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11th Annual Thomas Wolff Memorial Lecture
in Mathematics
November 14, 16 and 22, 2011

4:15 p.m.  151 Sloan

Stanislav Smirnov
Professor of Mathematics
Université de Genève

St. Petersburg State University

## DISCRETE COMPLEX ANALYSIS AND PROBABILITY

It is well-known that discrete harmonic functions can be defined on any graph, e.g. by requiring the mean value property, i.e. the function at a vertex is the mean of the values at its neighbors. Such discrete functions share many properties of their continuous counterparts and have been very extensively studied. The theory of discrete analytic functions also has a long history, but is less developed. One starts with a planar graph, e.g. the square lattice, and then asks a function to satisfy some discretization of the Cauchy-Riemann equations. There are many possible discretizations, and some of them have deep connections to integrable systems.

It turns out that much of the usual complex analysis can be transferred to the discrete setting, albeit with some difficulties. Sometimes discrete theories can be applied to the continuous case and even lead to easier proofs of several results, including the Riemann uniformization theorem. There are also exciting connections to combinatorics, probability, geometry and even computer science.

We will discuss discrete complex analysis and some of its applications to probability and mathematical physics: discrete analytic functions appear in lattice models of statistical physics (such as percolation, Ising model, self-avoiding polymers) and allow one to derive many of their properties.

Stanislav Smirnov, MS '95, PhD '96, received his undergraduate degree at St. Petersburg State University, then studied under Professor Nikolai Makarov at Caltech. He worked at Yale University and the Royal Institute of Technology in Stockholm before joining the faculty of the University of Geneva in 2003. Smirnov's interests include analysis, dynamical systems, probability and mathematical physics. In 2010, Smirnov received the Fields medal for "the proof of conformal invariance of percolation and the planar Ising model in statistical physics". He was previously awarded the Salem Prize, the Clay Research Award, and the European Mathematical Society Prize.

The Thomas Wolff lectures, sponsored by donations from his widow and his parents, memorialize Caltech’s great analyst who was tragically killed at age 46 in an automobile accident in July 2000. Wolff was a specialist in analysis, particularly harmonic analysis. Professor Wolff made numerous highly original contributions to the mathematical fields of Fourier analysis, partial differential equations, and complex analysis. A recurrent theme of his work was the application of finite combinatorial ideas to infinite, continuous problems.

His early work on the Corona theorem, done as a Berkeley graduate student, stunned the mathematical community with its simplicity. Tom never wrote it up himself since several book writers asked for permission to include the proof in their books where it appeared not long after he discovered it. After producing a number of very significant papers between 1980 and 1995, he turned to the Kakeya problem and its significance in harmonic analysis, works whose impact is still being explored.

Peter Jones, professor of mathematics at Yale, described Tom’s contributions as follows: “The hallmark of his approach to research was to select a problem where the present tools of harmonic analysis were wholly inadequate for the task. After a period of extreme concentration, he would come up with a new technique, usually of astonishing originality. With this new technique and his well-known ability to handle great technical complications, the problem would be solved. After a few more problems in the area were resolved, the field would be changed forever. Tom would move on to an entirely new domain of research, and the rest of the analysis community would spend years trying to catch up. In the mathematical community, the common and rapid response to these breakthroughs was that they were seen not just as watershed events, but as lightning strikes that permanently altered the landscape.”

Tom Wolff was noted for his analytic prowess, the depth of his insights, and the passion with which he nurtured the talents of young mathematicians. We miss him.

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