Before we look at the life and work of **Chidambaram Padmanabhan Ramanujam** we must warn the reader that this article is on Ramanujam, NOT Ramanujan the number theorist who worked with G H Hardy (there is only a difference of one letter in their names!).

Ramanujam's father was C S Padmanabhan who was an advocate working in Madras, India, at the High Court. C P Ramanujam was educated in Madras, first at Ewart's School, where he had his primary and the first part of his secondary education, and then at the Sir M Ct Muthiah Chetty High School at Vepery, Madras. His interests on the academic side were in mathematics and chemistry while on the sporting side he was an enthusiastic tennis player. Chemistry experiments were particularly fascinating to him and he made a chemistry laboratory in a room in his home. There he would spend happy times carrying out experiments with one of his friends. In 1952, while still only 14 years old, he passed his final High School examinations and entered Loyola College in Madras.

Ramanujam's achievements at High School had been outstanding and he had shown that he was extraordinarily gifted, so he entered Loyola College with great expectations. He continued his interest in chemistry but it was mathematics that he specialised in, taking Mathematics Honours after obtaining his Intermediate qualification. He was awarded a B.A. with Honours in Mathematics in 1957 but, strangely for such an outstanding student, he only obtained a second class degree. This may have been a result of starting his university education at so young an age before he was really ready, for the second class degree no way reflected his remarkable mathematical abilities. On the other hand it may have resulted from a lack of belief in himself which haunted Ramanujam throughout his life.

He had been taught mathematics by Father C Racine in his final honours years at Loyola College and he encouraged Ramanujam to apply for entry to the School of Mathematics at the Tata Institute in Bombay. In his letter of recommendation Father Racine wrote:-

He has certainly originality of mind and the type of curiosity which is likely to suggest that he will develop into a good research worker if given sufficient opportunity.

In Madras there was another prestigious Institute, the Ramanujan Institute of Mathematics. In 1957 Ramanujam learnt deep results in analytic number theory from the former director of this Institute (who had retired three years earlier) in the months before he left Madras for Bombay to begin his studies at the Tata Institute. At the Institute, Ramanujam quickly became an expert in many different mathematical areas. His wide expertise made him a natural person to write up lecture notes from courses given by visitors to the Institute and in 1958-59 Max Deuring gave a course on the theory of algebraic functions of one variable which was expertly written up by Ramanujam. He seemed able to soak up huge amounts of deep and difficult mathematics and he gave many talks showing what a deep understand he had of many topics. What he was not doing was producing original mathematical advances while some of his less able colleagues were being much more successful.

Ramanujam felt that he did not have what it takes to solve the big problems of mathematics, and he had no wish to solve small routine problems. Again, as in his undergraduate course, it would appear to be a psychological problem rather than a mathematical one but for Ramanujam it was a very real problem and he became more and more frustrated. He decided that his strengths were in teaching mathematics rather than producing original mathematics, and consequently he began applying to a variety of universities and colleges for a teaching position. His applications failed so reluctantly Ramanujam remained at the Tata Institute.

At this stage K G Ramanathan, the author of [4], began working with Ramanujam. He directed Ramanujam to work on some generalisations of the Waring problem to algebraic number fields. On this topic Ramanujam produced some outstanding results, generalising methods due to Davenport to attack certain questions which had been posed by Carl Siegel. For his deep results in number theory he was promoted to Associate Professor at the Tata Institute. It was not a position he easily accepted, arguing strongly that he was not worthy of such a post. However his friends and colleagues persuaded him to accept.

There is a fine line between whether someone behaves in a certain way because they have an illness or whether it is just their personality which determines their behaviour. Up to 1964 Ramanujam's lack of belief in his own abilities could have been described as part of his personality, but in 1964 he was struck with an illness which was diagnosed as severe depression and schizophrenia. Again feeling totally inadequate as a research mathematician he applied for university teaching posts.

During 1964-65 I R Shafarevich visited the Tata Institute and lectured on minimal models and birational transformations of two dimensional schemes. Ramanujam took notes at the lectures for publication and, as he had done previously he showed his deep understanding of mathematics in doing this task. On seeing the notes which Ramanujam had written, Shafarevich wrote to the Institute:-

I want to thank[Ramanujam]for the splendid job he has done. He not only corrected several mistakes but also complemented proofs of many results that were only stated in oral exposition. To mention some of them, he has written proofs of the Castelnuovo theorem... of the chain conditions ..., the example of Nagata of a non-projective surface ... and the proof of Zariski's theorem ...

In July 1965 Ramanujam was offered a Professorship at the Punjab University in Chandigarh. He accepted and began teaching there. However his depression returned and [4]:-

... amidst tragic circumstances he had to cut short his stay there after about eight months.

Back in the Tata Institute, Ramanujam received an invitation to spend six months at the Institut des Hautes Études Scientifique in Paris. Again his illness forced him to return from Paris before the end of the six months. However his ability to do mathematics seemed as remarkable as ever outside his periods of illness. In 1967-68 David Mumford visited the Tata Institute and again Ramanujam wrote up his lectures for publication. In the Introduction to *Abelian Varieties* Mumford wrote:-

... these lectures were subsequently written up, and improved in many ways, by C P Ramanujam. The present text is a joint effort. ... C P Ramanujam continuing my lectures at the Tata Institute lectured on and wrote up notes on Tate's theorem on homomorphisms between abelian varieties over finite fields.

Severe depression struck Ramanujam frequently. On one occasion he tried to take his life with barbiturates but was quickly treated and recovered. In February 1970, while again suffering depression, he resigned from the Tata Institute. The Director refused his resignation but later in the year he again resigned and went to the 1970-71 Algebraic geometry year at the University of Warwick in England. Mumford was also at the meeting and writes in [2]:-

... we were together in Warwick where he ran seminars on étale cohomology and on classification of surfaces. His excitement and enthusiasm was one of the main factors that made "Algebraic geometry year" a success. We discussed many topics involving topology and algebraic geometry at that time, and especially Kodaira's Vanishing Theorem. My wife and I spent many evenings together with him, talking about life, religion and customs both in India and the West and we looked forward to a warm and continuing friendship.

As a result of his work with Shafarevich and Mumford, Ramanujam went on to make contributions to algebraic geometry which Mumford describes in [2]. These include a characterization of *C*^{2}, a version of the Kodaira vanishing theorem, a study of the automorphism group of a variety, a study of the purity of the discriminant locus, a proof that the invariance of the Milnor number implies the invariance of the topological type, and a geometric interpretation of multiplicity. The work on the Milnor number was done in collaboration with Le Dung Trang.

Back in India after his year at the University of Warwick, Ramanujam asked for a Professorship at the Tata Institute but be based in Bangalore where a new branch dealing with applications of mathematics was being set up. This was agreed and he taught analysis in Bangalore but, again in the depths of depression caused by his illness, he tried again to leave the Institute and obtain a university teaching post. While waiting for an offer of such a post from the Indian Institute at Simla he took his life with an overdose of barbiturates.

In [3] Ramanan pays this tribute to Ramanujam:-

For sheer elegance and economy, I have come across few mathematicians who were C P Ramanujam's equal. he made so many remarks which clarified and threw light on different branches of mathematics that personally I derived immense pleasure from his company.

It was a stimulating experience to know and collaborate with C P Ramanujam. He loved mathematics and he was always ready to take up a new thread or pursue an old one with infectious enthusiasm. He was equally ready to discuss a problem with a first year student or a colleague, to work through an elementary point or puzzle over a deep problem. On the other hand he had high standards. He felt the spirit of mathematics demanded of him not merely routine developments but the right theorem an any given topic. He was sometimes tormented by 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 genuine original stamp.

**Article by:** *J J O'Connor* and *E F Robertson*