A Different Kind of Mind

KRP: I completed the first five years of my schooling in Thanjavur, a town which is about 250 miles south of Chennai. After that, I was in Chennai where I was looked after by my uncle and grandfather. I studied in P.S. High School in Mylapore, Chennai (which was called Madras at the time) from 6th standard till the completion of my SSLC. Our house was just adjacent to the school.

KRP: In school, we had two wonderful teachers, both of whom were named Narasimhacharya. They were extraordinarily good teachers and they posed interesting problems in school. The senior Narasimhacharya, who taught me in the 10th and 11th standard, sometimes uttered mathematical phrases and told us to repeat the same phrases to the elders in the family and examine whether they were aware of such phrases. I learnt a lot about the Euclidean geometry of the plane, mainly about circles and triangles in the school. After I completed my SSLC, I completed my intermediate studies, which was a two-year course, in the Vivekananda College.

After my intermediate studies, I again enrolled in Vivekananda College for a mathematics honours course, which was a three-year program that I completed in 1956. Again, we had some wonderful teachers here. For pure geometry, we had a teacher, Raghava Shastri, who was very famous in the city for his deep knowledge of plane geometry and projective geometry. We also had a very interesting teacher by the name of Krishnamurty Rao who taught us mechanics and even introduced the notion of the Hamiltonian during the honours course, even though the concept of Hamiltonian was not in the syllabus. We were quite excited about this. Unfortunately, before he could proceed further, in the middle of our course, he developed a carbuncle and died. It was a great loss.

We were taught analysis by another extraordinarily good teacher named K. Subramaniam, who was nicknamed “Cauchy” for his analytical prowess.

Furthermore, we had very good classmates. I had a classmate by name S. Ramanan who has now distinguished himself as a specialist in algebraic geometry and differential geometry at TIFR. Among my seniors, I had two friends namely R. Sridharan and Ranga Rao, both of whom became eminent mathematicians. Sridharan is now teaching in CMI. We also had a mathematical club run by Subramaniam where we listened to seminars on varied topics, for example, on the proof that a fifth degree equation doesn’t have solutions in radicals. So we had a good time.

We also had courses in hydrodynamics and astronomy. We did astronomy without ever looking at the sky! We learnt the hydrodynamics which John von Neumann called the hydrodynamics of “dry water”.¹

After I completed my honours in 1956, my family was reluctant to send me out of Madras but I had a friend who was studying in Presidency college who came and informed me about the Indian Statistical Institute in Calcutta (now Kolkata), and that admission to this institute is only possible through an entrance examination. I wrote this entrance test which was held in one of the Madras University buildings and got admission in ISI Kolkata. Incidentally, 1956 was the Silver Jubilee year of the institute.

KRP: Well, first of all, we didn’t have a good hostel. We all lived in sheds that had an asbestos roof and bamboo partitions, but the surroundings were green. There were ponds in which we could swim everyday and enjoy ourselves, but there was always this difficulty in getting clean water. Many students did fall sick. But at ISI, for the first time, we were attending classes where we could listen to proofs of theorems originally proved by our own teachers which was in stark contrast to our college, where we thought that all theorems came from Newton, Euler, Gauss and such people. They had written books from which our courses were taught, and so it was a completely different experience.

For the first time, we could listen to proofs of theorems proved by our own teachers

Who were your teachers?

KRP: Well, in the hostel, my friends were Ranga Rao, who was one year senior to me, and Veeravalli S. Varadarajan, who was directly admitted as a research scholar. In fact, those who had studied the statistics honours course were directly admitted for research but those with no statistical background were admitted into a program called the “Three-year Advanced Statistician’s Course”.

In that course, we had examinations with unlimited time where we were allowed to use our books and notes. The questions themselves were written on a long blackboard and there were no individual question papers distributed. Some problems involving numbers were written on the blackboard and we used a hand calculator facit machine to carry out matrix computations for computing regression, analysis of variance, and such things. In the exam hall, we used to sometimes sit down for five or six hours at a stretch, during which we were regularly supplied singara [a Bengali variant of a samosa] and tea. The exams had no strict pass or fail rule as the questions were taken from research papers. So even if we made some little progress, we were evaluated on the basis of that.

We used to sit for five or six hours at a stretch and were regularly supplied singara and tea

We also had a book grant which was unheard of in our college. I used my first book grant to buy General Topology by John Kelley, Measure Theory by P.R. Halmos, Mathematical Methods of Statistics by Harald Cramér, and Advanced Statistical Methods in Biometric Research by C.R. Rao. Apart from that, there were also notes compiled by Jogabrata Roy, R.G. Laha and Indra Mohan Chakravarti, which were practical manuals containing extraordinarily good problems and were published much later in a Wiley series.² So we had a very good education in statistics. Later in the third year, we were supposed to carry out projects. I chose to work on what is called the “moments problem” in probability theory—how do moments determine a probability distribution? Is there uniqueness? And so on. Because of this, I was very much attracted to problems on convex sets and that interest has sustained till date.

To elaborate, you are given k moments that make a positive definite sequence up to k and consider all probability distributions with those first k moments fixed. That is a convex set. You study this convex set for its extreme points and such things. This problem is linked with continued fractions, so there was a very good opportunity to deal with all these things. My project supervisor was Varadarajan, so it was a wonderful time.

KRP: In those days, there was no television, no telephone and we were all afraid of stepping out of the ISI campus because B.T. Road looked quite dangerous. So most of us just lived on the campus and in the nights all of us assembled in a common room with a big radio and listened to the news. Every Saturday we used to listen to the national program of music and there were also some students who were extraordinarily good in music. The artist on the radio would have barely begun and there would be tremendous competition among a group of students to identify the raga [scale] and then the tala [time signature of rhythm]. In those days, they did not announce the raga or tala at the outset and so even at the time of the aalap, students had an opportunity to identify the raga. I was not very knowledgeable in music, but nevertheless, I admired the enthusiasm of my friends.

Varadarajan, B. Ramachandran and another student by name H. Krishnamurty, who later became an IAS officer, were all very knowledgeable about music. Later, in the mess, we discussed all good literature so that was how we came to know about Bengali novels. Strangely enough, we were around 45 students in the batch and I think about 20 came from Andhra Pradesh, 20 came from Tamil Nadu, three or four from Bengal and one or two from Maharashtra and Uttar Pradesh. The mess served only vegetarian food with rice, sambar and idli, dosa and pesarat and only south Indian tiffin. The north Indian and Bengali students had a difficult time, and the languages spoken were mostly Tamil and Telugu.

R. Thyagarajan, who founded the Shriram Group, studied at ISI and was one of my juniors. He believes strongly in the development of mathematics and statistics, and he gave a lot of money to run the Chennai Mathematical Institute, and helped to buy land, construct its library and so on. So, ISI came a full circle much later with respect to mathematical education in this country.

KRP: When we were in this course, during the first three years, we did not have any direct interaction with him. He was deputy chairman of the Planning Commission at the time, and used to travel between Delhi and Calcutta. Whenever Mahalanobis came to Calcutta, every now and then, he was keen on giving a lecture to all students—research students as well as the students in the training program. He was a very practical minded person and was a professional economist who had just then completed a small book called The Approach of Operational Research to Planning in India. Of course, at that time I had no interest in this subject and was not reading these things but we had very distinguished economists visiting the institute. I had listened to lectures by John Galbraith, Wassily Leontief, and also some Russian and Polish economists. We were also lectured by the Dutch economist Jan Tinbergen, who was later awarded the inaugural Nobel prize in economics in 1969. Mahalanobis himself also lectured to us on the efficacy of sample surveys, sampling design, and so on.

We also had students from China at that time and our teachers also used to attend the lectures of Mahalanobis. C.R. Rao and Bahadur would be in the audience and the lecture would be a big monologue from Mahalanobis. Every now and then, he used to point to Rao and Bahadur in particular and ask, “Do you follow?” If the two of them remained silent, he would interject and say “Ha! My Chinese colleagues will understand.”

I remember one day, we had visitors from China, namely, Zhou Enlai and Chen Yi. Zhou Enlai was the Prime Minister of China and Chen Yi was the Foreign Minister. We didn’t understand why they were coming to ISI. I was being taught at that time by Sujit Mitra who later became my colleague. And, Sujit Mitra said, “I am writing some figures on the blackboard. All of you keep your facit machines ready and as soon as our dignitaries enter, compute the mean and variance of the data.” We were amused by this but we did what we were told to. The students enacted a kind of drama where we were doing statistics; then Zhou Enlai and Chen Yi walked into the room, looked at us, smiled and then left. I think some years later they even attacked India. [Laughs] So, this was the “Hindi–Chini Bhai Bhai” period that ended in a war.

My PhD supervisor at ISI was C.R. Rao. I remember I went to him and told him that I wanted a problem to work on. He simply said, “Learn information theory.“ I asked ”From where?” He said, “You ask [J.V.] Joshi.” Joshi was visiting us at the time. He had completed a PhD in France and was very fluent in French. He asked me to read the book Mathematical Foundations of Information Theory, published by Dover, written by A.Y. Khinchin. This book is extremely beautifully written. This was the same Khinchin who is famous for the Khinchin–Kolmogorov law of the iterated logarithm.

So, I was reading this book on the advice of C.R. Rao but I was not satisfied with one particular chapter. So, I read through it and rewrote that chapter. It involved learning some ergodic theory and about probability measures invariant under transformations. I wrote some papers on this subject. The importance was the richness of the ergodic measures in the class of all invariant measures, and in Khinchin’s program I found that for the shift transformation on the countable product of a finite set, the ergodic measures constitute an everywhere dense G_\delta under a weak topology in the space of all shift-invariant probability distributions in that space.³ This was a surprise, and I sent it to Joseph L. Doob, who then gave it to John C. Oxtoby, a measure theorist. Oxtoby wrote a nice report on that paper and he also read it in a meeting of the American Mathematical Society⁴ and they⁵ were published in a book Benchmark Papers in the Information Theory at that time.

I showed Oxtoby’s paper to C.R. Rao. In those days, there were no regular scholarships and whatever Rao decided was the scholarship amount. So, when I showed him this report he doubled my scholarship. This was the kind of administration we were working in at that time but I think we were given good encouragement. Then, meanwhile in 1956, Varadarajan proved Prokhorov’s compactness theorem on his own, where he gave a different proof and he explained it to us. He also proved Kolmogorov’s consistency theorem, which he derived from Riesz representation theorem in C[X]—the space of continuous functions over X, where X is a compact metric space—and he developed this to a very sophisticated level. He submitted this paper to Matematicheskii Sbornik, a Russian journal.⁶ The Russians translated Varadarajan’s thesis into Russian and published it. Later on, I re-translated it into English, which was published as part of the American Mathematical Society Translations⁷ so it had an amusing journey from English to Russian, then back to English.

KRP: No, the original manuscript could not be traced. Varadarajan went to Princeton and I think he didn’t take care of the original typed version. In those days, they were all cyclostyled on rice paper. There were no photocopy machines or anything. I don’t know what happened. Varadarajan asked me to translate it to English, so on the basis of my translation he edited it and then gave it to the AMS because AMS wanted to translate it. So it is included in AMS translations. It’s a very important paper. Even now it is in great use.

Varadarajan was sent to Princeton to work with Samuel S. Wilks, a statistician. But by that time, Varadarajan had already decided to get out of probability and measure theory and wanted to enlarge his knowledge. He met Harish-Chandra and George Mackey. It was George Mackey who told him, “If you want to learn more about representations, go and learn the formidable techniques of Harish-Chandra on Lie algebras. Only then you will be able to make progress.” Mackey had already studied representations from the purely measure- and Borel-theoretic point of view but not from the Lie-theoretic point of view. So, Varadarajan told me that he met Harish-Chandra and then said, “I want to learn representations.” Harish-Chandra told him to “learn the basics and then come”.

During the period that Varadarajan left for the US for his first visit to Princeton, S.R.S. Varadhan joined the Indian Statistical Institute. The remaining three of us wrote some papers on probability theory where Ranga Rao gave us a whole course on topological groups based on Pontryagin’s book. We also decided to prove, for example, limit theorems in probability theory from Gnedenko and Kolmogorov’s book. We transplanted it to the context of groups, and we did it for locally compact groups. Varadhan later on completed the program for Hilbert spaces and that was the first thesis on that subject, i.e., the Lévy–Khinchin theory for an infinite-dimensional vector space, where he got the Lévy–Khinchin representation theorem. The Fourier transform is not a good technique in the Hilbert space, an infinite-dimensional space, so he used the technique of concentration functions and wrote his thesis on this subject.

Then, Ranga Rao went to the US and I went to Moscow, after which Varadarajan came back to Calcutta. Varadarajan and Varadhan overlapped for a short period here and they began working on the representations of sl(2,\mathbb{C}) and sl(3,\mathbb{C}). Varadarajan has written about all this in a volume honouring Varadhan and Stroock on their 60th birthday. One can find these historical details there. Varadhan left for Courant institute in 1963, which was the same year I came back from Moscow.

Then, Varadarajan, Ranga Rao, and myself were together in Calcutta, where we embarked on a study of complex semi-simple Lie algebras and their representations. In all, we worked for two years on this. Varadarajan first gave a whole course on the famous paper of Harish-Chandra on the universal enveloping algebra of Lie algebras which was published in 1951.⁸ Towards the end of his stay at ISI, Varadarajan also gave a full course on the mathematical foundations of quantum mechanics based on the Chicago lecture notes of George Mackey. It is from this course that I picked up the foundations of quantum mechanics. Then we all went our separate ways.

KRP: What happened was, in February 1962, there was a convocation being thought of and Dr. Rao asked us to suggest who could be invited from the USSR. At this point, Varadhan and myself were there but Ranga Rao had already left. We told Dr. Rao that we would like to see Kolmogorov. Then C.R. Rao said, “No problem. We will get him.” Then I think he put a word to Mahalanobis, who put the right word in at the right time at the USSR Academy of Sciences, and Kolmogorov agreed to come but he wrote back saying that he can come only by boat. He would not fly since he had some problems with his ears.

So, how does one arrange a boat trip? That was the problem. Then, Kolmogorov said that he would join a Soviet oceanographic expedition at the time, and that when they reached Bombay, he would take a train and come to Calcutta. That was how he came, and on the boat, he wrote a very important paper from which emerged the famous concept of “Kolmogorov complexity” for algorithms. What is a random number? How to define it? How to generate random numbers and what is the complexity involved? Through those ideas, he came to the idea of complexity and whatever he wrote on the boat, he prepared a note and that was published in Sankhyā.

It is interesting in the history of Indian mathematics that at the birth of a very important concept, Sankhyā played an important role. Similarly, in the birth of quantum field theory, Arthur S. Wightman’s famous paper on the axiomatics of quantum field theory was published in the Journal of Indian Mathematical Society.⁹ Somehow, we lost such traditions of getting very eminent people to publish in our journals. Of course, the architecture of international science has become so complicated today that it is difficult to have such things. Maybe whenever we organize some international conferences, we have to see to it that our own journals get some papers from the visitors so that our journals still keep international contact. Particularly, whenever any famous and distinguished mathematicians come, we should have a paper from them in our journals. That would be a worthy pursuit.

KRP: I was there for a year. I was attending seminars of Eugene Dynkin on Markov processes and seminars of I.M. Gelfand on everything. Dynkin focused completely on Markov processes out of which this big fat book on Markov processes came, which later was translated into English and published in two volumes by Springer. It is a great book. At the Steklov Institute, I used to attend many probability seminars. Of course, they all were in Russian language and Gelfand’s seminars were always held on Monday evening. It would start between 6 and 7 pm, continue for three or four hours, and end around 11 pm.

In the course of these seminars, Gelfand used to walk around in the seminar hall.

I had a very interesting experience in the first seminar. I was sitting beside Yakov G. Sinai. Of course, at that time he was young—we are nearly of the same age. As Gelfand was giving the seminar, he came very quickly towards me and asked me something in Russian. I asked Sinai to translate it for me, and he said, “He is asking if have you come here by accident.” [Smiles] But I was a regular visitor to these seminars. I had attended some stray seminars by Mark Naimark. And at that time, V.I. Arnold used to participate in the Gelfand seminar. He was very young and had discovered the Kolmogorov–Arnold–Moser theorem, which made him famous. Russian newspapers printed his photograph. Of course, I didn’t understand much of those things but there was an atmosphere of excitement.Many visitors also used to speak at Gelfand’s seminars. I had the opportunity to listen to Evgeny Lifshitz, who authored the ten volumes of Foundations of Physics with Lev Landau. Sometimes there were lectures even in biology. Gelfand would give his comments in the middle of these lectures. It was a great experience, and for one year, I attended these seminars. Gelfand and Naimark had a long paper in Russian—more than 250 pages—on representations of classical groups. I used to look at this paper, and when I came back to Calcutta, I used to discuss with Varadarajan and he always used to call this paper “How is 1 to 288?” because I think it was 288 pages long—I am quoting from memory.

KRP: Well, I returned in 1963 and left in 1965. Varadarajan had got a letter from Joe Gani, who was setting up a new department for statistics and probability at the University of Sheffield. Varadarajan told me, “Why don’t you go there? It is a good opportunity.” I got married at the time and Varadarajan went to Los Angeles, so I thought I would have a change from B.T. Road because accommodation there was difficult at the time. ISI didn’t have campus accommodation in those days. We had to rent some buildings in the neighbourhood. For some time, Varadhan, myself, and Ranga Rao, all shared flats but later Varadhan took a separate house where he brought his parents. He lived in a small house behind ISI.

KRP: Yes. I got married to Shyama in Chennai. We moved to ISI, and at that point, ISI had some kind of campus. There was one T.A. Davis, who was working in biology with Haldane, and he gave us a room to stay. So we stayed in the campus for a few days before we returned to Chennai, and then went to Sheffield.

As soon as I went to Sheffield, I gave a course again on whatever work Ranga Rao, Varadhan and myself had done. I had polished that work to some extent and wrote an article with Sazonov improving on some of the results.¹⁰ Afterwards, while giving a course for PhD students, I wrote notes on this subject. I had a visitor from the US, Eugene Lukacs, who came to my office and saw my notes. There was no computer or photocopying facility at Sheffield, so through some facility there, I had brought and kept one copy on my table. Lukacs saw it and said, “I am the editor of a series run by the Academic Press. Can I have these notes for publication?” That was how, in 1967, the book Probability Measures on Metric Spaces appeared.¹¹

Then after this book was completed, I think after about one and half years in Sheffield, I moved to Manchester as a professorship was available there for which Gani asked me to apply. He wanted to retain me in England because I was almost about to accept a position at Cornell University at that time. He asked me not to go and said that he would offer the professorship at Manchester. He quickly arranged an interview at Manchester and I became a professor there. He organized what was called the Manchester–Sheffield school of probabilists, so these two neighbouring cities in the UK were connected.
Then my children were born. My first son was born in Sheffield and my second son was born in Manchester. When my first son was about to reach five years of age, I told Shyama that we should probably think of going back to India but we didn’t take any steps. In that time, I made a trip to the Australian National University (ANU), where I was giving a course on the mathematical foundations of quantum mechanics. I was staying in the ANU guest house, where I had two very distinguished mathematicians for company in the guest house as well as at the mess. I used to meet Kurt Mahler, who was an eminent number theorist, and Marshall Harvey Stone. Stone was a regular visitor to India and was fond of Carnatic music. He was great friend of C.T. Rajagopal as well as N. Venkata Raman of Madras University. He was friendly with all the administrators in the University Grants Commission (UGC) and similar places.

Stone asked me `Suppose I arrange a position for you in India, how would you take it?’

Then, one day, I found Stone attending my seminar. When everything was over, he asked me at breakfast the next day, “Suppose I arrange a position for you in India, how would you take it?” Then I told him: “I don’t know. I think I may not be able to take a position in Calcutta.” He said, “No, no. You don’t worry. I will examine what are the possibilities.” Then we parted and my course at ANU was over. I stayed in Australia for about two months and gave this course. My wife stayed with her parents, and then for return journey I came back, picked her up and then we went back to Manchester.

Later, I had a letter from the UGC saying that we have a Centre of Advanced Study in Mathematics in Bombay, and Shrikhande would like to have you as a professor in Bombay University. Would you be interested? So, I said, “Yes, why not?” Then we went to Bombay, and Bombay University had arranged for our accommodation in Matunga, and the Centre of Advanced Study was situated in a building of the Bajaj Institute, opposite to the Bombay University. I used to teach mainly analysis and measure theory at the Centre of Advanced Study. While I was there, I got a student, Inder Kumar Rana, and also two people from Matscience [The Institute of Mathematical Sciences] in Chennai joined me. They were both physics students. One of them went on a holiday and then, unfortunately, committed suicide. The other one left for the US. Only Rana stayed.

Then in 1973, C.R. Rao visited Bombay and told me that a Delhi Centre for the Indian Statistical Institute would be created, so I should try to move to Delhi. So, after three years in Bombay, C.R. Rao arranged for a position for me in IIT Delhi. Before I went to IIT Delhi in 1973, I saw that they had very spacious grounds and I thought it would make a very good playground for my children. So without any hesitation, I accepted and moved to IIT Delhi. C.R. Rao had moved from Calcutta and was living in a building just opposite IIT Delhi. Now and then, we used to meet in the lawns of IIT Delhi during evening walks.

Dr. Sujit Mitra had also moved to Delhi but there was no ISI Delhi Centre yet. They were teaching some courses at the planning unit of ISI in Yojana Bhavan. They had rented some rooms and they were running some classes.

Then, C.R. Rao wanted to start a M.Stat. course, and I used to go from IIT to Yojana Bhavan by a van in the evening and take a course in probability theory there. Meanwhile, in our present campus of ISI Delhi Centre, they had just the first floor of the hostel and they had set up some administration and one block of classrooms. So, we started teaching the M.Stat. courses within the ISI campus. Of course, there was a grand inauguration of the campus in 1974 by Indira Gandhi.

In our first M.Stat. batch at Delhi Centre, we had R.B. Bapat as a distinguished student. Apart from Rana, who moved to IIT Delhi, I had two research scholars. One was S. Ramasubramanian who is now at the ISI Bangalore Centre, and the other was Rajendra Bhatia. We had one postdoc, S. Gupta, who used to lecture to us on finite groups and their representations, while I used to lecture on Brownian motion and linear algebra. We were running a seminar in our office room for just the five of us.Then, after that in 1976, when the main campus building was ready, C.R. Rao came to my house and said that Gopinath Kallianpur had become the director of ISI Kolkata. C.R. Rao was looking after only the Delhi Centre, and the two decided to offer me a position at the Delhi Centre. So my group and I moved to ISI, along with one administrative assistant, V.P. Sharma, from IIT Delhi. Sharma was the first person in ISI to learn \TeX\, for typesetting, when computers were introduced in the Delhi Centre. So we were very thankful to Sharma for learning this because from that time onwards, all research papers, books and everything else were being typed by Sharma. He still lives in Delhi.

KRP: We wanted to build the campus and have a reasonable mathematics department. In these efforts, we were joined by T. Parthasarathy, and then Prakasa Rao. B. Ramachandran, Kalyan Sinha, and V.S. Sunder also joined.

Sharma was the first person in ISI to learn TeX for typesetting

We invited Pavaman Murthy from Chicago hoping to offer him a position in ISI Delhi. We also got Ananda Swarup Gadde and Kalyan Mukherjea. So, we had a substantial group of mathematicians. Mukherjea used to teach the theory of manifolds, calculus on manifolds and tensor algebras, and was also in charge of the library. We had a very weak library till then and he spent a long time in building up the library. The research students also helped us. We had a grant from Ford Foundation which allowed us to get all the volumes of the Bulletin of the American Mathematical Society in our library gratis. So, we had a strong group, and in ISI Delhi, all these three people—Rana, Ramasubramanian and Bhatia—completed their theses. Mukherjea gave a very crucial idea for completing Bhatia’s thesis by using exterior products for computing Jacobians of matrix maps.

Varadhan visited us and had a talk with Ramasubramanian and he suggested a problem. Ramasubramanian’s thesis was exactly on that problem—recurrence of diffusions. Rana had worked on Pompeiu’s problem and he wrote a thesis.

I think all of them went abroad, and spent some time there. Mahendra G. Nadkarni offered Rajendra Bhatia a position at the Centre for Advanced Study in Mathematics, and he worked for some time in Bombay. Finally, when new centers of ISI were created, Ramasubramanian joined in Bangalore after spending some time with Rabi Bhattacharya in the US. Rajendra Bhatia spent a year in Berkeley and then came back and joined ISI Delhi.

I gave a series of lectures in the Bangalore Centre of TIFR. At that time, K.G. Ramanathan was the head of the centre, so I stayed at the IISc campus guest house and gave a course on perturbation of linear operators which was attended by two good students. One of them was Muthuramalingam and the other was Tara Nanda. Nanda left for the US and Muthuramalingam told me that he would like to come to ISI Delhi. So, I brought Muthu to ISI Delhi.

M.D. Srinivas from Madras University sent his student, Krishna Maddaly. So, Krishna, Muthu, Kalyan Sinha and I, ran a regular seminar on quantum mechanics and perturbation of linear operators. I gave Kalyan Sinha a strong course on the theory on Brownian motion and also probability theory in general. I used to visit Britain for three months every year and work with my English friends, and whenever I came back, I lectured on quantum probability. Kalyan and I wrote many articles together, and it is during that period that we introduced quantum probability theory in the M.Stat. program itself. You [B.V. Rajarama Bhat], Anilesh Mohari, Debashish Goswami, and later Parthasarathi Chakraborty, learnt the fundamentals of non-commutative analysis and probability theory from Kalyan and me.

KRP: Quantum stochastic calculus started in a very interesting way. The details are very neatly given in an article by Robin Hudson, which you published for my 70th birthday.¹²

What happened was that Robin and I were attending a regional meeting of the Royal Statistical Society in Leeds. I gave a lecture and mentioned that a particular paper of Robin Hudson was a very attractive one or something like that. Then after the meeting was over, Robin Hudson came and met me and we began discussing it. Robin then invited me from Sheffield to Nottingham to give a talk, after which he posed some problems. So in the 1970s, during my periodic visits to England, we began discussing problems of quantum probability but still quantum stochastic calculus was not born.

Robin Hudson and his student had proved the central limit theorem for q and p, the position and momentum pair, and they had also written a paper on quantum Wiener process for this.¹³ At that point, I was on a visit to Warwick, where he proposed that we should look at creation and annihilation processes and he proposed a conjecture dA\,dA^{\dagger} = dT. But even at that time Robin and Raymond Streater wrote a short note with a heuristic proof of this conjecture because they were in a great hurry to announce the dA\,dA^{\dagger} = dT. So, I said that this is not enough from the mathematical point of view and we have to do it more carefully.

KRP: Ah. That happened this way. I think Meyer visited India, mostly Delhi, on two or three occasions. The first visit was during the golden jubilee of the Indian Statistical Institute, which was celebrated with the organizing of several conferences in 1981 and 1982. We had many eminent probabilists coming to ISI, like Kiyosi Itô, Hiroshi Tanaka, Daisuke Fujiwara, Hui-Hsiung Kuo, and also Meyer.

At the conference, Robin Hudson gave a lecture titled “Quantum diffusions”. Meyer was impressed by the introduction of the role of creation and annihilation operators. After the conference was over, Itô, Tanaka, Meier and Kuo, all came to Delhi. Meyer came to my office and told me that he wanted a change and he was going to start a seminar on the elements of quantum probability in Strasbourg. The seminar he organized was called “éléments de probabilité quantique”. A series of papers, by different authors, many of which were expository accounts, appeared in the Strasbourg seminar.

Meanwhile, my 1984 paper with Robin had also appeared. Then, Hans Maassen discovered his kernel approach to quantum probability in quantum stochastic calculus. From then onwards, I always made a visit to Strasbourg whenever I visited Europe. I used to give long seminars, and it was there that I talked about my work and introduced the phrase “Schürmann triple”. These things have appeared in the Strasbourg seminars.

I forgot to mention one very important fact in the history of quantum probability. Luigi Accardi of the University of Rome organized a conference on quantum probability in Villa Mondragone to which I was invited. I had not met Accardi before. M.D. Srinivas was also invited, and we were waiting with our luggage for a train from Rome to Frascati. I had probably met Srinivas before that, but anyway, we recognized each other and were the only two Indians there. We traveled together to Frascati, which is famous for its wines. We were put up in some hotels and the conference was held in a large hall that was the villa of the Pope. This was the villa—Villa Mondragone—from which the Gregorian calendar, as we follow today, was announced. This was a very big hall but there was no blackboard or anything. There was only a white screen and I was not yet accustomed to lecturing without a blackboard. I spent all night writing my transparencies for the whole lecture and I presented a talk on quantum diffusion.

Accardi later visited Delhi and he asked me to teach him the proof of Mackey’s imprimitivity theorem. I gave a course on this theorem and its role in quantum mechanics, and how you can derive all the fundamental observables of quantum theory from it. Accardi took a liking to these things and so I began to attend every quantum probability conference. I was attending all the conferences till 2006, when I had a heart attack and also developed acute glaucoma. After that, I decided not to travel at all.

KRP: At that time, Waldenfels, Vittorio Gorini and Accardi were the editors of the first volume of Quantum Probability which appeared in 2006.¹⁶ Waldenfels invited me to Heidelberg and then we became good friends and spent a full month lecturing on this subject. At that time, Michael Schürmann was Waldenfels’ student. I even remember having casually told Schürmann that he should try to prove the Lévy–Khinchin theorem for infinitely divisible states on a Hopf algebra. I think he and Michael Skeide further developed this subject.

KRP: Martin Lindsay and David Applebaum were students of Robin Hudson. So, whenever I went to Britain, I visited the University of Nottingham under different schemes like the Commonwealth Professorship and the British Council Professorship. I used to help Martin and Applebaum during their PhD. We had many free discussions, which was how I came to know these two people. We had many lunches and dinners together and also many beers in local pubs.

KRP: Yes. You see, as I told you before, my career itself started with information theory by reading Khinchin’s book. I sent the first papers I wrote to Jacob Wolfowitz, who said that this was a completely wrong idea of information theory, and that Khinchin did not have the correct approach. So he returned the papers. Then I sent the same papers to Doob, who said it looked fine and published them in the Illinois Journal of Mathematics.¹⁷

There was a summer school in Calcutta on information theory in 1961, and in that summer school Varadhan, Shrikhande, Kesava Menon and I were present. We had a good time. Varadhan gave a course on Markov processes—that is his subject even today—and I gave a course on information theory. I wrote cyclostyled notes which became popular. There were even offers to convert it into a book but then because of the very strong influence of Varadarajan, I shelved the whole thing.

I presented my work on quantum stochastic calculus at the International Congress of Mathematicians (ICM) at Zurich in 1994. I was part of the IMU Executive Committee before the ICM at Berlin in 1998, which had David Mumford as the chairman. Peter Shor was given the Nevanlinna Prize at the 1998 ICM for his work on a polynomial time algorithm for factorizing an integer into its prime factors by using a quantum computer. I attended his lecture, and though I didn’t understand much, I just jotted down the key phrases. Then after coming to Delhi, I began reading papers on this topic and then presented some of the ideas on quantum computation in Calcutta. I gave one seminar in Delhi in 2000, and news spread that something in quantum computing was going on.

I think some people at TIFR Bombay took interest in this subject because they have a department for computer science. Raman Parimala was the chairman of the School of Mathematics. She wrote a letter asking if I could come and give a course on this subject in TIFR Bombay. This was in the year 2000, and by this time, Nielsen and Chuang’s famous book had appeared.¹⁸ I was teaching at CMI at this point, and when my teaching and exam duties were over, I went to TIFR and gave a course. There, we had a very enthusiastic participant by name Amitava Bhattacharya, who took down notes. I gave him my scribblings on the subject and he worked very hard and prepared notes, where we had simplified the arguments of Shor and others. The algorithm for polynomial time factorization was included in these notes and then Jaikumar Radhakrishnan asked me to include a section on classical information theory and Shannon’s theorem. Then I wrote that part also and the whole thing was published as part of the TIFR Lecture Notes series and it was put on the web.¹⁹

I also taught the same course at IISc Bangalore, on J. Pasupathy’s invitation; at IMSc, Chennai; as well as Chungbuk National University in South Korea, on Un Jig Ji’s invitation. So, I have given this course in four places. I extracted from these notes the parts about coding theorems of classical and quantum information theory and wrote a little book for Rajendra Bhatia’s TRIM series in 2006.²⁰ There were many errors in it which I revised, but by that time my eyesight was very poor. Ajit Iqbal Singh helped me a lot by reading it line by line and in correcting all the mistakes. The second edition has much fewer errors. I am very fond of this little book, which is red in colour. They used to call Mao Zedong’s book “the little red book”, and I used to call my book also as the little red book!

KRP: I think my happiest result would be the Quantum Itô’s formula with Robin Hudson. The next one, if you wish me to say, would be the theory of the Lévy–Khinchin formula in locally compact Abelian groups, which I developed with Ranga Rao and Varadhan.²¹ Then there are my little papers on information theory which were published as benchmark papers in a volume by Oxtoby. Whenever I liked the basic results, I always wrote a book by giving a course to students, like the book on probability measures in metric spaces from the course I gave in Sheffield.

KRP: That’s a difficult question. I found the Peter–Weyl theorem for classifying the irreducible representations of compact groups, followed by Hermann Weyl’s trace formula for irreducible representations of compact semi-simple Lie groups, to be the most inspiring theorems. I think that is the most beautiful formula, but later on you can see the beauty of this in Harish-Chandra’s trace formula. Now I have more or less forgotten most of these proofs, but the Peter–Weyl theorem is very simple. Once you learn it, you will never forget it.

V.S. Sunder: You didn’t talk about Klaus Schmidt. When did you interact with him?

KRP: My interaction with Klaus Schmidt played a very important role in the evolution of quantum stochastic calculus.

I was teaching in Manchester. Klaus Schmidt had written a thesis on number theory under Edmund Hlawka in Vienna. One fine morning, Klaus Schmidt wrote to me that he would like to come to Manchester and work with me. Another Austrian, Karl Sigman, also wrote to me stating he would like to come to Manchester and work with me. So, both of them visited Manchester. Karl Sigman worked in ergodic theory; he soon left Manchester and worked on applying probability and ergodic theory in biological models.

My happiest result would be the Quantum Itô’s formula with Robin Hudson

But Klaus Schmidt stayed with me. I was running a regular weekly seminar on quantum mechanics in Manchester, where I used to lecture Klaus Schmidt, Roger Plymen and his students. Then one day Ray Streater, who was at Bedford College, invited me to London to give a seminar on infinitely divisible distributions. There, Streater told me that Araki had dealt with infinitely divisible positive definite functions, which was contained in a gist of the famous paper on the Araki–Woods embedding theorem. Streater mentioned that he was a physicist and he gave a heuristic account of this thing. Then I came to Manchester.

There is a book by Gnedenko and Kolmogorov on limit theorems for sums of independent random variables.²² In Kai Lai Chung’s translation, there is an appendix by Doob where he remarks that Gnedenko and Kolmogorov seem to be using only positive definite functions and they don’t seem to need any probability at all. I was impressed by this. So if they are using positive definite functions, why not try positive definite kernels?I told Klaus that we should look at infinitely divisible positive definite kernels. We worked quite hard on this and we had also read this paper by Araki and Woods. You combine the two ideas and it naturally leads to the idea of a continuous tensor product. And then because you are also interested in positive definite kernels which are invariant under a group action, it automatically throws up co-cycles, and through these co-cycles, we got a different proof of Lévy–Khinchin formula which had no notion of probability at all. We thus satisfied the condition put by Doob, which is that positive-definiteness alone should imply the central limit theorem and Lévy–Khinchin formula.

So, Klaus and I wrote a paper for Acta Mathematica on this,²³ and then there was a statistical interpretation for infinitely divisible positive definite functions. So, when you apply this to the Heisenberg group, you get all the necessary quantities which crop up in quantum theory. So it played a very important role that infinitely divisible positive definite kernels are invariant under a group action. That again can be thought of as Gaussian covariance, so all these things are linked.

I later came to Calcutta and gave a course on these ideas. Nadkarni was a regular participant in that seminar. He was also interested in the imprimitivity theorem, which led to his paper with Bagchi and Mathew in Acta Mathematica.²⁴

I was staying in the guest house at Calcutta while giving this course. At that time, Calcutta was facing Naxalite troubles. I remember one day I was sitting in the guest house with a pencil and paper and working, when, suddenly, somebody from the adjoining football ground threw a big stone that landed right on the dinner table. We heard gunshots later in the night.

The State of Mathematics Research in India

What is your opinion about the mathematics research scene now in India?

What is your message to students?

This is my final question—what is mathematics?

Family Life

RB to Shyama, KRP’s wife: Is he a very systematic person?

Did your children get interested in mathematics?

Audience: Do you have any preferred time of the day for your work?

Audience: What motivates you to do research after these many years in the field?

Audience: So according to you, the best way to learn a new subject is by giving seminars to each other?

Gelfand used to say, “Read the theorems but not their proofs”

acknowledgement We thank Shilpa Gondhali and Tiju Cherian John for their assistance in preparing this interview.

Footnotes

1. Early fluid dynamics disregarded the viscosity term, as the main focus was the mathematical structure of this approximated problem. This did not really describe the true behaviour of water, and was therefore called ”dry water”. ↩
2. Indra Mohan Chakravarti, Radha Govira Laha, and Jogabrata Roy. 1967. Handbook of Methods of Applied Statistics. Wiley Series in Probability and Mathematical Statistics. ↩
3. K.R. Parthasarathy. 1961. “On the Category of Ergodic Measures”. Illinois J. Math. 5: 648–656. ↩
4. J.C. Oxtoby. 1961. “On Two Theorems of Parthasarathy and Kakutani Concerning the Shift Transformation”. In: Ergodic Theory, Proc. Internat. Symp. Tulane University, Academic Press, New York, pp. 203–215. ↩
5. K.R. Parthasarathy. 1962. ”A Note on Mixing Processes”. Sankhyā. Ser. A. 24: 331– 333 and K.R. Parthasarathy. 1963. “Effective Entropy Rate and Transmission of Information through Channels with Additive Random Noise”. Sankhyā Ser. A. 25: 75–84. ↩
6. V.S. Varadarajan. 1961. “Measures on Topological Spaces”. Mat. Sb. (N.S.) 55(97): 1, 35–100. ↩
7. A.M.S. Translations, No. 48, 14–228 (English). ↩
8. Harish-Chandra. 1951. “On Some Applications of the Universal Enveloping Algebra of a Semisimple Lie Algebra”. Trans. Amer. Math. Soc. 70: 28–96. ↩
9. Arthur Strong Wightman. 1960. “Quantum Field Theory and Analytic Functions of Several Complex Variables”. J. Ind. Math. Soc. 24(3-4): 625–677. ↩
10. K.R. Parthasarathy and V.V. Sazonov. 1964. “On the Representation of Infinitely Divisible Distribution on a Locally Compact Abelian Group”. Theory Prob. Appl. 9: 118–122. ↩
11. K.R. Parthasarathy. 1967. Probability Measures on Metric Spaces. Academic Press, New York. ↩
12. R.L. Hudson. 2012. “The Early Years of Quantum Stochastic Calculus”. Comm. Stoch. Anal. 6(1): 12. ↩
13. C.D. Cushen and R.L. Hudson. 1971. “A Quantum-Mechanical Central Limit Theorem”. J. App. Prob. 8(3): 454–469. ↩
14. R.L. Hudson, P.D.F. Ion, and K.R. Parthasarathy. 1982. “Time-Orthogonal Unitary Dilations and Noncommutative Feynman–Kac Formulae”. Comm. Math. Phy. 83(2): 261–280\\ and R.L. Hudson, P.D.F. Ion, and K.R. Parthasarathy. 1984. “Time-Orthogonal Unitary Dilations and Noncommutative Feynman–Kac Formulae II”. Publ. Math. Res. Inst. Kyoto. 20: 607–633.. ↩
15. Robin L. Hudson, and K.R. Parthasarathy. 1984. “Quantum Itô‘s Formula and Stochastic Evolutions”. Comm. Math. Phys. 93(3): 301–323. ↩
16. Accardi, Luigi, Alberto Frigerio, and Vittorio Gorini, eds. Quantum Probability and Applications to the Quantum Theory of Irreversible Processes. Proceedings of the International Workshop held at Villa Mondragone, Italy, September 6–11, 1982. Vol. 1055. Springer, 2006. ↩
17. K.R. Parthasarathy. 1961. “On the Category of Ergodic Measures”. Illinois J. Math. 5: 48–656 and K.R. Parthasarathy. 1961. “On the Integral Representation of the Rate of Transmission of a Stationary Channel.” Illinois J. Math. 5: 299–305. ↩
18. Michael A. Nielsen and Isaac L. Chuang. 2000. Quantum Computation and Quantum Information. ↩
19. K.R. Parthasarathy. 2006. Quantum Computation, Quantum Error Correcting Codes and Information Theory. TIFR Lecture Notes, Narosa Publishing House, New Delhi. ↩
20. K.R. Parthasarathy. 2006. Coding Theorems of Classical and Quantum Information Theory. Texts and Readings in Mathematics, Hindustan Book Agency, New Delhi. ↩
21. K.R. Parthasarathy, R. Ranga Rao, and S.R.S. Varadhan. 1963. “Probability Distributions on Locally Compact Abelian Groups”. Ill. J. Math. 7(2): 337–369. ↩
22. Gnedenko, Boris Vladimirovich, et al. 1968. Limit Distributions for Sums of Independent Random Variables. Vol. 233. Reading, MA: Addison-Wesley. ↩
23. K.R. Parthasarathy and K. Schmidt. 1972. “Factorisable Representations of Current Groups and the Araki–Woods Imbedding Theorem”. Acta Math. 128: 53–71. ↩
24. S.C. Bagchi, Joseph Mathew, and M.G. Nadkarni. 1974. “On Systems of Imprimitivity on Locally Compact Abelian Groups with Dense Actions”. Acta Math. 133(1): 287–304. ↩