A bit about my research in mathematics

It seems natural in old age to reflect on one’s past and that has been the case with me recently, in particular with reference to mathematics. one used to study, pass examinations with the hope of getting a stable government job at some stage. That was the case with me and I was not particularly interested in any topic. Then suddenly at the age of 15 Professor V.Krishnamurthy in Loyola College, Madras started teaching a topics course once a week. he explained about rational numbers, real numbers, that there are many more real numbers than rational numbers using Cantor’s diagonal argument. I got hooked. It seemed to be subject one could understand which has ideas and not just calculations. From then on I wanted to study mathematics with the idea of ending up as a teacher somewhere and continue to learn and enjoy mathematics. But after a few years, just learning became more of the same and there were natural questions around what one was learning and some of them seemed worth pursuing. So learning slowly turned in to a mixture of learning and research. But I did not like teachers except one or two like Krishnamurthy and did not want to be guided in research. So I did not acquire expertise in any topic and what I leaned became a mixture actual knowledge together with some statements which I assumed without knowing proofs and kept thinking about them off and on trying to figure out how to prove such things. Meanwhile the subject kept changing and one had to learn new things alone though in an institutional setting. One such event was the arrival of William Thurston. One had to learn what he was doing or give up the subject. But I did not have the background to understand any of it. One way of entry seemed to be through old complex analysis and Mobius transformations leading to some hyperbolic geometry and limit sets and such. It is at this stage I got acquainted with Bernard Maskit. I was wondering about limit sets and he showed me the thesis of one of his students Perry Susskind. It was exactly what I was looking for and then I realised that they could be extended to all dimensions using Margolis Lemma. Perry kindly agreed to collaborate and it was exactly the kind of stuff I liked, learning combined with a bit of research. This knowledge of limit sets also led to some work with Peter Scott on the geometric finiteness of some Kleinian groups using famous work of Cannon and Thurston. This way, I slowly started understanding bits and pieces of Thurston’s work and at the same time writing papers and surviving in the profession. This acquaintance with Maskit, learning about Susdkind’s thesis etc seemed a crucial part of my continuing in the profession when survival seemed difficult after the arrival of Thurston. Another strand in my career was the theory of JSJ decompositions in which I entered accidentally. While passing through London in 1977, I found a copy of Hempel’s book on three manifolds. I found that there was no reference to me in the book though some work similar to mine made the references. I started looking at the problems at the end of the book and solved one of them on the train to Southampton. I was wondering whether it was publishable or not and later realised that it was a special case of a well known result of Johannson with a long proof and duly published it. I slowly acquired the reputation of having some expertise on JSJ without really reading any of the papers or knowing any thing. Then around 1996, Walter Neumann organised a conference and asked me to write a survey paper on JSJ for the conference. I foolishly agreed but found it impossible to read the relevant papers. After a month I told Walter that it is becoming more of research project and I would do it if he collaborated with me. Peter Scott said “An elementary proof of the existence of the JSJ decomposition was given by Neumann and Swarup in Their arguments greatly simplified the subject by concentrating only on embedded annuli and tori.” It immediately got in to text books. But since it is a joint paper and I had this unfortunate habit of not reading the parts written by collaborators, I am still struggling with the topic and again planning to write a proof of the deformation theorem for my own understanding.