## Not just a probability | A mathematician’s journey from Chennai to Padma Vibhushan

Mathematician S. R. Srinivasa Varadhan, who was recently named for the Padma Vibhushan, India’s second-highest civilian honour, is no stranger to awards.. Prof. Varadhan, who is currently the Frank J. Gould professor of science and a professor of mathematics at New York University’s Courant Institute of Mathematical Sciences, was awarded the Abel Prize given to outstanding international mathematicians in 2007; the Padma Bhushan in 2008 and the National Medal of Science, United States, in 2010, among others.

Born in Chennai in 1940, Prof. Srinivasa Varadhan graduated with an Honours degree in statistics from Presidency College, Chennai in 1959, after he which he joined the Indian Statistical Institute, Kolkata, where he completed his Ph.D in 1963.

In an email interview with *The Hindu*, he describes his journey from Chennai to the United States, his specialisation in probability theory, maths for students, and more.

**Can you tell us a little about your journey as a mathematician and a scientist from Chennai to New York?**

I was good in math as far back as I can remember. [I]was above my class level, and was fast tracked. During grades 6-11, my father was the headmaster of the school I attended, a government school in a small town twenty miles north of Chennai. [I]did well, especially in math and science. I had an uncle in Tambaram, and we tried, and succeeded, in getting a seat in “intermediate” a two-year programme before branching off into medical, engineering, science or humanities [courses].

I stayed with my uncle, along with two of my cousins, and had to decide after two years what to pursue. I was not interested in engineering. Majoring in mathematics was not appealing, due to [a] lack of job opportunities. I needed to get into an “Honours” programme in the basic sciences that led to a Master’s degree in three years rather than the four years in a regular programme. I was eligible to apply for established programmes in mathematics, physics, chemistry and an emerging new programme in statistics. It was hard to get in. The number of seats were under 20 in the whole State in all colleges combined, for physics, chemistry or statistics and slightly more in mathematics.

I was told statistics had good job prospects, and so I tried, and was lucky to be admitted as one of 13 students. We were a close-knit group and stayed together for three years. After 60-plus years we are still in touch and get together on Zoom every two weeks. The college that we attended was Presidency College facing the Bay of Bengal and we could watch the ocean from our classroom. It was a long commute for me. My father had retired, and took up a position as headmaster in a private school in Tambaram and after one year, I persuaded my parents to let me stay in the hostel. Those two years were idyllic. I would spend early evenings on the beach gossiping with my friends. I did not find schoolwork demanding and could take all the exams without any preparation. I graduated after three years in the Honours programme having scored the highest marks and was awarded a prize as the best student to graduate that year.

The question now was, what next? The big attraction for many of my classmates was high-level administrative positions in the government, banking, insurance etc. If you did well in a competitive exam you were selected for a high-level position and in addition to a great career, you also had the satisfaction of serving the country. The other option was to be trained as a professional statistician and work in the industry. One final option was research. I thought I would try it. [I] had no idea what it would involve. The idea was, if I did not like it, I was young enough to take the competitive exam! I was selected to do research at the Indian Statistical Institute [Kolkata], where I showed up in August, 1959. I was given a desk and there were a couple of courses that we could attend that lasted three months. I thought of doing research in applied statistics and was quickly disillusioned.

Two senior research scholars, [K. R.] Parthasarathy and [R.] Ranga Rao, persuaded me to work with them in probability theory. They had worked earlier with [Veeravalli S.] Varadarajan, who had left soon after I arrived in Kolkata, to go to the United States. It was great for me. Probability was, for me, the right mixture of mathematical rigour and statistical intuition. The three of us worked together for over two years. [Then] they both left, one to go to the United States and the other to Russia. Varadarajan returned and we worked together for a year, before I left for the United States.

Meanwhile, I had finished my thesis and Andrey Kolmogorov, a great Russian mathematician [who contributed significantly to probability theory] and father of the modern theory of probability, was visiting ISI [in 1962]. My advisor, [renowned statistician] Dr. [C.R.] Rao wanted him to be one of the readers of my thesis. I was to give a one-hour presentation of my work, which I did, except [that] it went on for nearly two hours. I got carried away in my presentation. When the talk ended, the audience who were at the end of their patience, got up and started to leave. Kolmogorov, who had stood up with a piece of chalk to say something, threw the chalk down and walked out in anger. My immediate reaction was: There goes my PhD. Along with fellow students, I rushed behind Kolmogorov, apologizing profusely for talking too long. He was calm and said he did not mind the long lecture but was annoyed because, as he put it, “When Kolmogorov speaks, people should listen.” He provided a very nice report on my thesis.

In the spring of 1963, I wanted to go to the United States for postdoctoral work, and Varadarajan wrote to Peter D. Lax at the Courant Institute [of Mathematical Sciences – New York University] suggesting my name for a “visiting member” as they were then called at the Institute. After some time, an offer came and I ended up as a visitor at the Courant Institute. I never left!

**Your work on large deviations is considered to be one of the cornerstones of modern probability theory and it has applications in fields as diverse as insurance risk modelling to predicting outlier weather events. Can you explain what the ‘art of predicting rare events’ is?**

You start with a model of randomness that determines the probabilities of various events. But it is hard to estimate the probabilities when they are very small. How to measure how small is small? We may have an event with small probability under our model. We can change the model and make this event, one with significant probability. By how much have we changed the model? That can be measured. There may be many ways to change the model and we can optimise. That provides a more exact measure of how small the probability is. To give a simple example, if you toss a fair coin 1,000 times, how small is the probability that you get 800 heads? If the probability of getting a head were 0.8 instead of 0.5, 800 heads would be normal. How far is 0.8 from 0.5 for this problem? There is a measure called relative entropy that serves the purpose.

**Yours is a storied career, adorned with accomplishments and honour. In that context what does the conferring of Padma Vibhushan, India’s second-highest civilian honour, mean to you?**

I consider myself lucky. The world is full of people with extraordinary talent. But only a few are recognised with high awards. Part of it is being at the right place at the right time and choosing the right problem to work on. I had a very encouraging environment with great colleagues to work with. We had a constant stream of visitors to our Institute who provided new perspectives.

**Math, for many, is an abstract, intimidating subject they study in schools. How do you think we can make the teaching of mathematics more engrossing for school-going children?**

Students should be encouraged to solve puzzles. They need to think of problem-solving as a game, rather than as a task. Each student can find their own level.

**At a personal level, what about mathematics means the most to you? Is it the process of solving a problem, or the discovery of the inherent beauty and elegance of the solution?**

First, it is the challenge of trying to prove something. Once you succeed, you are pleased for a couple of days, and then on to the next problem. Eventually you see a pattern emerging and all the odd pieces that were down below appear part of [a] grand structure when looked [at] from above. Very satisfying!