**‘ 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 20

^{th}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 10

^{th}at our home and he agreed. But unfortunately on the 9^{th}he breathed his last while at the hotel. I dedicated my book to his great name. The book was published in 1990.
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