In September this year I was announced as one of the recipients of the Shanti Swarup Bhatnagar Award for my work in Mathematics. What follows below is essentially what I gave in response to queries from reporters.
Motivation for research
In High School I became interested in Mathematics and Physics. I had a number of good science and mathematics teachers. At home my parents had a number of "popular science" books. We also had access to the (cheap) MIR publications books on Mathematics from USSR.
When I secured a rank in JEE my parents and my sister insisted that I follow my own interests. They said I should not be pressurised into joining Electrical Engineering (which was the most popular discipline at that time) if I wanted to choose M.Sc. in physics or mathematics.
IIT Kanpur in the late 1970's and early 1980's was a very active undergraduate campus. My teachers and co-students were not only interested in getting good grades but also in discussing mathematics, physics, computers (which were somewhat new then), music, politics, philosophy, society and life in general. There was an emphasis placed on being a "fundu" rather than "just another 10-pointer". "Bull" sessions were the norm; where anything and everything was discussed.
In 1978/1979 I went to NCERT Summer Schools in IIT Madras/Kanpur where I met a number of other students who motivated me to study more. Some of the names that come to mind are:
- Nitin Nitsure (currently Professor, TIFR)
- Dipendra Prasad (currently Professor, TIFR)
- K. Vidyanath Rao (currently Professor, Ohio State University, USA)
- Siddhartha Sahi (currently Professor, Rutgers University, USA)
- Jugal Kishore Verma (currently Professor, IIT, Mumbai)
- Usha Balasubramanian (who was then a student of Pune Vidyapeeth)
- T. N. Venkataramana (currently Professor, TIFR, Mumbai)
In 1979 I learnt LISP programming from Keshav Nori (currently TCS, Pune).
In 1979 Madhav Nori, V. Srinivas and Arvind Banerjee visited IIT Kanpur and gave motivating lectures on Algebraic Topology, Number Theory and Geometry.
In 1980 at a summer school in Mysore I learnt a lot of mathematics from the teachers there which included Kalyan Mukherjea (ISI, Kolkata) and S. Ramanan (TIFR, Mumbai).
In 1981 at a summer school in Pune I learnt a lot of mathematics from the teachers there which included N. Mohan Kumar, R. V. Gurjar and Balwant Singh (all from TIFR, Mumbai) and C. Musili from University of Hyderabad.
TIFR, Mumbai in the 1980's was a very lively atmosphere for learning mathematics. Questions of all kinds were raised and discussed at all times from breakfast till midnight tea-time! I count many of those who were there at that time among my teachers. In addition to those from the TIFR mentioned above I would single out R. R. Simha (currently Emeritus at Mumbai University) and D. S. Nagaraj (currently Professor, IMSc, Chennai) as being ever willing to discuss questions in any part of mathematics.
My Ph. D. was under the guidance of S. Ramanan. One chapter of the thesis was done under the guidance of Madhav Nori. In addition, I received a lot of help and guidance from Mohan Kumar and V. Srinivas.
In 1990-91 I visited the University of Chicago for one year. There I had to teach courses UG as well as PG; I found the experience very enjoyable. The faculty there, especially Pavaman Murthy, Madhav Nori, Raghavan Narasiman, Niels Nygard, Spencer Bloch and William Fultaon gave me all the input that led me to the study of Algebraic Cycles. The AMS meeting on Motives led to interactions with Dinakar Ramakrishnan as well.
I later visited University of Utah for a workshop on Higher Dimensional Geometry. Interactions with Kollar, Shokurov, Mori, Corti and Alexeev were really fruitful.
On my return to TIFR in 1991, Srinivas and I started a seminar on motives and algebraic cycles. The entire group there including youngsters like Rajan, Indranil Biswas, Kirti Joshi was very dynamic. A brief visit in 1993 to Essen where I worked with H. Esnault and E. Viehweg was also very fruitful.
In 1993-95, I was in Bangalore where I worked with V. Pati of ISI, Bangalore. Through discussions with Pati, I learned to appreciate differential geometry. In collaboration with A. Sitaram and Gadadhar Misra we started a seminar to study the work of Harish-Chandra. These people also pushed me to look at the problem of finding simpler expositions of advanced topics.
On joining IMSc, in 1996, I was fortunate to find G. V. Ravindra, Gautham Dayal as students who wanted to take up the challenge of studying algebraic cycles. Later Jishnu Biswas as a post-doc and Shreedhar Inamdar as co-faculty joined this group which became Seminar on Algebraic Cycles in Chennai (SACCHE = truthful in Hindi).
Academic pursuits of people close to me
My father, the late Dr. H. K. Paranjape, was a well-known economist who taught at Sydenham College, Mumbai, IIPA, New Delhi and worked for the Govt of India on the MRTP Comission and Rail Tariff Enquiry Commision.
My mother, Manak Paranjape was a Professor of French at numerous places including Elphinstone College, Mumbai and JNU, New Delhi. She has written a book on teaching French to working scientists and engineers which I used during my research years.
My mother's elder brother (my sakkha mama) Dr. Madhav Varde is a Ph. D. in Chemistry who worked for BARC and RCF in Chembur, Mumbai. He also was very interested in making practical things based on scientific principles which he taught us.
My wife, Sudeshna Sinha is a Professor of Physics at the IMSc. We have numerous discussions (arguments!) on everything under the sun. She has been the force that has kept me on track to complete the projects that I undertake.
Area of Research and Work Done
Broadly speaking, I work in Geometry---more specifically Algebraic Geometry and even more specifically in Algebraic Cycles. I have written about 20 papers in these areas.
I have worked on the connections between algebraic cycles and Hodge Theory. The "Hodge conjecture" has been chosen as one of the Millenium Prize Problems by the Clay Mathematics Institute. One part of my work has been to exemplify and refine the implications of this conjecture. In particular, a paper I wrote (in the journal: Annals of Mathematics 1994) shows some interesting implications of the conjectures for the study of algebraic cycles on hypersurfaces. This paper has been very well received.
After that work was done, I have studied the connections of these questions with problems in Number Theory and have written a number of papers on the interconections. I am particularly proud of the paper on "A Geometric Characterisation of Arithmetic Varieties" published in the Proceedings of the Indian Academy of Sciences (Mathematical Sciences), 2002.
Of late I have been working with Dinakar Ramakrishnan trying to find some geometric representatives of modular forms.