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12th Annual Thomas Wolff Memorial Lecture
in Mathematics
May 8, 13 & 15, 2013

4:00 p.m.  151 Sloan

Horng-Tzer Yau
Professor of Mathematics
Harvard University

Universality of random matrices
Lecture 1

Abstract: Eugene Wigner's revolutionary vision predicted that the energy levels of large complex quantum systems exhibit a universal behavior. These universal statistics represent a new paradigm of point processes that are characteristically different from the Poisson statistics of independent points. This fundamental vision has never been proved for true interacting quantum systems, but several classes of matrix ensembles have recently become mathematically accessible. A prominent example is the Wigner-Dyson-Gaudin-Mehta conjecture asserting that the spectral statistics of random matrices with independent entries depend only on the symmetry classes but are independent of the distributions of matrix elements. Another class is invariant matrix models for which the eigenvalue distributions are given by a Coulomb gas with a potential V and inverse temperature beta. In this lecture, we outline the recent solution to these conjectures and discuss related topics. All basic notions of random matrix theory needed will be reviewed and no prior knowledge on the subject is assumed in this lecture.

Dyson Brownian Motion and the proof of Wigner-Dyson-Gaudin-Mehta conjecture
Lecture 2

Abstract: Dyson proposed to study the dynamics of the eigenvalues of random matrices if all matrix elements evolved by Brownian motions. Dyson observed that this dynamics approaches equilibrium in two time scales: a slow relaxation for global modes and a fast one for local fluctuations. We will show that this fundamental observation can indeed be made rigorous and it is in fact the cornerstone of the recent solution of Wigner-Dyson-Gaudin-Mehta conjecture. We will also discuss two other key elements in the proof: The first one is the local semicircle law which extends Wigner's semicircle law to the scale near the typical spacing of the eigenvalues. The second one is the Green function comparison theorem which compares the Green functions, and, in particular, eigenvalues, of two matrix ensembles.

Coulomb Gases and De Giorgi-Nash-Moser theory of parabolic regularity
Lecture 3

Abstract: In this lecture we will sketch the proof of the universality of Coulomb gases both in the bulk and at the edges of the spectrum. For both cases we will link the universality problem to the decay of correlation functions of Coulomb gases. We will show that such a decay follows from a Holder regularity of a discrete parabolic equation with random coefficients. The parabolic regularity will be established partly using the recent argument of Caffarelli-Chan-Vasseur, which is a De Giorgi-Nash-Moser type method. The singularities in random coefficients pose major challenges; we will use optimal level repulsion estimates in random matrices to control them.

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Horng-Tzer Yau is one of the world’s leading probabilists and mathematical physicists. He has worked on quantum dynamics of many-body systems, statistical physics, hydrodynamical limits, and interacting particle systems. Yau approached the problems of the quantum dynamics of many-body systems with tools he developed for statistical physics and probability. More recently, he has been the main driving force behind some stunning progress on bulk universality for random matrices. With Laszlo Erd?s and others, Yau has proven the universality of the local spectral statistics of random matrices, a problem that was regarded as the main challenge of random matrix theory. This argument applies to all symmetry classes of random matrices. In the special Hermitian case, Terence Tao and Van Vu proved bulk universality concurrently. Yau’s work has been extended in many directions, for instance in his recent results on invariant beta ensembles with Paul Bourgade and Laszlo Erd?s.


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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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