‘ Tapas’ of Mathematical Research: 1956-62
I joined Annamalai University as Assistant Lecturer in Mathematics in August 1947. I served there for three years. I came into contact with original mathematicians like Dr. V. Ganapathy Iyer and Dr. M. Venkatraman. The idea of ‘Research’ took roots in my mind. In 1950 I moved over to Thiagarajar College, Madurai as Lecturer. As an affiliated College, with a small department of mathematics and a very commonplace library it did not provide by itself any means of advanced study. Dr. V Seetharaman was the Head of the Department. Looking at the stature of these PhD’s that I had come across, I developed a great urge to do some advanced studies. Just two books were located in the library of Thiagarajar College that might be useful to me for advanced study. These were: Hardy: Pure Mathematics and Cramer : Mathematical methods of Statistics. In my three attempts at IAS examinations (in 1949, 1950 and 1951) I had chosen my third optional subject (besides the two regulars, namely Pure Mathematics and Applied Mathematics) differently each time. Once it was Statistics; another time it was Sanskrit and another time it was Advanced Indian History. Each time I succeeded in the written exams but was flunked at the interview stage by being awarded the same 75 marks (out of 300) each time. Anyway the study of Statistics for this purpose gave me an impetus to choose Cramer’s book of Mathematical Methods of Statistics for my proposed advanced study. But the first few chapters of that book was all high analysis and that floored me. It was this time (summer of 1953) that I had an opportunity to go to the Indian Statistical Institute, Calcutta for a two month period during the summer of 1953, for a so-called training in Statistical Methods. The training was nothing but some practice in the operation of hand calculators. But what I gained was some good mathematical grounding in Real Analysis, particularly Measure Theory (of which I knew nothing till then) through the lectures of Dr. R. Vaidyanathaswamy who was also there at the Institute at that time, fortunately for me. So, later when I came back to my college in Madurai, Cramer’s book made sense to me. But before I could make any substantial progress in the book, so many other things happened in my personal life and career that the so-called advanced study was still (as late as 1956) only in the future-to-be.
In the year 1954 I authored and also published on my own a text book of Trigonometry for the Intermediate classes of South Indian Universities. It was received well but in the year 1956 the University of Madras scrapped its Intermediate curriculum and programme and introduced a one-year Pre-University programme followed by a three-year degree programme. This made me rewrite my Trigonometry book, include the portions of the degree programme, cut off the portions that went into the Pre-University and thus produce a three-year degree textbook of Trigonometry – which I did by around the middle of 1956. My father was writing his magnum opus – Gitamritamahodadhi, in Sanskrit – but it was written by him in Grantha script and so when he wanted to send it to the Kanchi mutt it had to be transcribed into devanagari. I had the good fortune of doing that job of transcription – in the process my acquaintance with advaita philosophy started –and this was finished by the end of 1955 and the mss. was sent to the Kanchi mutt, where it is still in their library. Children Ravi and Balaji had serious recurrent medical problems, sometimes simultaneously, – which required hospitalization, several, several visits to the doctor, and consequent money crunch and what not. My father passed away in January 1956. All these contributed to my complete non-activity in advanced studies in mathematics. But my father had always been telling me: ‘Your better times will start, perhaps, after I am gone’!
Usha was born in April 1956. Around that time I saw an advertisement that Govt of India was offering research scholarships to teachers, particularly in affiliated colleges, to move away on leave or lien to University departments and do wholetime research for three years. I applied for the scholarship at Annamalai University to work under Dr. V. Ganapathy Iyer . And I got it. By that time I had been earning around Rs.400 or so per month at Madurai which included both my salary and income from private tuitions. But the scholarship that was being advertised was only Rs.200 and legitimately no engagement in other tuitions etc. will be allowed. With a family of four children it was going to be impossible to make both ends meet. For a whole month around May-June 1956, we debated intensely among ourselves, very often with Mr.Uppili, my neighbour and colleague (of the English Department) and my father-in-law, who was visiting us at the time. I consulted my St. Joseph’s College Professor T. Totadri Iyengar, who had just then joined as Principal, Madura College. He strongly advised me to take it, for three reasons: 1. In the ensuing decades no college teacher can expect to thrive without a Ph.D. 2. Dr. V. Ganapathy Iyer was internationally famous and to work under his guidance would be a rare opportunity. 3. I was only 29 then and so my whole future can be made in these three years.
But during our discussions at home, in addition to the paucity of living means that the experiment would entail, we repeatedly hit at the problem: What if, at the end of three years I had not done enough to merit a Ph.D? At the time I was being known as an extremely successful teacher, I was already an author of one book, and was readying to become so of a second book and the tuition-world was opening out a volley of financial better times for me. So am I throwing away the brightful present for a doubtful and risky future? These discussions were going on endlessly.
That is where my wife Kamala came to our rescue and decided things for me! She had heard me speak very often how great a Ph.D. degree was, how knowledgeable Dr. Seetharaman of our department was, how Dr. Ganapathy Iyer with his D.Sc. was famous in the mathematical world and how just a B.A.Hons. converted into an M.A. was nothing before such great achievements. So she decided that she will do all in her power and even beyond her power to help me get this Ph.D. for which the opening is just this scholarship. All other problems of shifting the family to Chidambaram, running the household with less than half the present emoluments, -- all these challenges, she said, could certainly be met and the memories of that struggle, if any, will be buried in the past once I get my Ph.D. And that was her decision. With the support of such a brave and far-sighted lady as she was (and is !) we all accepted her decision and coupled with Prof.T.T’s advice, I cast the die!
Thiagarajar College granted me a study leave without pay for three years and a lien on my lecturer’s post. I joined the Annamalai University as a Government of India Senior Research Scholar on September 1, 1956. I met my Professor the very same day at his house (Usually he meets his research scholars at his own home in a small room where he works on his research, without paper and pencil !). Two books were suggested to me for a beginning: Titchmarsh’s Theory of Functions and Hardy and Wright’s Theory of Numbers. He told me Keep reading these books; when you have doubts come to me. That was all. I came home, a house that I had rented in Chidambaram town, with the two books, picked from the Library. But the moment I reached home on that day, I was down with fever and in a day or two it was diagnosed as chicken-pox. So challenges for Kamala started the very first day. The chicken-pox spread to the three children, Padma, Ravi and Balaji in the next few days. Only after a month the life at home came back to normal. I was able to resume my duty only by the 15th of September or so..
Thereafter the routine was the same on almost all the days. I cycled to the Campus and spent most of the day at the Department or the Library, reading almost everything in Mathematics that I can lay my hands on. It was like a poor man enjoying all the riches he could never have dreamt of. Of course periodically, probably once in three or four days I go the Professor and ask doubts. But I may go with a score of doubts, thinking that this might take probably a whole day. But he would clear them in just ten minutes. But what is more, any mathematics book that I may take to him, any doubt that I may ask, he will not only clear it but he would start telling me how that subject matter of the book develops, what its connection is with the rest of Mathematics, where it is applied and used and what problems are usually trending in that area. That area itself may be far removed from his specialization – Analysis – but that would not deter him. In fact by such memorable interactions with him, in about six months, I became familiar with quite a lot of branches of mathematics, having read at least the first chapters of the main books in that branch of mathematics. I would not know at that time how much good this was going to be in my future as a mathematician.
Well, in spite of all that excitement, I did not make any progress with respect to any problem of research. Wherever I could lay my hands, whether it was Function theory or any other topic , whenever I suggested a problem and took what seems to be the direction of proceeding with the problem, in my discussion with him it turned out to be either fruitless or one which demanded far more techniques and understanding that I still had to acquire. At home I would spend as much time as possible to mathematics; in fact I would keep studying mathematics books and articles even while I had to be engaged in rocking my little child Usha in a cloth hammock when her mother was busy otherwise. Kamala was working very hard to see that the household runs within the means that the scholarship was giving me. She did not even emgage a maid servant to help her wash dishes or sweep and clean the house, even when the maid servant was only asking a monthly salary of Two Rupees! This is one remarkable instance of how she was meeting the ‘challenges’ that she knew she would face and have to meet!
A greater challenge was erupting for me! The Annamalai University would not recognise my Madras University M.A. (of 1947) as equivalent to the M.Sc that all Universities had just introduced in 1956; because it was said that the M.A. that I had got after a three-year honours was only by efflux of time and not by a course taken in a university or by an examination! – though all along since the twenties the Universities had been approving it as equivalent to a post-graduate degree. The system was introduced by Dr. A L Mudaliar, following some British Universities of that time. Anyway the bottom line was, the Annamalai University told me that I have to do an MSc by research and submit a thesis for M.Sc not earlier than one year from the time of my registration. Since my registration for research was in September 1956, I was expected to submit an M.Sc. Thesis by September 1957 and then work, in residence at the University, for at least two years to submit my Ph.D. Thesis. All this meant I had to quickly get some results of research done to merit an M.Sc. Thesis. But so far, even as late as April 1957, I had no success.
In the meantime I had picked up enough French and German at least to the extent I could read written Mathematics matter in these languages, though I would not pronounce or read them in French or German. What I learnt to do, was to read them in English, by mentally translating the matter and this I could do so fast that somebody looking at me reading a paper in French or German would think I had the English translation right before me! I picked up this competence in about six months and this helped me enormously. I began reading Bourbaki’s books (all in French) and also read almost the full book (in German) of Uniformisierung by Nevanlinna and actually did my own translation of the book of 200 pages by the end of 1958. While all this was a great achievement, what the MSc requirement wanted was one or two research papers containing original results.
Well, matters were really discouraging. My father had left in his many handwritten notes certain mantra tips for various kinds of redresses and benefits. One of them was a pair of shlokas from the Valmiki Ramayana Balakanda very first sarga. The prescription says: If you repeat these two shlokas 256 times everyday morning facing east for six months you will become master of all learning. I embarked on this project around January 57, but by the time I reached April 57 my tempo of studies became so intense and time-consuming that I had no more any time for this ritual of mantra repetition. Thus the six-month japa stopped almost midway. I was able to submit a paper in the area of Entire Functions to the Proceedings of the American Mathematical Society. It got published in April 58. And this became my first of two papers constituting my MSc thesis, submitted to the University in September 1957 exactly one year after registration.
But now appeared the second challenge. The result of the MSc thesis (sent for evaluation to three Professors, Dr. Shah in Aligarh, Dr. R.P.Boas in Northwestern University, USA and to Dr. V. G. Iyer) took its own usual time and was made available only in June 58. Now the University stipulated that residence for my PhD work will be counted only from this date and so I should put in two years of residence from this date. But this would require me to stay in Annamalai University beyond the termination of three-year leave period from my College. Thus started my second obstacle to my path to PhD.
In January 1958 the Indian Science Congress had its annual session in Madras (now-Chennai) and Dr. V. G. was invited by Prof. A M Shah of Aligarh to take the leading part in a symposium on Riemann Surfaces at the Conference. Professor conveyed his inability to attend the Congress because of health reasons; but he recommended me to Prof. Shah saying that ‘V.K. is competent to talk on Uniformisation of Riemann Surfaces since he has recently been reading Nevanlinna’s Uniformisierung, the only book available on the subject and that too being in German’. This resulted in my speaking at the Indian Science Congress session for one hour on a topic with which even the professors were only vaguely familiar and coming with the blessings of Dr.V.G., I had a great approbation at the symposium.
Throughout 1958 and early 1959 I had been studying among many other things, the new subject of Linear Topological Spaces from the only book then on the subject by Bourbaki and this one was in French. It turned out that this study prompted me and finally enabled me to generalise the several papers on Spaces of Entire Functions by Dr. V.G over the years 1950 to 59. I was also able to generalise some results of Taylor and Halberg on state diagrams of linear operators on normed linear spaces to Linear Topological Spaces. All this constituted three papers from me one in Mathematische Annalen (Germany), one in Journal of the National Institute of Sciences of India and one in the Journal of the Australian Math Society. And thus was I ready to write my PhD thesis incorporating these three papers. But the beauracratic hurdles created history.
First I will not be allowed to submit my thesis until I had put in two years of residence after getting my MSc. So I will have to wait till June 1960. But my three year leave from Thiagarajar College will require me to be back there by September 1959. In the meantime the College had created a post of Additional professor of Mathematics, which would naturally go to the second in command in the Department. I would have got it if I had been there and I would get it now if I go there now. So a big dilemma bothered me as well as all my well-wishers, particularly my family. 1. If I continue to stay at the Annamalai University beyond the three years, what would be my subsistence? Who will pay me? My scholarship was only for three years. 2. If I do not go back now to my College, the new post of Additional professorship would go to somebody else. 3. But if I go back now, I cannot submit my PhD. Thesis and the very purpose for which I had been struggling for three years would be lost. And 4. If I do not go back now the lien on the existing post of my lecturership would itself evaporate and I would be nowhere at the end of the fourth year unless of course the College extends my leave and lien. But will they?
Again I went back to my father’s notes and found a mine of a help. For consecutive nine days one is supposed to read the two chapters numbered 104, 105 of Ayodhya Kanda of Valmiki Ramayana (where Rama advises Bharata in a famous speech called ‘Rama-Gita’ not to insist on his request that Rama should immediately return to Ayodhya) three times each day at the end of one’s daily puja and then each day feed four Pundits who know the vedas. This I did from April 9 to April 17 (1959), finishing on the Sri Rama Navami day. The effect of this ritual has been declared in my father’s notes as ‘sarva-kArya-siddhi’ (= Success of all Tasks). In this project full credit has to go to Kamala my wife who gladly took the trouble of feeding four Pundits with a specialised menu for nine days continuously and simultaneously manage a family of five children plus her own father who was visiting us at the time!
Did a miracle happen? Yes, it did. There was a Fellowship announced by the National Institute of Sciences of India (NISI) of Rs.400 per mensem for postdoctoral work for two years at any University in India. I applied for it and I got it by about June 1958. But now the new dilemma was: Is a Fellowship for one or two years better than a sure professorship that was readily waiting for you now, or was the Professorship the better option? There were many elderly friends who recommended the latter option. I made a frantic trip to Madurai, met my Principal, the Correspondent of the College and over and above him the Founder-President, Sri Karumuthu Thiagaraja Chettiar. After these interactions I had two assurances from them. They will extend my leave and lien by one year and the additional professorship will wait for me to take it up the next year! Thus I continued in Chidambaram for one more year, but now as NISI Fellowship-holder. and in this fourth year I was a Mathematics free-lancer, in the sense, I studied as many advanced books in varied sub branches of Mathematics as possible and gave myself a thorough base in the wide Mathematics Ocean! This foundation proved to be a valuable asset in my future career as a Professor of Mathematics. Of course I submitted my PhD Thesis around July 1960 and got the degree in due time. I was pleasantly surprised and happy to know later that one of the three evaluators of the thesis was Prof. Dr. G. Koethe (Germany) himself, an international authority on the subject and the author of a seminal voluminous book in German on Linear Topological Spaces. This is the book which happened to be my bible for my later work on the subject. My paper in my PhD thesis was quoted by him in his second edition of the book, now available in English since 1964.
Just one more story and this will complete the tapas! I was in Thiagarajar College, as Additional Professor in 1960 -62. The ‘God’ of this story is Prof. Marshall H. Stone of the Chicago University, at that time. Prof. Stone needs no introduction to mathematicians, but here I have to say that Stone was one of the major ace-setters and founders of 20th century Mathematics. He was also a connoisieur of Carnatic music and so he used to visit Madras, particularly, in the music season, almost every year. Every time he visited India he usually paid a visit to Chidambaram to pay his respects to Prof. V. G. Iyer, whom he considered as a mathematics ‘sage’. In 1962 March, he visited Madras University and the Dept. of Maths. under the headship of Dr. V. S. Krishnan, presented to him a five-day programme of mathematical presentations by research scholars and faculty of the Department. Dr. V. G. Iyer wrote to me to go to Madras and be present at these presentations and also meet Prof. Stone. Accordingly I was there in Madras University for all the five days. I could not get any chance to present my researches to Prof Stone, because of two reasons: I did not belong to the University dept. and secondly when I asked Dr. VSK to give me a chance he showed to me his schedule and pleaded there was no vacant slot for me. On the third day or so Dr. Krishnan threw a grand party to Prof. Stone, in the lawns of the University. During the party when I had the opportunity to talk to Prof. Stone, he enquired very caringly about Prof. V.G.’s health and asked me about what work I had done under him. I mentioned just the word ‘Locally convex spaces’. He immediately said : Why don’t you also present your work like these people are doing? I told him there was no vacant slot for me according to Dr. VSK. He immediately walked over to where VSK was standing and told him, Give Krishnamurthy the slot that you have fixed for me (Stone) tomorrow afternoon. And that was done. The consequence was I had the opportunity to present my PhD work particularly that portion connected with locally convex spaces the next day. And to my pleasant surprise, it so happened that professors and other mathematicians from the city colleges were also present for the lecture, because it had been announced earlier in all the colleges of the city that Professor Stone was going to address the meeting that afternoon! Well, at the end of my lecture, Prof. Stone who was sitting in the last bench stood up and gave me a standing ovation! It was due to his recommendation that year that I was able to go to the University of Illinois at Urbana-Champaign for a three-year assignment as a faculty member. The coming to Chidambaram from Madurai to do research was one turning point for the good in my life. The next turning point was this my visit to the University of Illinois, followed by the visit of Kamala, with Ravi, Balaji and Usha in the next year to stay with me in 1963-65. These two major events in my life not only shaped my life but rewarded the rest of the family with the exposure to the world outside of Tamilnadu and outside India and also accelerated the shape of things to come in their own lives.
Later, during the first week of 1989, Prof. Stone visited Madras (for the music season) and was staying in Woodlands Hotel. I was then settled in Madras after my retirement in 1988. I visited him at the hotel on the 7th January and he asked me about my forthcoming book: Culture, Excitement and Relevance of Mathematics. The original motivation for this book must be traced back to my fourth year of my research period in Chidambaram which broadened my acquaintance with Mathematics in no small measure. It was also sharpened by the Mathematics 'Cocktail' lectures by Professors organized by Maths students of BITS Pilani in the seventies. I described to him the book chapter by chapter and he listened with warmth and gave me a few suggestions. My wife invited him for a lunch on the 10th at our home and he agreed. But unfortunately on the 9th he breathed his last while at the hotel. I dedicated my book to his great name. The book was published in 1990.