# Conceptual strength and ageless beauty

Kristian Seip, the chairman of the Abel Committee, gave this speech just after the announcement at the Norwegian Academy of Science and Letters 22. March:

Mr. President, ladies and gentlemen:
The Abel Prize for 2007 is awarded to Srinivasa S. R. Varadhan for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviations.
Probability theory is the mathematical tool for analyzing situations governed by chance. Systems in the real world will almost always be subject to what we at best can comprehend as random effects. Thus, probability theory is one of the foundations of modern science, technology and economics.
The law of large numbers was discovered by Jacob Bernoulli early in the eighteenth century. This law shows that the average outcome of a long sequence of coin tosses is usually close to the expected value. Yet, as we all know, the unexpected happens, and the question is: how? The theory of large deviations studies the occurrence of rare events and has concrete applications to fields as diverse as physics, biology, economics, statistics, computer science, and engineering.
The law of large numbers states that the probability of a deviation beyond a given level goes to zero. However, for practical applications, it is crucial to know how fast it vanishes. For example, what capital reserves are needed to keep the probability of default of an insurance company below acceptable levels? In analyzing such actuarial “ruin problems”, the Swedish mathematician and statistician Harald Cramér discovered in 1937 that standard approximations based on the Central Limit Theorem (as visualized by the bell curve) are actually misleading. He then found the first precise estimates of large deviations. It took 30 years before Varadhan discovered the underlying general principles and began to demonstrate their tremendous scope. Varadhan’s theory of large deviations, partly worked out in collaboration with Monroe D. Donsker, provides a unifying and efficient method for clarifying a rich variety of phenomena arising in complex stochastic systems, in fields as diverse as quantum field theory, statistical physics, population dynamics, econometrics and finance, and traffic engineering. It has also greatly expanded our ability to use computers to simulate and analyze the occurrence of rare events. Over the last four decades, the theory of large deviations has become a cornerstone of modern probability theory.
Varadhan has made key contributions in several other areas of probability theory. In joint work with Daniel W. Stroock, he developed a martingale method for characterizing diffusion processes. This new approach turned out to be an extremely powerful way of constructing Markov processes, for example of the type that arise in population genetics. Varadhan has also made important contributions to the analysis of hydrodynamical limits describing the macroscopic behaviour of very large systems of interacting particles. His ideas also had a strong influence on the analysis of random walks in a random environment. His name is now attached to the method of “viewing the environment from the travelling particle”, one of the few general tools in the field.
Varadhan's work has great conceptual strength and ageless beauty. His ideas have been hugely influential and will continue to stimulate further research for a long time.