mathematics has become more important since 1950, due largely to the achievements of the School of Mathematics of the Tata Institute. I shall discuss the early days of the Tata Institute since they mark this change. A great deal of research comparable to the best work of the people discussed below was done at the Tata Institute, but I shall pass over this in silence. A full treatment would require an article longer than the present one and it might be difficult for me to be objective.

I shall not discuss scientists who worked for most of their productive lives outside India. Thus, I shall leave out of the discussion some of the best scientists of Indian origin; the most obvious of these are S. Chandrasekhar and Harish-Chandra.

Nor shall I deal with Ramanujan. A great deal has been written about him and his work recently, especially in connection with the “Centennial Year” (Ramanujan was born in 1887). Nevertheless, I believe that Ramanujan still awaits his mathematical biographer. It is very difficult to assess his work and put it in proper context in the mathematics of today.

Most aspects of life in India were controlled by the British in the nineteenth century and the first third of the twentieth; this certainly was the case with education. In 1857, the British established three universities, at Calcutta (which was the capital of India from 1833 to 1912), Bombay and Madras. Their aim was not the encouragement of intellectual activity. Rather, it was to produce a sizeable group of Indian “civil servants,” Indians who would learn the English language and be sufficiently familiar with western ways and thought to be of real assistance to the British in administering territory of the size of the Indian subcontinent. The Civil Service offered the best paid jobs with the greatest security as well as a certain degree of authority over others. That this was attractive to relatively poor Indians who were often part of a large family is certainly understandable. But this system of education had no room for independent thinkers who might dedicate themselves to scientific research. The British did not want any challenge to their authority which independent thinkers might well have made. Thus the educational atmosphere was not conducive to a real academic career as we picture it now. Teaching was so abysmally paid that it could not hope to attract able people. The age of retirement was 55, and active people had to look for other positions when they were perhaps past their prime.

The university system which was set up was not centralised. It was not like the system of rather large centres (often state universities) that existed in Europe or that one finds in the United States today. The system in India was modeled on the University of London (according to Sir Charles Wood’s Education Despatch of 1854), and consisted of a large number of small colleges affiliated to some university; some of these colleges, such as Presidency College in Madras and St. Stephen’s in Delhi, predate the creation of the universities. Thus, when mention is made in what follows of a scientist being at some university, it should be remembered that this often meant that he was at one of the colleges where the teaching was onerous and the opportunity to influence students limited. There were some university positions outside the colleges, but except in Calcutta, the people who occupied them did little teaching, and their influence on the students was, at best, through their research and by example.

There was another way in which the British had a negative effect on mathematics in India. Mathematics in Britain was in a lamentable state in the second half of the nineteenth century. Despite the work of men such as Cayley and Sylvester, the British took almost no part in the great expansion of ideas which was occurring in Germany, France and Italy.

With the emergence of G.H. Hardy (1877–1947) and J.E. Littlewood (1885–1977), real analysis and certain aspects of number theory and of complex analysis were being practised at very high levels. But until the forties, there was no one in Britain who understood the work of, say, Poincare or Hilbert. In his presidential address to the Mathematical Association at London in 1926, Hardy discussed the state of research in the country and characterised it as follows: “Occasional flashes of insight, isolated achievements sufficient to show that the ability is really there, but, for the most part, amateurism, ignorance, incompetence, and triviality.”

The men who came from Britain to India to take charge of the universities were the less successful products of the British system. As far as I am aware, only one real British mathematician accepted a position in India; this was W.H. Young, who was at Calcutta for part of each year from 1913 to 1916, and I shall say something about him in a moment. But the large majority of the mathematicians who came to India did no research of their own, encouraged none in others, and transmitted a constipated view of mathematics to those with whom they came into contact. These same men were also often placed at the head of entire education programs for the provinces (large states into which India was then divided).

**Courtesy** Convergence Portrait Gallery, MAA

W.H. Young began to do serious research in mathematics rather late in life; he and his wife, G.C. Young, who was also an excellent mathematician, began to do fundamental work in real analysis in Göttingen where they lived from 1897 to 1908. Young’s permanent residence was outside Britain from 1897 till his death in 1942 (in Switzerland). Young, Vitali and Lebesgue were led, independently of each other, to define the Lebesgue integral, but it was Lebesgue who proved the main convergence theorems and other basic properties which make this integral indispensable in real analysis. Young’s work on Fourier series (the so-called Young–Hausdorff inequality comes readily to mind) was, in some ways, even more original.

Young was not appreciated in Britain; his candidacy for a professorship at Cambridge was not taken seriously and he accepted a chair at Calcutta in 1913. He spent the winter months of three years at Calcutta, leaving the university in 1916. He was also Professor of the Philosophy and History of Mathematics in Liverpool where he lectured during the summer months. He moved to Aberystwyth as Professor of Pure Mathematics in 1919.

To return to our discussion, I believe that the negative effects of the British system in India were made worse by the Indian mentality. There was a passiveness, not to say fatalism, among even the intellectual elite. However well Indians might follow a trail already blazed, there were few who were ready to take the risks of striking out on their own. One meets this attitude in India even today.

Fortunately, there were exceptions to what I have just said. (Sir) Asutosh Mookerjee (1864–1924), following the lead of Mahendra Lal Sircar, was among the first to propose the idea that a university should be a centre of independent intellectual activity, of research, and of teaching of high quality at all levels. Mookerjee came up through the British system and was appointed a judge of the Calcutta High Court in 1904.

This remarkable man had always been interested in mathematics; he wrote some interesting papers dealing with aspects of plane algebraic curves and of differential equations in the years 1883–1890. He became Vice-Chancellor of Calcutta University in 1906 and began systematically to improve the university. He brought talented people to Calcutta (S.Mukhopadhyay and N.R. Sen among them; see below). He founded the Calcutta Mathematical Society in 1908. He did his best to encourage talent and used the Society to bring the few researchers in India together and to publish their work (in the Bulletin of the Society).

The mathematical work of Asutosh Mookerjee is forgotten. His pursuit of the idea that Indians could, and should, undertake serious scientific investigations, that they should have, and raise, academic standards will not be forgotten so quickly.

Other universities were established from 1882 on.^{1} Some of them began to attract individual mathematicians who did their best to do research and to improve the quality of teaching. In doing this, these men were certainly making personal sacrifices. They were generally able people and could easily have passed the examinations leading to the civil service, which, as already mentioned, meant good pay and an assured future. Instead, they chose to pursue an academic career which was ill paid, involved a great deal of drudgery and where advancement, even for the most talented, was problematic. Their dedication kept mathematical tradition alive and made conditions somewhat better for those who followed.

Among the places where some good work was being done (in the period 1900–1950) were the following: Calcutta (and Dacca), Benares, Lucknow, Allahabad, Aligarh, Punjab, Delhi, Poona and Bombay, Bangalore, Annamalai, Andhra, and Madras. I shall attempt to describe briefly some of the pure and applied mathematicians who functioned in these places. Calcutta and Madras were the most important of them, and I shall take them up at the end.

Before starting, let me try to locate these places geographically. I shall assume that the reader knows where Calcutta, Delhi, Bombay and Madras are (corresponding roughly to the east, north, west and south respectively). Benares, now known as Varanasi, is a holy city on the Ganges river about midway between Calcutta and Delhi. Allahabad is not far from Benares, to the west. Lucknow is some 200 miles northwest of Benares. Aligarh is a little southeast of Delhi. Punjab University was in Lahore which is now in Pakistan, just west of the Indo–Pakistan frontier. It was moved to Chandigarh after the partition of India. This city was designed and built by Jawaharlal Nehru, LeCorbusier, Maxwell Fry and the latter’s wife Jane Drew; because of this, it is better known in the west than other Indian cities of its size. Chandigarh is about 150 miles north of Delhi. Poona is near Bombay, to the southeast. Bangalore is situated some 200 miles west of Madras. Annamalai University is on the outskirts of Chidambaram (an ancient centre of Hindu religion) which is some 100 miles south of Madras. Andhra University is in Waltair on the east coast of India, about 350 miles northeast of Madras.

### Allahabad and Benares

One of the earliest influential figures in Indian mathematics was Ganesh Prasad (1876–1935). After early work in India, he went to Cambridge, England and then to Göttingen and came under the influence of Klein and Hilbert. He wrote some papers on differential geometry, mainly concerning surfaces of constant curvature, soon after his return to India. The bulk of his work is, however, on potential theory and the summability of Fourier series.

After a year or so in Allahabad, he was in Benares from 1905–1914 and 1917–1923. He founded the Benares Mathematical Society. He went to Calcutta in 1923 and finished his career there. He also succeeded Asutosh Mookerjee as President of the Calcutta Mathematical Society.

Despite the fact that a large part of Ganesh Prasad’s work (summability) was in the British tradition, he was one of the first Indian mathematicians to feel continental influence. He was a very powerful figure in mathematical circles. It is a little hard to escape the feeling that he could have had a more positive impact on Indian mathematics than he in fact did.

The career of B.N. Prasad (1899–1966) had some similarities to that of his teacher, Ganesh Prasad. He went to England (Liverpool) and then to Paris. Much of his work deals with the summability of Fourier series. But Prasad was not happy with the British system as it worked in India. He strongly advocated creating schools modeled on the École Normale Supérieure in Paris. He lived to see the Tata Institute flourish, but his hopes of heading an institution of higher learning and research remained unfulfilled.

B.N. Prasad is a clear example of the sacrifices made by able people who went into teaching and research. His work was well thought of by such mathematicians as Titchmarsh and Denjoy. Nevertheless, he remained a lecturer at Allahabad for many years (1932–1946) and was then made a Reader. He became Professor on a temporary basis in 1958; the appointment was made permanent in 1960, shortly before his retirement.

B.N. Prasad also founded the Allahabad Mathematical Society, which publishes its own journal even now. Two others in Benares and Allahabad who must be mentioned are V.V. Narlikar (1908–1991) and A.C. Banerji (1891–1968). Both were applied mathematicians. Narlikar was principally at Benares and was well known, at least in India, for his work on relativity. Banerji taught at Allahabad from 1930 to 1952 and became Vice-Chancellor. He was a very broad-minded man and he worked hard to expose his students to new ideas.

**Courtesy** Indian National Science Academy

### Delhi, Lucknow, Aligarh

Two men who were active at St. Stephen’s College in Delhi were Ram Behari (1897–1981) and P.L. Bhatnagar (1912–1976). Ram Behari’s work was almost entirely in differential geometry. He studied at Cambridge and Dublin and was a student of J.L. Synge. His early work was on classical questions (ruled surfaces and the like). In later years his work, done partly in collaboration with some of his students, became broader and more modern. He moved to the University of Delhi in 1947.

P.L. Bhatnagar worked both in pure and in applied mathematics. He studied with A.C. Banerji and B.N. Prasad. He was at Delhi from 1940 to 1955; he then moved to Bangalore to head the department of applied mathematics^{2} at the Indian Institute of Science. He, too, did some work on the summability of Fourier series, but the major part of his work was in such fields as astrophysics and fluid dynamics. He was also seriously interested in ancient Hindu mathematics. Bhatnagar was very influential in developing applied mathematics in India.

B.R. Seth (1907–1979) too spent several years in Delhi (1937–1949). His work was mainly on questions of elasticity and fluid dynamics. In 1950, he moved to the newly created Indian Institute of Technology at Kharagpur (near Calcutta). He founded something of a school of applied mathematics there.

A.N. Singh (1901–1954) spent most of his professional life at Lucknow. He worked hard to attract students to the subject; for example, he organised a mathematics exhibition regularly in the department in an attempt to make the subject more popular. Most of his work was on differentiability properties of functions of a real variable. He was also deeply interested in the history of Hindu mathematics, and even wrote a book on the subject.

Lucknow also had R.S. Varma (1905–1970) on its faculty from 1938 to 1946. He worked on classical analytic questions (integral transforms, special functions) till he moved to Delhi. He worked for the Defence Science Organisation from 1947 to 1962 and dealt with problems of ballistics and operations research.

The Muslim University at Aligarh had a very famous mathematician on its faculty from 1930 to 1932: André Weil. Sylvain Uvi, the well known Indologist, was approached by Syed Ross Masood (to whom E.M. Forster’s *A Passage to India* is dedicated). Masood asked Levi if he could recommend someone who was expert in Romance languages and in French culture and history. Weil was very interested in India and its culture, he knew Levi, and was willing to spend some time in India. Masood had become Vice-Chancellor at Aligarh in October, 1929 and offered Weil a Chair in mathematics as being the only one available. Thus, Weil came to India.

At Aligarh, Weil came into contact with T.Vijayaraghavan and D.D. Kosambi; I shall discuss them later. He had a powerful impact on Vijayaraghavan, and helped to broaden his view of mathematics (as Vijayaraghavan himself told C.P. Ramanujam and me more than once).

Weil clearly understood what needed to be done in the colleges to improve the quality of the students and to make them better prepared for research. His outline of the lecture on this subject that he delivered to the Indian Mathematical Society in 1931 is printed in vol.1 of his collected works. It is sad that Weil’s remarks about the steps that need to be taken apply with as much force today as when he made them.

The three men mentioned above were at Aligarh for rather short periods. S.M. Shah (born 1905) was a junior colleague of Weil’s in 1930 and remained in Aligarh till 1958. He then left for the United States, where he has been since.^{3}

Shah worked extensively on functions of a complex variable, in particular on entire functions. His work deals largely with the growth properties of entire functions and their Taylor coefficients. He and Ganapathy Iyer were among the few Indians who were aware of the revolutionary ideas of Rolf Nevanlinna concerning meromorphic functions and Picard’s theorem.

### Punjab

There are three well known number theorists to come out of Punjab.

S.Chowla (born 1907) had his early training at Lahore. He then studied in Cambridge and came under the influence of Littlewood. He taught for a while at Benares and at Andhra University in Waltair.^{4} He then became a professor at Lahore. After the partition of India, he went to the Institute for Advanced Study in Princeton in 1948 as a visiting member. He held professorships in Kansas, Colorado and Penn State, retired in 1976, and lives now in the United States.^{5}

Chowla’s work was very extensive. He obtained very interesting results in analytic number theory (L-functions, Waring’s problem and so on). He also worked on combinatorial problems. The best known of his results is part of work he did in collaboration with Atle Selberg on the so-called Epstein zeta function. It is of importance in the study of the class number of imaginary quadratic fields, and has led to further important work, for instance, by B.Gross.

Chowla was also very active in encouraging and promoting students. His best Indian student was R.P. Bambah (born 1925), who has done very interesting work on the geometry of numbers. Bambah has spent most of his life in Punjab; he is at present Vice-Chancellor of the university at Chandigarh.^{6}

**Courtesy** Indian National Science Academy

The other number theorist from the Punjab referred to above is Hansraj Gupta (1902–1988). Gupta was profilic, and has written on several aspects of number theory. The most important part of his work is on partitions. He did, for example, some numerical work extending tables of the number of partitions made by P.A. MacMahon which has proved to be very useful.

### Bombay, Poona, Bangalore, Annamalai

The University of Bombay had no research department in mathematics, at least until rather recently (although it did in, for example, chemistry). However, D.D. Kosambi (1907–1966) was active at nearby Poona. He was at Fergusson College from 1932 to 1946 before he moved to the Tata Institute. He had spent a few years at Benares and at Aligarh before 1932.

His purely mathematical work was in differential geometry, in particular, on path spaces which are generalisations of spaces with an affine connection. He had studied at Harvard at both the school and the college level, and his view of mathematics was much broader than was common in India at the time.

He also became very interested in statistics, and in applying its methods to fields such as numismatics and Sanskrit literature, but particularly to Indian history; these subjects became his main interests. He isolated himself almost completely from other mathematicians when he was at the Tata Institute. Despite his incisive intelligence, his influence on Indian mathematics was small.

The Indian Institute of Science was founded in Bangalore by Jamshedji Tata, one of the great industrialists of India. (Sir) C.V. Raman (1888–1970) spent some years (1933–1947) at this Institute before moving to an institute named after himself. Raman had resigned a civil service position to pursue science in Calcutta. Asutosh Mookerjee appointed him to a professorship in physics in 1917; he held the position till 1933. It was here that he did the work in optics, partly in collaboration with K.S. Krishnan (1898–1961), for which he received the Nobel Prize. He and Krishnan were very active in many of the efforts being made to improve teaching and research in the sciences, including mathematics. It might be added that Mookerjee was also responsible for bringing Krishnan to Calcutta.

Another physicist who was to have a profound influence on the development of mathematics in India also spent some years at the Indian Institute of Science in Bangalore. This was Homi J. Bhabha (1909–1966). He had been working in Cambridge with Dirac and was in India in 1939, meaning to return to Cambridge, when the second world war broke out. He went to Bangalore, and while he was there, he founded the Tata Institute of Fundamental Research (TIFR) in Bombay. Bhabha was prophetic in his recognition of the inevitability of the development of nuclear energy. He wanted to assure the existence of Indians sufficiently close to the cutting edge of modern work in physics to take advantage of this development. He also realised that this required supporting mathematical research at the highest level. With financial help, again, from the Tata family and from the provincial government of Bombay, TIFR was created in 1945.

B.S. Madhava Rao (1900–1987) spent much of his professional life at Central College in Bangalore. His own work was in applied mathematics, but his view of the subject was very enlightened and he encouraged students to work in a wide variety of fields. He sent several of his students to TIFR in the late fifties, some of whom have done outstanding work. When he retired, he went to the Defence Science Organisation in Poona where he is reported to have been exceptionally effective.

A.Narasinga Rao (1893–1967) was at Annamalai University from 1929 to 1946. He moved to Andhra University in 1946, and to the Indian Institute of Technology at Madras in 1951. He worked mainly on aspects of Euclidean geometry, but was also interested in other fields (e.g. aerodynamics). He did his best to encourage young mathematicians. He was responsible for attracting Ganapathy Iyer and S.S. Pillai to Annamalai. He was the first editor of *The Mathematics Student*, a journal (founded by the Indian Mathematical Society) to which mathematics teachers and students could send their contributions. (He also instituted a medal, named after himself, awarded through the Indian Mathematical Society to outstanding young people.)

V.Ganapathy Iyer (1906–1987) too was a professor at Annamalai. He was an excellent mathematician, and wrote papers on many aspects of both real and complex analysis. His best work was on the theory of entire functions; in some of this work he applied methods from functional analysis to spaces of entire functions. As mentioned in connection with S.M. Shah, he was among the few in India who understood Nevanlinna theory. Given the high level of his accomplishments, it is rather surprising that he did not have more of an impact on the younger generation.

### Calcutta and Madras

Calcutta had the first, and for a long time, the most important university in India. There was always a strong tradition of intellectual pursuit in Bengal. The poet Rabindranath Tagore, the botanist Jagdish Chandra Bose and the political leader Subhash Chandra Bose all came from Bengal. Among scientists, one of the best known is S.N. Bose (1894–1974) who worked at Calcutta and Dacca; K.S. Krishnan and T. Vijayaraghavan were colleagues of his at Dacca. He was a long-standing member of the Calcutta Mathematical Society. He is best known for his discovery of the so-called Bose–Einstein statistics which describes the statistical behaviour of certain types of particles. (Particles with certain spin characteristics are now called bosons in his honour.)

Calcutta played host to many distinguished scientists. I have already mentioned W.H. Young, C.V. Raman, and K.S. Krishnan who all spent fruitful years at Calcutta; there were many others.

Of the mathematicians and statisticians who were more permanently associated with Calcutta, I shall cite only a few. They are: S.Mukhopadhyay (1866–1937), N.R. Sen (1894–1963), R.N. Sen (1896–1974), P.C. Mahalanobis (1893–1972), R.C. Bose (1901–1987), and the German mathematician F.W. Levi (1887–1966).

Levi fled Nazi Germany and came to Calcutta as professor of mathematics in 1935 where he remained until 1948. He then retired from Calcutta and moved to the Tata Institute.

Levi played an important part in the acceptance of algebra in university curricula all over India

His influence on mathematics in India was considerable. He introduced Indians to algebra (which used to be called modern algebra). He was an active participant in mathematical meetings. Levi played an important part in the acceptance of algebra in university curricula all over India.

Levi seems to have been less effective when he was at the Tata Institute (he was there only about three years); perhaps a university milieu suited him better. He retired in the early fifties and returned to Germany.

Syamdas Mukhopadhyay was brought to the university in Calcutta by Asutosh Mookerjee to organise the mathematics department. He was clearly an independent man, and was not much influenced by the British view of mathematics. He wrote on the differential geometry of curves, both in the plane and in n-space; he considered global properties of ovals in the plane. His work was much better known in France than at home. Hadamard is supposed to have had a high opinion of his work on plane curves. (Note: The name Mukhopadhyay is derived from Sanskrit; its Bengali equivalent is Mookerjee, with variations in its English spelling. Thus our mathematician is referred to sometimes as Mukherji.)

Nikhilranjan (= N.R. ) Sen too was brought to Calcutta by Asutosh Mookerjee. He worked mainly in applied mathematics and physics (relativity, cosmogony, fluid dynamics) but he also did useful work in potential theory and probability. He was very helpful to younger people. Rabindranath (= R.N. ) Sen came to Calcutta in 1933. His work was mainly on differential geometry, and especially on parallelism.

**Courtesy** Indian National Science Academy

P.C. Mahalanobis was a very influential statistician. He did important theoretical work to which he was led by analysing actual data obtained in the field. This connection between theory and application was important to him and he tried to foster the relationship in the work of his colleagues. He did important statistical work for both the state and the central governments.

Mahalanobis founded the Indian Statistical Institute (ISI) in 1931. He brought excellent scientists to the institute, for instance, R.C. Bose. One of the best statisticians to come out of ISI is C.R. Rao who was at the institute from 1944 to 1979. He then left India to go to the United States; he is currently at Pennsylvania State University.

R.C. Bose is very well known internationally for his work in statistics and combinatorics. He was a student of Mukhopadhyay, and his early work reflects this influence; it is largely on geometric questions, including hyperbolic geometry. Mahalanobis brought him to ISI and induced him to work on statistical questions. Bose responded by applying geometric ideas to statistics.

Bose is perhaps most famous for having disproved a conjecture of Euler on “mutually orthogonal latin squares” in collaboration with his student S.S. Shrikhande (who headed the mathematics department at the University of Bombay during the sixties). Bose went to the United States in 1949, where he remained till his death. He seems to have had more Indian students there than in India.

ISI produced four excellent pure mathematicians V.S. Varadarajan, R.Ranga Rao, K.R. Parthasarathy, and S.R.S. Varadhan. They have done impressive work on measure theory, stochastic differential equations, and especially on the representation theory of Lie groups. This was done on their own initiative.

Madras has never had a centre of mathematics of the quality of the department at Calcutta University. This applies not only to the university itself, but also to two other institutions meant to foster research, the Ramanujan Institute and Matscience. I shall say something about the first of these later. No serious work was produced at the second until recently.

Nevertheless, there was always a strong tradition of scholarship and learning in and around Madras. Certainly, the best pure mathematicians to emerge in India during the period 1900–1950, even leaving Srinivasa Ramanujan out of consideration, came from the Province of Madras. The men I shall now discuss are: K.Ananda Rau (1893–1966),R.Vaidyanathaswamy (1894–1960), T.Vijayaraghavan (1902–1955), S.S. Pillai (1901–1950), and S.Minakshisundaram (1913–1968).

Ananda Rau’s early work was done in Cambridge. He was a student of Hardy, and there is no mistaking the influence of Hardy and Littlewood on this work. Ananda Rau had a critical and independent mind and he was conscious of the vast extent of mathematics and the lack of interest in India in many of its branches. He knew Ramanujan in Cambridge and spoke of him with great admiration, but without a trace of the mysticism or romanticism of many others. On his return to India, he took up a position at Presidency College, where he remained until his retirement. He was in the Indian Educational Service, and was thus better off than most other mathematicians. (The IES was abolished shortly after his appointment.)

Ananda Rau’s first successes concerned summability by general Dirichlet series (including power series). He established very general “tauberian theorems” which had resisted the efforts of some of the best analysts. One of the methods of summation to which he contributed some of his most original ideas, Lambert summability as it is called, is intimately connected with the distribution of prime numbers. In later years, Ananda Rau was occupied with modular functions and the representation of integers as sums of squares. It has always seemed to me a great pity that neither Ramanujan nor Ananda Rau ever came into contact with someone like Erich Hecke when they were young. Hecke was a master at combining hard analysis with the arithmetic behaviour of modular (and automorphic) functions, exactly the two topics central in the work of these two Indians.

Ananda Rau had a strong influence on his students. He taught them, besides mathematics, the value of intellectual independence

Ananda Rau’s work is of great depth and elegance. He had a strong influence on his students: Vijayaraghavan, Pillai, Ganapathy Iyer, Minakshisundaram, Chandrasekharan and others. He taught them, besides mathematics, the value of intellectual independence.

Vaidyanathaswamy studied in Edinburgh with E.T. Whittaker, then visited Cambridge and worked with the geometer, H.F. Baker. When he returned to India, he was something very new on the scene. He was greatly interested in “structures”. He emphasised general principles over special results, however deep. Even in his work on properties of plane algebraic curves, he was fascinated by general properties of homogeneous forms and of birational transformations. He also worked on number theory, but was more interested in the general behaviour of multiplicative arithmetical functions than in special properties of even the most important among them.

Vaidyanathaswamy was the first Indian to study mathematical logic, set theory and general topology; he even wrote a book on this last subject. He was very influential. For over two decades, he was the motive force behind the Indian Mathematical Society. Even if one cannot point in his work to individual results of the depth of those obtained by Ananda Rau, his opening up of a different view of mathematics was most beneficial. He avoided the danger of empty generalities, just as Ananda Rau focused on special problems that would illustrate general behaviour. He lectured on topics in pure mathematics at the Indian Statistical Institute in Calcutta for about two years after his retirement from Madras.

It was Ananda Rau who recognised Vijayaraghavan’s talents; he even had to intervene to get him admitted to Presidency College. Vijayaraghavan began to work independently even as a student and, following Ramanujan’s example, sent some of his work to Hardy. This led, albeit with some delay, to his proceeding to Oxford to work with Hardy. On his return to India, he was a colleague of André Weil’s for about a year, taught at Dacca (near Calcutta; now in Bangladesh) for a few years and then went to Andhra University.

In 1949, Alagappa Chettiar created the Ramanujan Institute in Madras, and brought Vijayaraghavan there as Director. Based on some fairly extensive contacts with this institute since 1954, my impression was that it existed mainly as a place of work for Vijayaraghavan and his younger colleague, C.T. Rajagopal, at least till the latter’s retirement in 1969. Vijayaraghavan would have liked having at his disposal the funds necessary to bring young researchers and outside scholars to the Ramanujan Institute. In this, he was disappointed. But he was very helpful and kind to the students who did go to him. Thus, for well over a year, up to the time of his death in 1955, Vijayaraghavan received C.P. Ramanujam (about whom more later) and me regularly. He helped us with the material we were trying to study, gave us a few lectures, and encouraged us to work on our own. I need hardly add that these activities were not part of his normal duties at the Institute.

Vijayaraghavan’s work, like that of Ananda Rau, began with an impressive theorem about summability, in this case, Borel summability; the theorem was suggested by work of Ananda Rau as Vijayaraghavan himself has said. He then did a piece of work which is typical of his abilities, disproving a conjecture of E.Borel about the growth of solutions of non-linear ordinary differential equations. This brought him to the attention of the mathematical world; G.D. Birkhoff, for example, was impressed and was instrumental in having Vijayaraghavan invited to the United States as Visiting Lecturer of the American Mathematical Society in 1936.

His third major investigation was the distribution of the fractional parts of the numbers \alpha \cdot \xi^n (\alpha and \xi being fixed real numbers) as n \to \infty. This led him very naturally to a class of numbers which, for a short time, were called Pisot–Vijayaraghavan numbers. The distribution is far from being understood even now, despite major advances in diophantine approximation.

When Vijayaraghavan died, C.T. Rajagopal became the director of the Ramanujan Institute. For many years, the institute housed the library of the Indian Mathematical Society, and Rajagopal was Librarian. Shortly before his retirement, the Ramanujan Institute was absorbed into the University of Madras as an Institute attached to the Department of Mathematics.

S.S. Pillai studied with Ananda Rau. He also came into contact with Vaidyanathaswamy, but his own inclination was towards important specific problems and Ananda Rau’s influence on him was the greater. He was a lecturer at Annamalai University, and for a brief period, at Calcutta where he went because of the interest taken in him by F.W. Levi. He obtained a stipend from the Institute for Advanced Study in Princeton (for the year 1950/51). He was to go to the Institute and, from there, take part in the International Congress of Mathematicians at Harvard. Tragically, the plane carrying him crashed near Cairo.

Pillai’s work on Waring’s problem remains his greatest achievement

The work for which Pillai will always be remembered concerns Waring’s Problem. The question itself is easily understood: Given an integer k \ge 2, what is the smallest integer g(k) with the property that every natural number n can be written as a sum n = x_1^k + \cdots + x_s^k of s k-th powers of natural numbers x_j with s \le g(k)? It is, of course, far from obvious that this can be done at all, viz. that g(k) is finite. A famous classical theorem of Lagrange tells us that g(2)=4.

In 1909, David Hilbert proved the existence of g(k) for all integers k \ge 2. By a remarkable analytic argument, Hilbert first proved a beautiful algebraic identity stated by his friend A.Hurwitz; Hurwitz himself had only been able to check this identity in a few special cases. Hilbert then added an interpolation argument to complete the proof. Hardy and Littlewood, using, in part, techniques introduced by H.Weyl, brought powerful analytic methods to bear on this problem. By improving Weyl’s techniques, I.M. Vinogradov sharpened these results very considerably; his methods have been central in analytic number theory since he introduced them.

But none of these methods enables one to actually compute g(k). What Pillai did was to evaluate g(k), at least when k \ge 7; L.E. Dickson independently obtained very similar, and perhaps slightly stronger results at about the same time. The value depends on the behaviour of the fractional part of (3/2)^k. It should be added that the problem is not completely solved; it is strongly suspected that the behaviour of this fractional part is such as to produce a single simple formula for all values of k \ne 4, but this remains conjectural. Further, the value of g(4) was only determined recently (in 1986) by R.Balasubramanian, J.M. Deshouillers and F.Dress. In principle, g(k) can now be computed for any given value of k.

Pillai wrote many papers on the theory of numbers. Each of them shows analytic power and originality, but his work on Waring’s problem remains his greatest achievement.

Minakshisundaram began as a research student at Madras University. After his doctorate, when he could not find a job, Father Racine (about whom I shall say something later) helped him earn a living coaching students. Minakshisundaram was then appointed Lecturer in Mathematical Physics at Andhra University. Thanks to the interest taken in him by Marshall Stone, he was able to spend a few years (1946–1948) at the Princeton Institute for Advanced Study; he retained his position at Andhra University during this time. When he returned to India, he could still not find a job better suited to him. Although he was promoted to a professorship in Mathematical Physics, circumstances at the university made it impossible for him to strike out on his own. This is another clear example of the sacrifices that were necessary in pursuing a career of intellectual achievement in India.

Minakshisundaram’s early work, influenced by Ananda Rau, was on the summability of Dirichlet series and eigenfunction expansions. Then he met Fr. Racine and M.R. Siddiqui. Siddiqui, who was in Hyderabad (some 300 miles north of Bangalore), had been a student of Lichtenstein in Leipzig and had done good work on the initial value problem for parabolic equations, extending results of Lichtenstein. (He became Vice-Chancellor of the University of the Northwest Frontier Province in Pakistan, and later, President of the Pakistan Academy of Sciences.) Minakshisundaram thus became interested in partial differential equations, and was led to study the zeta-function associated to boundary value problems. This culminated in his fundamental joint work with Å.Pleijel (done when they were both in Princeton) on the eigenvalue problem for the Laplace operator on a compact Riemann manifold. His idea of using the heat equation in this study has proved very fruitful and extremely powerful. It led to work of Atiyah, Bott and Patodi on a new approach to the so-called index theorem for elliptic operators, and is being used even today in what has come to be known as the index theorem for families of elliptic operators.

Minakshisundaram did quite a lot of other work. For example, he and K.Chandrasekharan worked extensively on subtle analytic properties of the so-called Riesz mean and applications to multiple Fourier series. It is sad that Minakshisundaram’s exceptional abilities did not have the impact on Indian mathematics that they should have had.

Fr. C. Racine (1897–1976) came to India in 1937 with a Jesuit Mission. He had studied analysis in Paris with such people as E.Cartan and J.Hadamard. He was a friend of many of the outstanding French mathematicians: Leray, Weil, Delsarte, Lichnerowicz, H.Cartan and others. These men were making fundamental discoveries; they were also making fundamental changes in the way mathematics is taught and learned. In fact, some of them were in the process of creating Bourbaki. Thus Fr. Racine came to India with a view of mathematics completely unlike the prevailing one. In the classroom, but especially in personal contacts outside, he communicated this dynamic view of the subject. He sent students interested in mathematics to the Tata Institute to experience this kind of mathematics for themselves when that institution was beginning to function well.

Fr. Racine was deeply committed to India. He went back to France on his retirement as was expected of him, but returned to Loyola College, Madras to live out his life among the people for whom he had done so much.

C.T. Rajagopal (1903–1978) was mentioned in connection with Vijayaraghavan. He too was a student of Ananda Rau. He taught at Madras Christian College just outside the city till he went to the Ramanujan Institute in 1951. Rajagopal’s work was almost entirely devoted to summability and related questions in analysis, although he wrote some papers on functions of a complex variable. In his last years, inspired by work of K.Balagangadharan, he became interested in the history of mathematics in medieval Kerala (southwestern coastal state of India). Rajagopal’s work, while not as deep as that of Ananda Rau, had some outward similarity to it.

The last person from Madras I shall mention is V.Ramaswami Aiyar (1871–1936). He did some teaching in different colleges and then became a civil servant, but he was a true amateur (in the French as well as the English sense of the word) of mathematics. It was Ramaswami Aiyar who founded the Indian Mathematical Society in 1907, and started publication of its Journal. He worked hard to foster mathematical research in the country. Ramaswami Aiyar was also the first of several Indians to recognise the exceptional talents of Ramanujan before Ramanujan wrote to Hardy; he tried to get modest financial support for Ramanujan to enable him to continue his work, but in this he was only partially successful.

I have not treated Andhra University separately, but it will undoubtedly have been remarked that several of the mathematicians mentioned above spent some time there. This was no accident. Andhra had two very enlightened Vice-Chancellors in S. Radhakrishnan (the philosopher and statesman) and his successor, C.R. Reddy. In acting to help good scientists who were in some difficulty, they followed the advice of C.V. Raman. At Waltair, as at Calcutta, the qualities of the Vice-Chancellor were crucial in enhancing those of the university. The situation in Madras illustrates the opposite side of the coin. But for the prejudices of its long-time Vice-Chancellor Lakshmanaswami Mudaliar, Madras might well have had one of the better research departments in mathematics in the country.

The picture of mathematics in India during 1900–1950 that I have tried to describe is the following. Indians were gradually shaking off the effects of an outmoded system and a narrow view of mathematics. There were a few truly outstanding scientists, and many more good ones, in various centres distributed all over India. They formed scientific societies and published journals which formed a good outlet for work being done in the country. These societies often had an international membership, and foreign members contributed papers to these journals. All this was done under difficult circumstances; for instance, the publication of journals was continued even when the second world war created terrible paper shortages. The contributions of the people I have mentioned, and those of their many colleagues whom I have not, should not be completely forgotten.

It will not have escaped the reader’s notice that all the Indians who have been mentioned so far were men. Social conditions in most parts of the world form one of the main reasons that it is difficult for women to become seriously interested in mathematics at an early age. The conditions are even more difficult in India which has a system of arranged marriages and near ostracism of unmarried or divorced women and widows. Nevertheless, there have been women mathematicians in India. S.Pankajam and K.Padmavally are among the earliest of whom I am aware. The former was active in the forties, the latter in the fifties. They were both students of Vaidyanathaswamy. There have been several women at the Tata Institute. One of the best women mathematicians the country has produced is Bhama Srinivasan, who is currently at the University of Illinois at Chicago. Her work on group representations is well known.

We come now to the Tata Institute of Fundamental Research (TIFR). As stated earlier, the Institute was the brainchild of H.J. Bhabha and was founded in 1945. Originally, it received no financial support from the central government, but this support increased gradually, and the Institute is now financed almost entirely by the Government of India. However the Institute continues to function with a large degree of independence.

Bhabha brought F.W. Levi and D.D. Kosambi to Bombay; neither of them was particularly effective there. Upon consultation with John von Neumann, André Weil, and Hermann Weyl, he approached K.Chandrasekharan in Princeton to try and bring him to Bombay to develop mathematics at TIFR. Bhabha had already heard about Chandrasekharan several years earlier, probably from Kosambi.

Chandrasekharan was a first rate analyst. He was a student of both Ananda Rau and Vaidyanathaswamy; he also came into close contact with Vijayaraghavan later. His early work was on intuitionistic logic and on functions of a complex variable. He then began, on his own initiative, important work on multiple Fourier series and related questions of analysis.

In 1944, Marshall Stone was in Madras and wanted to meet the best young mathematicians there, and especially Chandrasekharan and Minakshisundaram of whom he had heard through Kosambi. With Stone’s help, Chandrasekharan went to the Institute for Advanced Study in Princeton. This was, for him, a turning point. Hermann Weyl had a profound influence on him which, I think, would be difficult to exaggerate. He was, in fact, Weyl’s assistant for one year. He also met, and became friendly with, many of the outstanding figures in the subject; I shall mention only C.L. Siegel, A.Selberg and S.S. Chern as having had the most powerful influence on his thinking. His mathematical output was very extensive. He began a fruitful collaboration with S. Bochner, with whom he studied Fourier series and transforms in several variables, and Dirichlet series.

Bhabha offered Chandrasekharan a position at TIFR with the understanding that he would have Bhabha’s full support in building up the school of mathematics. Chandrasekharan insisted that he must have a free hand in running the school, and to a very large extent, he did.

The school of mathematics at TIFR was not modeled on any existing institution, although I am sure that the Institute in Princeton, and the vast experience of Oswald Veblen and Hermann Weyl provided Chandrasekharan with ideas for what the shape should be. Within a few years of his arrival at TIFR in 1949, he managed to form a sizeable group of people and to create a remarkable atmosphere. Let me make some more comments on this.

Members of the school were free to learn and to do research with no distractions (except of their own making). There were no restrictions on the field of work, no unfashionable subjects. This was specially important to people newly arrived from a restrictive university environment. These newcomers were given a stipend sufficient to cover necessities and encouraged to test their mettle. The older members (those who had been there longer) provided advice, information, and encouragement to the newer ones. Members of the school did not have to give lectures, nor did they have to undertake administrative duties, unless, of course, they wanted to.

There was continuous contact with the best minds in mathematics. Outstanding mathematicians from all over the world, including the Soviet Union, came regularly to TIFR and lectured on the most diverse branches of the subject, presenting connected accounts of topics of current interest. These courses were written for publication by one or two members of the school. This often led to independent work by the “notes-takers”, and sometimes, to collaborations with the lecturers.

Material success and advancement in TIFR followed one’s work; it was fair and rapid, based solely on the merits of the work done. When I joined the school of mathematics at TIFR in 1957, the atmosphere there was heady. Nothing seemed as important or as exciting as mathematics. New subjects were being talked about constantly and trying to learn them was a challenge. Listening to a colleague try out his ideas and attempting to understand and improve on them was the best instruction one could have. And then there was the excitement of working on problems oneself.

One quality of the greatest importance in a leader of such a group is the ability to recognise significant mathematics and important problems even in fields far removed from his own areas of specialisation. In my opinion, Chandrasekharan had this quality to an extraordinary degree; otherwise the Institute would have been very different.

Chandrasekharan was helped in all the work that creating this school entailed by K.G. Ramanathan. (Ramanathan was a number theorist who worked for a few years with Siegel.) He made the major decisions himself, as, for instance, the decision to organise periodic International Colloquia on important subjects. There is little doubt that the running of the school involved personal sacrifices on the part of Chandrasekharan. Let me mention only the most obvious of them. He continued to do excellent mathematics when he was directing the school; he even broadened his interests and began a series of papers of some importance (in collaboration with others) on analytic number theory. But there can be little doubt that he would have done much more personal research if he had not spent a great deal of time on “trivial details” so that others could work uninterrupted.

The school of mathematics became famous internationally. In many fields such as algebraic geometry and complex analysis, Lie groups and discrete subgroups, and number theory, it has produced a body of work of lasting value.

Chandrasekharan left India for Switzerland in September, 1965; he still lives in Zurich. Shortly after his departure, Bhabha died in a plane crash in January, 1966. It is my belief that the Tata Institute has been unable to absorb the loss of its two most visionary members, and that this has changed the atmosphere.

The reputation of the school of mathematics of TIFR is, of course, based on the outstanding work done by many of its members. With one exception, I shall not try to describe these members or their work; I do not know recent developments and conditions at TIFR very well, and the work with which I am familiar would take too much space to describe. I make an exception for C.P. Ramanujam because his was certainly one of the most powerful mathematical minds to emerge in India since the mid-fifties, and he was, in many ways, a singular figure.

Ramanujam came to TIFR in 1957 with a great deal of knowledge of deep mathematics. This would be unusual anywhere; in India, it was indeed exceptional. He was one of those rare people who feel completely at home in all branches of mathematics. Thus, he understood sophisticated analysis as deeply as he did Grothendieck’s view of algebraic geometry which appears very abstract, but illuminates the fundamental relation between geometry and arithmetic. Ramanujam helped many of the people at TIFR to understand difficult subjects. It was natural to turn to him when one reached an impasse in one’s work.

In his short mathematical life, Ramanujam did some very profound work. This included definitive solutions of well known problems (as with his solution of Waring’s problem for number fields) as well as the introduction of methods and results which formed the basis of progress by others (the Kodaira–Ramanujam vanishing theorem, characterisation of the affine plane, …).

He was diagnosed as being schizophrenic when he was 26. This meant that the time he could devote to mathematics in the ten years that remained to him was severely curtailed. He collected an impressive library of books dealing with the malady and read them carefully. He came to the conclusion that the condition was incurable and took his own life when he was at the height of his intellectual powers (at the age of 36 in 1974).

When Chandrasekharan left India in 1965, the state of mathematics in India was very different from the situation in 1950. There was substantial government support for mathematics, not only at TIFR, but through several other agencies as well. There was a very strong group of mathematicians, mainly at the Tata Institute, whose work was internationally acclaimed. Famous mathematicians from all over the world were ready to come to India and establish beneficial contacts.

It is hard to assess the situation today. Government support has certainly continued and even increased; a National Board of Mathematics exists with very substantial financial resources and a free hand to operate all over India. TIFR still attracts some outstanding young talent from throughout the country.

The colleges in India have, however, not improved very much. The curricula today are more modern than they used to be, but the quality of the teachers in these colleges has not improved substantially, and this is unlikely to change as long as they continue to be so poorly paid and to have so much uninteresting teaching to do. Where the civil service attracted the most talented, it is now industry, some of it private, much of it run by government, which competes with the lure of intellectual pursuit for the commitment of the best people.

It is to be hoped that these things will change sufficiently for the future of mathematics in India to be truly distinguished.

### Footnotes

- As mentioned earlier, the universities of Calcutta, Bombay, and Madras were founded in 1857. Punjab University was established in 1882, Allahabad University in 1887. The Benares Hindu University was founded in 1915, the Universities of Lucknow and Dacca in 1920, as was the Muslim University in Aligarh. Delhi University was created in 1922 by an Act of the Indian Legislature but consisted then simply of three existing colleges; funds were appropriated for a separate campus only in 1927. Andhra University was established in 1926, Annamalai University in 1929. Jamshedji Tata long wanted to create a research institute for technical training and donated land in Bangalore in 1898; the Indian Institute of Science was opened on this land in 1909 with funding from Jamshedji’s sons, Dorabji and Ratanji, five years after Jamshedji’s death. ↩
- Now the department of mathematics. ↩
- S.M. Shah died in 1996. ↩
- Now, Vishakhapatnam. ↩
- S. Chowla died in 1995. ↩
- R.P. Bambah retired in 1993. ↩