Abstract  :
A large system of particles seen from a distance looks like a cloud. There are equations that describe the motion of particles like classical mechanical systems described by Hamiltonians or stochastic systems of interacting particles described by a Markov process with a large state space. Continuous objects like clouds are expressed by suitable functions (density) in space and they evolve in time according to partial differential equations of one kind or other. They both describe the same object. The goal is to make the connection precise. Note: Only basic knowledge of differential equations and probability theory will be assumed.

Prof. Srinivasa S. R. Varadhan is currently a Professor of Mathematics and Frank J. Gould Professor of Science at the Courant Institute of Mathematical Sciences, New York University. Prof. Varadhan received his B.Sc. honours degree in 1959 and his M.A. the following year, both from Madras University. In 1963 he received his Ph.D. from the Indian Statistical Institute, Calcutta, with the distinguished Indian Statistician C. R. Rao as his thesis advisor. Prof. Varadhan began his academic career at the Courant Institute of Mathematical Sciences as a postdoctoral fellow during 1963 to 1966 and stayed at Courant ever since. He and Daniel Stroock were awarded the American Mathematical Society's Steele Prize. 15 years later he was appointed Director of Courant (1980-84), following Peter Lax. Prof. Varadhan followed Peter Lax both as Director of Courant and now also as an Abel Laureate. He came back to serve a second period as Director of Courant (1992-94). Prof. Varadhan has held visiting positions at Stanford University (1976-77), the Mittag-Leffler Institute (1972), and the Institute for Advanced Study (1991-92). He was an Alfred P. Sloan Fellow (1970-72) and a Guggenheim Fellow (1984-85). His awards and honours include the Birkhoff Prize (1994), the Margaret and Herman Sokol Award of the Faculty of Arts and Sciences, New York University (1995), the Leroy Steele Prize (1996). Prof. Varadhan received the Abel prize in 2007 for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviation. He also has two honorary degrees from Université Pierre et Marie Curie in Paris (2003) and from Indian Statistical Institute in Kolkata, India (2004). Prof. Varadhan was Invited speaker at the International Congress of Mathematicians in 1978 and 1994. He has been elected a member of the American Academy of Arts and Sciences and the Third World Academy of Sciences in 1988 and the National Academy of Sciences in 1995. He was elected a Fellow of the Institute of Mathematical Statistics in 1991, the Royal Society in 1998 and the Indian Academy of Sciences in 2004. Prof. Varadhan's research interests include probability theory, stochastic processes, partial differential equations. Prof. Varadhan's theory of large deviations 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, both pure and applied. In the 1970s, Varadhan and D.W. Stroock wrote an impressive series of papers on so called "martingale problems" culminating in their book "Multidimensional Diffusion Processes" of 1979. Their approach unified, simplified and extended the previous results in the area substantially. The basic idea is that instead of looking for solutions to quite complicated problems of mathematical analysis, "all" one has to look for is a probability distribution which turns certain processes into martingales.