Recently, somebody asked me me to write about my mathematical experiences. Here it goes, based on comment in Pramathanath Sastry’s post.
There is no precedent for mathematics in the family. I came from a farming family, though my father was a teacher who taught English and History. I started studying mathematics, a B,A. (Hons) course in Madras University since father thought it would be good for competitive examinations, particularly, I.A.S. Examinations where one could choose two optionals in mathematics. I was doing ok but did not understand what I was doing. Even calculus I started understanding only after I saw how real numbers were constructed. Slowly, it be came a passion, mainly learning. Also I was not in any sense trained, cutting classes and learning what I liked. When it came to research, I did not want to guided and continued learning what I could understand. So the background was always patchy and incomplete. There was no ambition or aim except that I settled on topology early after browsing Hausdorff’s ‘Set Theory’ in college. Some of the results seemed metaphysical, like the invariance of domain and dimension, and I wondered how such things could be proved at all. Luckily I found an unpublished lecture notes by Samuel Eilenberg in 1964 and saw the proofs. To this day Algebraic Topology has been the first love.
For a long time I was happy with abstract and formal thinking without worrying about examples. As late as 1992, a collaborator was exhorting that integers were a counterexample to what I was trying to prove. Anyway, with interest more in learning than career, learning continued. If I could not do a ph.d. there was always the possibility of going home and teaching in a mofussil college. After a few years, research came easy since there are always natural questions when you learn some thing. So it continues.
Then at some stage, around 1978 or so William Thurston came long, and learning became difficult. After a few years I went back to old books on Complex Analysis trying to learn the background to Thurston and organised a one year seminar in I.S.I. Delhi. Around this time Peter Scott advised that it would be difficult to learn this stuff without talking to others and invited me to a conference in Warwick which he organised. There I found that even senior professors were struggling with this and were trying to figure out even the definitions. In any case, I started talking to others and trying to learn through conversations.
Then luckily in the late eighties, Peter Scott decided to collaborate with me perhaps because he used some of my early work. That led to a different sort of approach. I was getting interested in other stuff like political economy and would not work in mathematics for months. But once in a while there would be queries from Peter related to joint work. It would take a week or so to understand the definitions and start thinking about the problems again. Sometimes, I did not understand my own papers and a colleague Lawrence Reeves kindly agreed to listen to me when I tried to read them again. Again talking to others helped. Then at some stage, some of the areas of research became more crystallised and one could start thinking about them without too much paraphernalia.
For the last 25 years or so, it seems more physical. You have these things floating around in front of you and keep staring and feel some thing emerging. Often the results are approximate but stable. Recently I found that several things in a 2003 paper were incomplete and since then there are few more papers based on that paper. But over all they seem correct. May be it depends on how much one is exposed to at a given stage. But in my case, keeping the problems in front of my mind and staring at them for days and months seem to help. But this is with somebody not particularly talented ( I know since I collaborated with some good mathematicians) and I do not know how it works for others. Somehow, the work in the last three four years seems to be the most satisfactory work I have done so far.
