Laurent Schwartz and Development of Mathematical Research in India

Ushering novelty and freshness

Laurent Schwartz was a great mathematician and an inspiring teacher. What made him even more unique were his human qualities—he fought against injustices, oppression and exploitation, and he played an active role in promoting the cause of the disadvantaged. Schwartz was an internationalist who had empathy towards developing countries and an understanding of their problems and of their potentialities. One example of his concern and commitment was the crucial support he brought to the development of mathematical research in India in the post-independence period.

In the early fifties, Komaravolu Chandrasekharan initiated a programme for promoting mathematical research at the highest level at the Tata Institute of Fundamental Research (TIFR) in Bombay. At his invitation, Schwartz visited TIFR in 1955 for the first time and this was a landmark in the history of mathematical research in India. He gave a course on complex analytic manifolds. Starting from the definition of a differentiable manifold, the course covered—among other topics—currents, de Rham’s theorem, elliptic partial differential equations, Hodge theory, Kähler manifolds and the Riemann–Roch theorem for compact Riemann surfaces. His inspiring lectures not only introduced young mathematicians at TIFR to important topics and techniques which were hardly studied in India at that time, but also introduced them to a whole new way of thinking about, and of doing, mathematics. l was a research student at TIFR at that time. I wrote the notes of his lectures and this gave me an opportunity to get to know him. Conjeeveram S. Seshadri also attended and our common interest in the field of algebraic geometry can be traced to the impetus received from these lectures.

His visit to India made him aware of the potential for the development of mathematics in India. He proceeded to bring concrete support to the efforts of those In India who were already engaged in the task of developing mathematics there. He persuaded a galaxy of French mathematicians to visit Bombay for extended periods and to give extensive courses. Amongst the early visitors were Francois Bruhat, Jean-Louis Koszul, Jacques-Louis Lions and Bernard Malgrange. In this way, young Indians were introduced to areas at the forefront of contemporary research.

Schwartz also arranged, with the help of Henri Cartan and others, for some young Indian mathematicians to visit Paris for long periods. This enabled them to work with eminent French mathematicians. I myself worked in Paris with Schwartz, and Seshadri with Claude Chevalley. We used our stay in Paris to familiarise ourselves with the tremendous development that was then taking place in algebraic geometry, and naturally this played an important role in our future research.

Within a few years, groups of first rate mathematicians were formed in India, working in algebraic geometry, differential geometry, complex analysis, Lie groups, algebra and number theory. In a letter written in 1971, Schwartz said that the development of mathematical research in India during this period was a remarkable achievement. In this, it must be said that his role was crucial. Schwartz kept in close touch with Indian mathematicians and visited India several times.

I am personally grateful to him for my mathematical development and the inspiration he provided. My family and I got to know Madame Schwartz as well and we have very fond memories of her kindness and friendship. During my stay in Paris I had some serious health problems and Schwartz made special efforts to see that l received the best medical attention. During my recuperation, he gave me, as “light reading”, the huge manuscript of Kodaira and Spencer on deformations of complex structures—in retrospect, this turned out to be a very perceptive choice of gift. The work of Kodaira and Spencer played a crucial role in my future research.

M.S. Narasimhan is presently associated with the Centre for Applicable Mathematics of TIFR in Bengaluru, and with the Department of Mathematics, Indian Institute of Science, Bengaluru. He was a professor at the Tata Institute of Fundamental Research in Mumbai for nearly three decades. He worked with Laurent Schwartz in Paris from 1957 to 1960.

This article was first published in Gazette des Mathématiciens Édition Spéciale (2003) pp. 37–38. We thank the French Mathematical Society (SMF) for the permission to reprint in Bhāvanā. We reprint it, with minor edits, in this issue of Bhāvanā along with the memoirs of Schwartz to provide context for some of the themes dealt with in the memoirs and to highlight his role in transforming the mathematical landscape of post-independence India.