‘One should approach maths the way one approaches a jigsaw puzzle’

January 29, 2015 12:41 am | Updated 12:41 am IST

Srinivasa Varadhan of the New York University’s Courant Institute for Mathematical Sciences.

Srinivasa Varadhan of the New York University’s Courant Institute for Mathematical Sciences.

Renowned probabilistSrinivasa Varadhanwas in Chennai recently. In 2007, Prof. Varadhan was awarded the Abel Prize for his work on large deviations in probability theory. He is now with New York University’s Courant Institute for Mathematical Sciences. He explains the importance of his work toShubashree Desikan.Excerpts:

How does one convey the importance of probability and statistics, especially statistical inference? 

We often have to make decisions based on incomplete information. It involves making an “educated guess”. Statistical inference is an attempt to incorporate the available information in a scientific manner in the decision process. Evaluate the potential risk and arrive at the proper choice. Since it requires quantifying uncertainty, probability theory is involved.

You have received many awards, including Padma Bhushan, the National Medal of Science, USA, and the Abel Prize. How is the Abel Prize different from the others?

Any prize or award like Nobel or Abel is recognition by one’s colleagues for the work done. There are not enough prizes to award to everyone that deserves it. There is a certain amount of luck involved. I was happy to receive it and it was a wonderful occasion for me and my family.

What is the global impact of Indian Statistical Institutes today? You got your PhD from ISI Kolkata.

Over the years, ISI has produced excellent probabilists and statisticians. Even today, many of the PhD students in probability or statistics in the U.S have had their master’s level education at one of the ISIs.

Can you tell us the importance of your work on large deviations and the prediction of natural disasters?

I am a mathematician and my work is mainly on methods for estimating small probabilities with some precision. It is very much model-dependent. The major issue in natural disasters is one of modelling and any application has to be robust with respect to model errors.

Estimating probabilities that are small is important for various reasons. Often, rare events occur. Large deviation theory, besides estimating the probability can also predict other rare things that might have occurred. [Also], what cause could have most likely produced this rare event and what other events could have been triggered by the same cause.  Of course, one must understand that these are model-based conclusions and are only as reliable as the model.

What are the areas in statistical mechanics where large deviation theory is applied?

Statistical mechanics has to do with how complex systems organise themselves.  Systems like to be in the lowest energy state. But when you observe it many may not be so. The number of different states at a given energy level varies with the level; there may be many more states at a higher energy level.

The number of states at a given energy level is controlled by entropy and there is give and take between energy and entropy, and an optimal balance is reached and that is what we will normally observe.

Suppose we have a biased coin which has 75 per cent chance of coming up head and only 25 per cent for tail.  Then if we toss the coin 100 times the sequence of all heads has more probability than any other sequence, because replacing a tail with a head improves the probability by a factor of 3. But throwing in a few tails increases the number of possibilities [which favours the entropy]. Just 99 heads and 1 tail has a hundred possibilities. The tail can be anywhere. The loss of 1/3 is more than made up by a gain of 100.  There is a balance point which is roughly 75 heads and 25 tails. 

How did you find your field of interest? Can you trace the trajectory your career has taken?

When I started my research in probability it was in an area called limit theorems, a more traditional branch in probability.

The theory of stochastic process was just getting started; Markov process and diffusion processes were areas I was interested in. When I started my postdoctoral research at NYU’s Courant Institute, I was introduced to various aspects of Markov processes and its relation to Analysis, especially partial differential equations. People like Donsker, Kac, Feller, McKean, Nirenberg, Lax, were a great inspiration.

I learnt a lot and was exposed to some of the open problems in these areas. I had also the good fortune to collaborate with great colleagues over the years. Stroock, Donsker, Papanicolaou, Olla, Yau, and some of my past students Quastel and Rezakhnalou.

Is there anything in particular you'd like to tell students and others interested in maths?

It is important for children not to be afraid of mathematics. One should approach it the way one approaches a jigsaw puzzle. It’s a challenging game that has to be played using one’s mind rather than one’s muscles.

Very often, we place too much emphasis on memorising the rules of arithmetic. That turns youngsters off. Maths can be fun!

You are an alumnus of Presidency College, Chennai. Have you been there since 1959? 

I was there [a few days back] for two occasions after nearly forty years. I was saddened by the state of the building. It is almost as if no one thought of having to maintain the  structure.  For the premiere institution that had produced Nobel Laureates S. Chandrasekhar and C.V. Raman and many other distinguished graduates, it was sad to see that the government could not provide enough resources to maintain the premises in a respectable state.

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