M.S. Raghunathan

* India has become a player of reasonable standing in the international mathematical arena — not quite a big power yet but with reasonable prospects of attaining that status. *

Very few, even among mathematicians, would recognise June 11 as a landmark date for Indian mathematics. It was this day 95 years ago, Ramanujan’s work — work done in India — was presented formally for the first time to a fairly big professional mathematical audience: G.H. Hardy spoke on some of Ramanujan’s (as yet unpublished) research at the London Mathematical Society at the monthly meeting of June 1914. Among others, two well-known Cambridge mathematicians who had been approached by Ramanujan but who did not respond, were present at that meeting. It can perhaps be considered the day on which India gave notice to the world that it had ambitions of joining the big league in mathematics.

Indian mathematics attracted international attention even earlier: Syamadas Mukhopadyaya’s paper proving the “Four vertex theorem” for convex curves published in 1909 is a case in point. But this was the first time Indian work was presented formally on a forum outside India. Also, the work Hardy spoke on was research of a higher calibre than work that had emerged from India till then. The anniversary of that event is as good a time as any to introspect on how far our ambition has got.

** Epochs in the past**

There were epochs in the past when India was a world leader in mathematics. Already in the Vedic Period (circa 1000 BCE), India embarked on the study of geometry. There is firm evidence that Pythagoras’ theorem was known by the eighth century BCE. The concept of zero as a number originated in this country (probably as early as 200 AD) as did the place value system of representation of numbers as is commonly used today. No other (mathematical) discovery has perhaps had a comparable impact on everyday life and, of course, science. Arab scholars from West Asia who came to India soon after Baghdad’s emergence as a great cultural centre were greatly taken up with this system that facilitated dealing with large numbers and performing arithmetical calculations with great ease, and helped spread it westward. When it first reached Europe, the Church resisted it but eventually good sense prevailed. What was in the Middle Ages at the frontiers of science is now part of any 10 year-old’s repertoire.

Aryabhatta (5-6th Century AD) and Brahmagupta (6-7th Century AD) were formidable minds who made great contributions to astronomy and trigonometry and were undoubtedly the greatest mathematicians in the world in their times. So was Bhaskara in the 12th century. The idea of dealing with manipulation of variable quantities rather than numbers per se — algebra in essence — is again of Indian origin. It was developed more fully by the Arabs and the Persians. Bhaskara also seems to have had a gift for poetry but, unlike Omar Khyyam (the Persian poet who was also a first-rate mathematician), he used the gift exclusively in the service of his mathematics. In his famous work, Lilavati, many of the mathematical problems are presented in verse. It would appear that in the past India gave the world a lot more mathematics than it learnt from outside. This is not entirely a matter of pride. Rather, this insularity eventually cost us our pre-eminence in mathematics.

The last period of glory for Indian mathematics was 200-odd years from the 13th to 16th century. An outstanding school had developed in Kerala and its most significant achievement was the anticipation by Madhava (the school’s leading figure) of some of the ideas of Calculus which Newton and Leibnitz came up with in the 17th century. The school’s work, unfortunately, does not seem to have received notice in Europe where the Renaissance had ushered in unprecedented intellectual activity.

After the demise of the Kerala School, mathematical activity more or less came to a standstill. It was the creation of universities in Madras, Bombay and Calcutta by the British in the mid-19th century that revived mathematical activity in the country and we owe a great deal to British academics for that. The initial years were a time of assimilating the developments that had taken place in Europe, leaving us far behind. By the turn of the century, however, India was embarking on the creation of new mathematics.

Hardy’s address to the London Mathematical Society ushered in a new level of impact of Indian mathematics on the international scene. Ramanujan’s contributions were mainly in Number Theory and related areas. It was inevitable that with his iconic status, Indian mathematicians in the post-Ramanujan era mostly pursued these areas. But starting mid-1930s, there were efforts to move into some other areas as well. Today, Indian mathematicians are at work in a broad spectrum of areas in all of which they have made a significant impact. Many are well recognised names internationally.

** Number Theory**

Number Theory continues to attract many adherents in the country. After Ramanujan, there was some work by S.S. Pillai (working then at Annamalai University) that attracted attention. He made impressive progress on what is known as Waring’s problem. Some 50 years after Pillai, R. Balasubramanian, currently director of the Institute of Mathematical Sciences (IMSc), Chennai, in collaboration with two Frenchmen, came up with the last word on that problem.

Among the Indian mathematicians working in India, again I would like to mention two persons who have performed at the highest level — M.S. Narasimhan and C.S. Seshadri. Narasimhan is a versatile researcher with very important contributions in several areas: Differential Equations, Differential Geometry, Algebraic Geometry, Lie theory and Mathematical Physics. Seshadri’s work is largely in Algebraic Geometry and his contributions have set the directions of important sub-areas in the field. The former is a recipient of the prestigious King Faisal Prize, an international award instituted by Saudi Arabia, while the latter has been awarded a prize given by the International Centre for Theoretical Physics in Trieste, Italy. A third major figure is C.R. Rao whose work has had a big impact in the way his research area — Statistics — has developed over the last several decades. Rao moved to the U.S. after his retirement.

There are other individuals working in the country who have achievements of the highest order to their credit and have received international recognition. Manindra Agarwal of IIT, Kanpur, received the Godel as well as Fulkerson prize for his work on “primality testing” in 2006, while Sujatha Ramdorai of the Tata Institute of Fundamental Research (TIFR) received the 2006 Ramanujan Prize (this is for mathematicians from developing countries). Several Indian mathematicians have received the Third World Academy of Sciences (TWAS) prize (again given only to mathematicians from developing countries). Two institutions have established themselves as centres of excellence internationally: the TIFR and the Indian Statistical Institutes (ISIs).

The diaspora too has performed at superior levels contributing to the reputation of Indian mathematics. Among those who worked largely outside the country, mention should be made of Harish-Chandra and S.R.S. Varadhan, both of whom have had a major role in the very evolution of their fields. Varadhan’s formative years were spent at the ISI, Kolkata.

In sum, it is fair to say India has indeed become a player of reasonable standing in the international mathematical arena — not quite a big power yet but with reasonable prospects of attaining that status. An indication is that India has won the bid to hold the International Congress of Mathematicians in August 2010. The event, by far the most important international mathematical meeting, takes place once in four years. This is the first time in over a hundred years (the first congress was held in 1897 in Zurich) that India will be hosting one. The venue for the 8-day congress is Hyderabad. The other Asian countries that have hosted an ICM are Japan (1990) and China (2002). An invitation to give a talk at the congress is considered highly prestigious. It is again a measure of our standing that since 1970, there has been at least one invited talk by an Indian working in India, a record not matched even by some West European countries or China.

The attendance in recent congresses has been around 3,500 and that is the kind of number expected at Hyderabad too. Some 700-800 Indian mathematicians are expected to participate. There will be plenty of possibilities for (postgraduate) students to participate, giving them an opportunity to interact with the finest mathematical minds and enlarge their horizon. There will also be programmes in the run-up to and during the congress reaching out to the general public, attempting to give them an idea of what mathematics is all about.

(Prof. M. S. Raghunathan, an eminent Indian mathematician, is Homi Bhabha Chair Professor at the Tata Institute of Fundamental Research, Mumbai.)