KC and I: My Fond Remembrance

Konrad Jacobs/The Mathematisches Forschungsinstitut Oberwolfach
Konrad Jacobs/The Mathematisches Forschungsinstitut Oberwolfach

Pray, do not mock me:
I am a very foolish fond old man,
Fourscore and upward, not an hour more or less;
And, to deal plainly,
I fear I am not in my perfect mind.
Methinks I should know you, and know this man;
Yet I am doubtful: for I am mainly ignorant
What place this is and all the skill I have

king lear, act iv, sc vii

I once heard Komaravolu Chandrasekharan’s (addressed as KC hereby) views on obituaries in general. “Why,” he asked me, while he was reading an obituary, “did they not say all the good things they wanted to say about him when he was alive, which would have made him happy, but waited for his death to shower all these praises?” Needless to say that it was a very reasonable statement, with which one has to agree!

This article is certainly not an obituary of KC, but about remembrances of my past and my interactions with, and understanding of, him during my TIFR days, which became warmer after he left TIFR to work at ETH, Zurich. I used to visit ETH often, and these visits started solely because of the interest that KC took in me by inducing Professor Beno Eckmann at the Forschungsinstitut für Mathematik at ETH to invite me for a year or more many times.

It was by mere chance that I joined the Tata Institute. To explain this, I have to go back a little further in time in my personal history. I had the good fortune to have had two great teachers in mathematics, one during my last two years at the Ramakrishna Mission High School (North Branch), T. Nagar, Chennai, and the other during my five year studies culminating in the B.A. (Hons) degree at Vivekananda College, University of Madras. The latter teacher was Professor K. Subramanian. Apart from being a very enthusiastic teacher, he was personally interested in me and convinced me that I should continue to do mathematics and become a mathematician by profession. I should be ever grateful to him for all the trouble he took to enrich my mathematical knowledge. For instance, he not only gave many lectures on advanced topics in mathematics and built a great enthusiasm in me for the subject, but also encouraged me to learn advanced topics in mathematics by inducing me to give extra curricular lectures.

One of the most wonderful things that he did for me, after I wrote my B.A. (Hons) examination, was to take me with him to the residence of Professor C.T. Rajagopal, who was then staying at Tambaram, Madras to seek his advice as to the best place for me to continue my study of mathematics. While Rajagopal was talking about some generalities on the state of mathematics at the University of Madras, I told him specifically that I had heard about the existence of a certain reputed Tata Institute in Bombay, and that I wished to join this Institute. I asked him whether this was a good idea. “No. You would not have any chance of being selected there,” was his categorical reply. “Indeed, I am told they only take students from Loyola College.”

I must explain how I had developed a great interest to join TIFR. A fellow student and good friend of mine at Vivekananda college, named Vishwanathan, knew very well the sons of a very respected retired Deputy Accountant general, Mr. Narayanaswamy Iyer. Mr. Iyer, who had missed the opportunity of doing mathematics himself, was very enthusiastic about young people doing mathematics and encouraged students to pursue its study. Inspired by Mr. Iyer and his sons, Viswanathan kindled my interest in joining the Tata Institute by telling me about it—in particular about a certain student, Seshadri, of Loyola college, who was then doing excellent work there. Viswanathan told me that it would be a very good place for me too if I was really interested in mathematics.

Therefore, I felt very depressed after talking to C.T. Rajagopal. However, I did apply to the Tata Institute sometime in 1955 after my exams and, to my surprise, was called for an interview there. I went to Bombay and, on the evening before my interview, stayed with a friend of my brother in a hotel near TIFR. The next morning, I went for my interview to TIFR, which was then situated at the Old Yacht Club building near the Gateway of India. It turned out to be a long wait at the Institute till the evening before my turn for the interview came. During this wait, I had a chance meeting with the ever-smiling and cheerful S.S. Rangachari. He, and his batchmate and good friend, S. Raghavan, who had been selected the previous year by the School of Mathematics, were both from St. Joseph’s college, Trichy. Thus, meeting them disproved the claim of C.T. Rajagopal! After I was myself selected, I realised that there were indeed many students from various colleges apart from Loyola College who were working at the School of Mathematics. (Incidentally, my first impulse, as soon as I had seen the ugly and turbid waters of the Arabian Sea near the Gateway of India the previous evening, was that I should not join TIFR even if they selected me! Strangely enough, however, I spent the next forty-five odd years of my life at TIFR in Bombay.)

The first time that I had the occasion to meet KC and his colleague K.G. Ramanathan was when I went for the interview rather late that evening. Though I think I did fairly well in my interview, I cannot say that it went off very well. At some stage, when KC tried to drag me into the theory of Dedekind cuts, real numbers, etc., I confessed that some of the ideas were not entirely clear to me and that I would rather prefer not to be examined on this aspect—probably my honest reply influenced their decision in my favour! I must add here that, at some point, KC asked me why I thought that I would do well as a mathematician. I rather shyly mentioned that I had read not only the mathematics usually covered in my curriculum but some mathematics on my own too; and that I had also, for example, given lectures on the proof of Bernstein of the Weierstrass approximation theorem. “All of us give such lectures too,” KC replied with a smile. “I do not find it a satisfying and sufficiently good reason!” I had no reply to give him! Anyway, very soon after the interview, I got a selection letter from the Registrar, TIFR, which came much earlier than my B.A. (Hons) examination results. This gesture on the part of TIFR still impresses me.

C.L. Siegel  TIFR Archives
C.L. Siegel TIFR Archives

That year, there were five students who got selected by the School of Mathematics. We were taught basic analysis, algebra and point set topology during the academic year 1955–56. Our progress was continuously monitored by K.G. Ramanathan, who used to call us periodically to his room to ask questions to test us. While analysis was taught by K. Balagangadharan, topology was taught by B.V. Singbal (who was incidentally one of the kindest persons I met at TIFR), and algebra was taught during the first half of the term by T.P. Srinivasan and in the latter half by Ramanathan himself. During this year, Prof. Carl Ludwig Siegel visited the School of Mathematics to give lectures on the Analytic Theory of Quadratic forms, which Ramanathan insisted personally that I attend. However, I found his first few lectures to be rather technical and hard to understand and stopped attending his course. Meanwhile, under the influence of seniors like M.S. Narasimhan, C.S. Seshadri, Singbal and others, I started studying Claude Chevalley’s Theory of Lie Groups, basic algebraic topology, homotopy theory, etc. Ramanathan used to come to the library very often where the students used to sit. When he saw me once reading Chevalley’s book, he was rather upset with me, though I explained to him why I dropped out of Siegel’s lectures.

My interview by Ramanathan and KC was a total disaster

A very basic activity of the School of Mathematics was a weekly colloquium preceded by tea, which all the members including KC and Ramanathan attended. There was a memorable incident connected with them during this colloquium. On one occasion, when V.C. Nanda (one of my seniors, who was always a smiling and jovial person), made a joke about a friend of his in the Physics department taking a book from the library for getting it photocopied. This was a book that was very often sought by students, and hence rarely available in the library. Hearing this, KC, a strict observer of rules, said to Nanda: “If you tell me his name and the section he works in, I will report it to the Head of his section, who will take appropriate action against him. Does he not know that it is illegal to take books from the library for xeroxing?” Nanda kept silent and fortunately nothing serious happened. I understood for the first time that KC was a strong stickler for some basic rules of behaviour.

Towards the end of our academic year, there were interviews for our batch to decide the confirmation of our scholarships. While I don’t know how my batchmates fared, I knew that my interview by Ramanathan and KC was a total disaster. I got confused, and I think the two of them were also confused, probably because of me, between the notions of neighbourhoods and open sets—and this was the principal reason why the interview ended in a mess. KC, who till then had been rather pleasant and smiled whenever he saw me, became very angry with me and used very harsh words and ended the interview saying that I will be thrown out of the Tata Institute in six months if I did not improve my mathematics in the meantime. I was quite hurt by his words and I thought that at least Ramanathan, who had been watching my progress throughout the year, should have realised that I needed to be treated better. However, when I talked to him after the interview, he too was not very sympathetic and said: “I am very disappointed with you. I thought you would do better.” After the interview, and even during it, I thought I was being unnecessarily intimidated and had half a mind to tell them that I would like to leave the Institute immediately. However, I didn’t do it for a non-mathematical reason—I had borrowed some money from a physicist friend of mine, which I had to repay. I knew I would never have been able to do it had I been thrown out of the Institute and been without a job!

Eilenberg courtesy Columbia University
Eilenberg courtesy Columbia University

Eilenberg’s visit in December 1956 was the most crucial event in my mathematical career

I, however, continued studying algebraic topology and decided to attend the lectures of C.H. Dowker, who was to visit the next term and lecture on sheaf theory. Dowker gave a very nice set of lectures and was indeed a very nice person too. Towards the end of his course, I went to him and told him of my interests in topology and requested him to suggest a problem in algebraic topology which I could work on. He was very pleasant to me but said he could not think of one immediately. He suggested instead that I read a paper of J.W. Alexander1 in the Bulletin of the AMS and told me that it might probably lead to a problem. He added that I could discuss this paper with Professor Samuel Eilenberg, who was supposed to visit TIFR soon to spend a few months.

Eilenberg was the senior mathematician who, along with KC, really took a tremendous interest in me and helped me grow mathematically during my TIFR days. In my view, Eilenberg’s visit in December 1956 was the most crucial event in my mathematical career. Indeed, following the suggestion of the very kind mathematician that Dowker was, I immediately started reading the paper of Alexander and began to look forward to meeting Prof. Eilenberg.

I can never forget my first meeting with Eilenberg in his office, which was more or less immediately after he arrived. I was really impressed by the extreme kindness with which he greeted me. When I told him about Dowker’s suggestion to me, his immediate reaction was that I should leave the journal with him for a day and meet him the next day to discuss the paper. I promptly presented myself following his suggestion, met him the next day, and asked him whether I should discuss with him the contents of the paper. “No, it is not necessary!” he smiled and said. “I have myself more or less read the paper already. You see, I do not believe that this paper is going to lead you to any research. However, I am going to be here for a few months and my lectures here will be on a new topic, and I am sure attending my course will lead you to many solvable questions. So, I suggest you attend my lectures.” I believe he also wanted to judge my interests in mathematics and my capacity to do mathematics on my own, as he added: “Incidentally, I need a proof of a result in topology in a paper I am writing. I would be very happy if you can fix a proof of it for me in a couple of days.”

Needless to say, I worked through the day on the problem he set for me and proved what he wanted. I went to his room the next day with my solution to the problem. He looked at the proof and asked me: “Did you take the help of someone in solving it or did you do it all by yourself?” I told him that the proof was indeed mine, but before presenting it to him, I had the proof checked by Singbal, my teacher in topology, to make sure it was okay. Eilenberg was very happy and asked me to get my proof typed and to give it to him the next day, so that he could use it. But the bureaucracy of the School of Mathematics did not allow me to get it typed. Let me not go into further details, but Eilenberg, being a very kind and understanding person, did not insist on it.

KC believed strongly in insisting on and maintaining academic standards

One pleasant consequence of meeting Eilenberg was that KC, who I believe must have been told by Eilenberg about my capacity to work independently in mathematics, once again became friendly. Indeed, there was a sea change in KC’s assessment of me and, from that time onwards, he did all he could to help me in every possible way. This led me to the realisation that KC believed strongly in insisting on and maintaining academic standards, and that he judged people purely on academic grounds.

I started learning homological algebra very seriously by attending the fascinating lectures of Eilenberg. In fact, my knowledge even of algebra itself owes much to his inspiring and beautiful lectures. I also used to meet Eilenberg regularly and he once asked me to help him by proofreading a paper of his, which later appeared in the Annals of Mathematics. He also gave me a hand-written manuscript of a joint paper of his with Alex Rosenberg and Daniel Zelinsky for reading and was wholly responsible for my getting a firmer grounding in algebra in general, for which I am most grateful. When KC perhaps came to know through Eilenberg of my progress, he also encouraged me in many different ways: In fact I was asked by him to write a review of a textbook of Fr. Racine on Algebra for the Mathematics Student in 1957, and later on a book by N. Jacobson titled Structure of Rings for the same journal in 1959. (KC was also the editor of the Journal of the Indian Mathematical Society at that time.)

In 1956, the School of Mathematics did not take any new entrants, and as though to compensate for this, took a large batch of students in 1957. Most probably, it was KC who decided to make me teach the first year algebra course to this batch. I have often wondered whether KC consulted K.G. Ramanathan at all before he took this bold decision, since, to my surprise, very soon after I had started my course, I saw Ramanathan sitting on a back seat of my classroom. Once, more or less as soon as I started my lectures, he took me by surprise by shooting a question related to the topic I was lecturing on! Though I answered his question immediately, I did not like his acting like a supervisor to find out whether I really knew the subject. Incidentally, in the following year, I was also asked to teach the first year graduate lectures on point set topology.

After Eilenberg left TIFR around 1957, I, along with two fellow students, N. Ramabadran and N.S. Gopala Krishnan, decided to work on some problems in homological algebra (we called ourselves “The Dimension Club”) and I wrote two papers on my own, and one jointly with these friends, which were all published. It was indeed really a very pleasant time we were having!

One day, in late 1956, KC called me to his room and, to my astonishment, told me that Eilenberg, who was the Dean of Mathematics at Columbia University, had written to KC saying that on his recommendation, Columbia University had decided to offer me the the prestigious “Higgins Fellowship” for doctoral studies for the academic year beginning September ’57. KC added: “It is indeed a very nice gesture of Eilenberg, which you certainly deserve. Shall I write to him that you will accept the offer? I am sure this Institute would be very happy to depute you to Columbia University with salary. Shall I write to him that it is okay with you?”

I was taken aback. I told KC that while it was indeed a very kind thought of Eilenberg to offer me a fellowship to do a PhD at Columbia, and that TIFR was also generous in willing to depute me, I was reluctant to accept the fellowship since I was already having some results working with two of the students at the School of Mathematics at TIFR. But more importantly, doing a PhD abroad would mean being away from my parents and my family for some years, which I did not like. KC heard me patiently and advised me to make whatever choice I was comfortable with, and that he would not like to force any decision on me. In the meantime, one of my batch mates, K. Varadarajan, who was studying the theory of Lie groups and Lie algebras, was interested in working with Harish-Chandra, who was then a faculty member at Columbia University. KC wrote to Eilenberg that since I had some family problems and did not want to accept the fellowship, he would like to send Varadarajan instead. Eilenberg was very kind and acceded to his request. Varadarajan accepted the fellowship and went to Columbia in September 1957 instead of me. After a while, I began hearing from him that as a graduate student at Columbia he had to secure 60 graduate credit points by passing various graduate lectures which meant that he would have to spend about three years before he could start his research work, and that he was very unhappy with the idea!

“Why don’t you stay for a few more months with us, relax, and do mathematics for your pleasure?”

In the meantime, KC sent me some other application forms which he wanted me to fill to apply for a Swiss government scholarship. I did fill them up but perhaps in such a clumsy way that he was very upset—he called me and told me that if I was not interested in applying, I should have directly told him instead of doing a clumsy job. He was indeed very angry with me! I apologised to him immediately. Anyway, Eilenberg, once bent on something, never gives up, and he repeated his invitation to me for the next academic year, 1958, offering the same Higgins Fellowship. This time, KC called me and told me: “Since Eilenberg is so interested in having you at Columbia, it is only fair that if you have a genuine problem, you should write a candid letter to him explaining the basic reasons why you do not want to go.” So I wrote a letter to Professor Eilenberg telling him that I was being told that I would have to clear 60 credit points before I can start my research work and that this would mean spending long number of years in the US, and that due to my family problems, I was considering declining the generous offer from him. There was an immediate wonderful reply from him to me saying that such administrative hassles would be best taken care of by him and that the only thing I should do was to accept the offer of the university! This showed me his magnanimity and the belief this wonderful person had in me, and so I immediately accepted the offer, much to the relief and happiness of KC.

I have discussed in somewhat great detail elsewhere2 my stay at Columbia, the remarkable kindness and the humane attitude of Eilenberg and many of my teachers there, so I don’t want to discuss it here once again—except to sum it up by expressing my gratitude for the great support and sympathy I received at Columbia, in particular from the faculty and the generosity of Professor Eilenberg. I should also add that KC wrote to me very encouraging letters when I had some health problems there.

There was one very special act of kindness from Eilenberg which I cannot forget. “You never relaxed during all your stay here,” Eilenberg told me when I had decided, after the completion of my doctorate at Columbia, to come back to India. “Why don’t you stay for a few more months with us, relax, and do mathematics for your pleasure?” I readily agreed. Through Eilenberg’s request, Columbia University gave some monthly financial support which was to continue for a few months. But unfortunately, my father fell ill in November 1960 in India, and I had to advance my journey back to India by a couple of months. I still remember Professor Eilenberg going with me in person to the finance department to request them to give me a stipend for the remaining months too. Of course, the finance department did not agree to his suggestion!

I did have the great pleasure of meeting Eilenberg in India twice after I came back to India. Once was at a conference in Delhi in 1981 and the second time when he suddenly turned up at TIFR looking for me, possibly around the mid-1980s. He spent an hour with me and went to Raman Parimala’s office as well. He heard Parimala lecture on some of her work and was very happy to hear it. He readily agreed to join us for lunch at the West canteen, after which he said goodbye to us. I asked him whether I could arrange for him a vehicle of the Institute to go back to wherever he was staying. “No, thanks!” he said smiling. “I always bring with me a vehicle ready for me to go back and it is waiting for me.” We bid him goodbye and that was the last time I saw this great man.

A very short while after my return to India in November 1960, I was asked to lecture on semisimple modules and related topics to the graduate student batch which had, amongst others, M.S. Raghunathan, who had joined the Institute that year. In the following year, I lectured on first year algebra during both terms to the new batch, which consisted of Amit Roy and two others. I remember very vividly my experiences during the meeting that was convened to decide whether these students should be allowed to continue. The meeting was attended by KC, K.G. Ramanathan, Seshadri and other senior members. During this meeting, Ramanathan was emphatic that none of the three were fit enough mathematically and they should all be sent away. Apparently, some of the senior members had interviewed them earlier and they were indeed not satisfied with the progress of these students during the year.

I, however, had taught them the algebra courses through the year and had a different opinion. In my assessment, Amit Roy was quite sound in algebra and had assimilated the contents of the course very well and I said so in the meeting. I asked Ramanathan what he thought of Amit Roy’s understanding of algebra. “It was really very poor,” Ramanathan said. “He could not even define the notion of
the product of two ideals of a ring.” But I did not agree with his assessment. “I cannot believe it,” I replied. “He has a really good knowledge of algebra.” KC, who was sitting at the far end of the huge table, said smilingly, “Gentlemen, let us not quarrel!”, and turned to Seshadri who was sitting next to him and asked, “What is your opinion, Seshadri?” I can never forget Seshadri’s reply even now, who in his usual calm and collected manner, said: “I have had no mathematical interaction with Amit Roy, so I cannot pass a judgement on him. But if Sridharan has a good opinion of him, I will certainly go with him.” The consequence was that Amit was allowed to continue at TIFR. After the meeting, Seshadri came to me and asked me to make sure that I guide Amit and lead him to some research.

KC, with Jamshed Bhabha on his left, speaking to JRD Tata TIFR Archives
KC, with Jamshed Bhabha on his left, speaking to JRD Tata TIFR Archives

KC was quite aware that the School of Mathematics could not exist in isolation

When I had come back to TIFR, I was thinking of characterising Clifford algebras, which was motivated by my thesis. I suggested it to Amit instead, and happily enough, he solved it and wrote a very nice paper on it which later became a part of his doctoral thesis. I must say in this connection that in spite of Amit’s mathematics being quite sound, and his excellence as an artist (he could sing beautifully, paint, etc), fate was very unfair to him and his stay at TIFR, though academically okay, was not a happy one. I feel terribly sad about this really very talented and many-sided personality. Perhaps too much insistence on productivity and competition, which was, as opposed to scholarship, the “general guideline” of the School of Mathematics of TIFR, led to the destruction of many a sensitive person—Amit was one such.

I turn to KC once again after this sad diversion. KC was quite aware that the School of Mathematics could not exist in isolation and that it needed the goodwill and support of mathematics teachers of the Bombay colleges and the Bombay University. Therefore, he saw to it that some academic members of the School interacted with mathematics teachers of the various colleges and, in particular, lectured to them. In fact, people like Jal Choksi, Singbal, I, and some others were encouraged to give lectures for teachers. We also supported the activities of the “Bombay Mathematical Colloquium”. The School of Mathematics even helped in framing the Master’s syllabus in Mathematics of the Bombay University.

KC wished to help increase awareness of higher mathematics among mathematics teachers all over India. This led to his organizing annual summer schools for mathematics teachers on fairly advanced topics in mathematics for a few weeks, where many us of have taught. I still remember the first such school organized in the early ’60s which was on Algebraic Topology, where, beginning with set theory, the lectures covered topics right up to singular homology. I still remember that I had to give the first lecture, which was on set theory, in this summer school and KC was sitting in the first row! This set up the tradition of the School of Mathematics having annual summer schools and programmes for teachers of India which stopped after many successful years—I do not know why. The notes of these lecture courses were revised and printed as pamphlets by TIFR, and they have been of much use for students and teachers of mathematics in Indian universities.

KC wished to help increase awareness of higher mathematics among mathematics teachers all over India

One such summer school on Algebraic Topology was attended by Prof. S.S. Shrikhande, a leading expert in combinatorics, who was then working at Banaras Hindu University (BHU). He disliked the mathematical atmosphere at BHU and was apparently looking for an opportunity to get away from there. It was indeed one of KC’s remarkable ideas to have induced the University Grants Commission (UGC) to start a “Centre for Advanced Study” in mathematics at Bombay University and to have influenced them to offer Shrikhande the Headship of the Mathematics department and the Centre. KC also had an agreement with the Bombay University that the School of Mathematics, TIFR, would depute two of the senior academic members from TIFR to work at the Centre, along with Shrikhande, as professors of the university for some years. It all worked out as he had planned, since he had great influence with both the Bombay University and the UGC. It was Balagangadharan and I who were deputed by TIFR to work as professors at the Centre along with Shrikhande, and we formed the first faculty of the Centre. Incidentally, KC managed an agreement with Bombay University that while one of the two academics to be deputed by TIFR would be paid a salary by TIFR, the other would be paid by the University. It so happened that my monthly salary as a professor, which under this agreement was paid by the University, turned out to be less than the salary I was getting at TIFR as a Fellow, which was the position I had in the year the Centre was formed. To circumvent this problem, it was agreed between the institutions that while my monthly salary would be remitted by the University to TIFR, my usual salary as a Fellow at TIFR would continue to be paid to me by TIFR. I mention this merely to show that a university professor’s monthly salary at that time was less than that of a “Fellow” of TIFR (in addition to the allowances, of course) at that time.

Working as a colleague with Shrikhande and Balagangadharan at the Centre was a pleasure. We had of course many problems to face with the UGC, but during the three years I was at the Centre, we managed to do a fairly good job academically. In this period, two students got their doctorates with me, while Prof. Shrikhande also had two students. More importantly, from the mathematical point of view, it was a good thing to have started the Centre, since Vijay Kumar Patodi, the remarkable mathematician who joined the Centre, did not even know of the existence of TIFR when he joined, and only later on switched to TIFR where he did excellent work.

I used to teach a course on projective geometry, which was part of the syllabus of the M.Sc. students. This led me to think of a mathematical problem that eventually led to some research, and more importantly the beginning of lifelong mathematical ties with mathematicians like Manuel Ojanguren, Max-Albert Knus, Parimala, Jean-Louis Colliot-Thélène, and others. This is a separate story which I shall not expand upon, except to say that I was essentially left to resign my position at the Centre for administrative reasons in 1967, when I left for ETH, Zurich.

My going to Zurich was directly due to KC. In 1965, there was a rumour in the Tata Institute that KC was planning to leave the Institute. Very soon, it became a reality and this came as a sudden and unpleasant surprise for many of us mathematicians at TIFR. The general impression that went round was that KC’s leaving was because of his differences of opinion with Homi Bhabha on many issues, which perhaps was correct. In this connection, I should say that due to chance, when the furniture of the Institute was changed in some of our rooms, my old table was replaced by a bigger table. When I was transferring some of my old correspondence and other papers to the empty drawers of this table, I found two small pieces of paper with some notes in ink. They turned out to be parts of some exchanges between KC and Bhabha. Obviously they were left there when KC’s table was being cleared. One of them was a note signed by KC addressed to Bhabha and was about the closing down of the Institute canteen on a certain afternoon to accommodate a meeting of the Department of Atomic Energy (DAE). KC had written that some of the faculty members had brought this intended closure to his notice as a complaint and Bhabha had replied saying that he (KC) should bring to the notice of the complaining faculty that DAE is the financial supporter of the Institute and therefore there is no point in such a complaint being made. KC seemed to have started a retort but had given it up in the middle.

Perhaps there were very significant differences of opinion between them and it is possible they did not see eye to eye on some issues. But, one possible reason which I have heard was that KC used to spend the summer every year in Zurich, Switzerland, and he was a very good friend of Eckmann and others at ETH who wanted him to settle in Zurich.

TIFR, however, suffered the even more tragic loss of Homi Bhabha in 1966, when he died in an air crash near Geneva. It is important to mention the rather pathetic attempt of the mathematicians at TIFR to recall KC back to TIFR by offering him the Directorship of the Institute after Bhabha died. In fact, KC was the Deputy Director, School of Mathematics, when he left the Institute. Raghavan Narasimhan and Jal Choksi left their jobs “as a protest” when M.G.K. Menon, Deputy Director, School of Physics, was appointed by the Council as the Director of the Institute.

Sridharan, at his eightieth birthday celebration in July 2015, with Max Knus and Manuel Ojanguren
Sridharan, at his eightieth birthday celebration in July 2015, with Max Knus and Manuel Ojanguren

Coming back to KC, I should first of all reiterate the almost caring affection he had for me after Eilenberg’s visit. He enquired through mail about my health when I fell ill at Columbia, and his concern continued even after I came back. For example, after the sudden death of my father, I fell ill for more than a month, during which time he was very concerned and made many enquiries about my health. He especially urged the TIFR doctor to take very good care of me and to give any kind of medical help that was necessary. At the same time, he also wanted to offer me academic assistance in various ways. I remember one particular instance vividly. After my return from Columbia, he called me once to his room and said that he got a letter from a newly started university in Australia, requesting a suitable person to whom they could offer a professorship in mathematics. He told me that they would accept immediately any suggestion he made. “Are you interested in trying out Australia?” he asked me. “While I would indeed be very sad to lose you, maybe, if you wish, you can go and try for a year or so. They are prepared to offer a fabulous salary.” I thought about it seriously and told him that I would rather like to remain at the Institute and of course he was very happy about my decision.

I mention this incident to emphasize that KC had a very kind corner for me. Hence, I was not surprised that he induced Eckmann, the Director of the ForschungInstitut für Mathematik, ETH to invite me for a year as soon as he joined ETH. This meant that I had to leave the Bombay University Centre. I spent a year and a few months in Zurich and a month in Pisa, Italy, on my way back. The most important offshoot of my visit was the start of my very fruitful and lifelong friendship and collaboration with Ojanguren, Knus, and many others. In fact, my work with Parimala grew more intense after my return to India from Zurich. Moreover, KC took great interest in Parimala as soon as he recognised her talent and always greatly supported her.

CMI, in spite of the great financial difficulties it faces, continues doing a wonderful job of teaching and research

When I retired from TIFR in 2000 and thought I would bid farewell to mathematics, Seshadri invited me to spend some time at the Chennai Mathematical Institute (CMI). Very reluctantly, I joined CMI as an adjunct professor, and thanks to Seshadri’s capacity to influence people, I still continue to stay at CMI even after sixteen years. CMI, in spite of the great financial difficulties it faces, continues doing a wonderful job of teaching and research. KC, who had an immense admiration for Seshadri and for his courage of conviction about the growth of CMI, was in the habit of writing to me till very recently. I cherish his appreciation of my small contribution to education here at CMI and my recent work on ancient Indian contributions to combinatorics, which he has often expressed in his letters—I keep these letters, which show his affection for me, as souvenirs and treasure them.

KC was also very touched by an article of mine on Hermann Weyl,3 which was supposed to be my presidential address of the Indian Mathematical Society a few years ago, and some other articles which I sent him. He too greatly admired Hermann Weyl—let me quote from three of his letters where he mentions incidents involving Weyl. In his first meeting with Weyl (1946–47), KC was taken aback when Weyl showed him a copy of his [Weyl’s] printed paper and told KC: “You quote Haar, but you don’t quote me on `Sturm–Liouville series’. Don’t worry; it was not published, because Hilbert said it was Haar’s thesis to be.” Weyl’s paper had already gone through the page proofs for the Math-Zeitschrift and he had to pay Springer for withdrawing the paper at that stage; KC adds that he learnt a lot from this example. On another occasion when he was at Princeton working as an assistant of Weyl, a person from the US Internal Revenue Service wanted to know what it meant to be an “Assistant”, to which Weyl replied: “He is indeed my intellectual companion.” KC wrote that he was moved by Weyl’s generosity. On yet another occasion, KC, as assistant to Weyl, was filing a mass of reprints which Weyl used to have and KC was asked to comment on anything that struck him as unusual. It once happened that there was a reprint of Weyl’s own lectures to the (future) US occupation forces in Germany. “This document,” KC apparently told Weyl, “could be very useful to anyone in India interested in building a research centre.” Weyl immediately said, “Take it and do what you like with it!” In KC’s words, “The rest is history.” Indeed, this was published under the title, “Universities and Sciences in Germany” (text of two lectures given in 1945 by Hermann Weyl) in the Math. Student, 21 (1953), Volumes 1, 2, p. 1–26.

More than two years ago, CMI celebrated my eightieth birthday by inviting some of my friends and students to participate and give talks. On this occasion, Knus brought from KC a present for me and, very touchingly, it was an English translation of the memoirs of Hermann Weyl about his first wife, Hella. It was accompanied by lovely birthday greetings from KC, which I cherish with gratitude. I shall end this article with the short but emotional “thank you” reply I wrote to him. I only hope that his health permitted him to read this letter dated 6 August 2015, which was my last and lasting tribute to him:

Dear Professor Chandrasekharan,

Your wonderful message (containing as ever profound and beautiful thoughts) was read out during the felicitation programme. Max Knus had already conveyed your greetings to me earlier when he saw me. They made me feel very small and I wondered whether I really deserve this beautiful praise from you. Mere language and words are not adequate to express gratitude to you, for your true generosity, kindness and your timely help at every stage of my life, which made my mathematical life as well as my little contribution to mathematics possible.

I would indeed be grateful if you could accept my deep respects and thanks.




  1. J.W. Alexander. Gratings and Homology Theory. Bull. Amer. Math. Soc. 1947. 53(3): 201–233.
  2. Sudhir Rao. Sridharan @ 80 – Polyglot, Polymath, Pure Math. Math. Newsletter. Sep 2015. 26(2): 44–59.
  3. R. Sridharan. An Encomium of Hermann Weyl. The Mathematics Student. 2010. 79(1): 1–11.
R. Sridharan is at the Chennai Mathematical Institute.