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9th Annual Thomas Wolff Memorial Lecture
Integrable systems: a modern view
The modern theory of integrable systems began with the solution of the Korteweg de Vries equation by Gardner, Greene, Kruskal and Miura in 1967. This led to the development of a variety of new mathematical techniques, and over time, and quite unexpectedly, these techniques have found applications in areas far beyond their dynamical origins. The applications include problems in algebraic geometry, numerical analysis, analytic number theory, combinatorics and random matirx theory, among many others. In these lectures, the speaker will present some of these techniques and describe some of their applications. Amongst these techniques, Riemann-Hilbert (RH) methods, Painlev'e analysis, and the theory of random matrices (RMT), figure prominently.
In the first lecture the speaker will discuss some of the history of these developments together with some recent, illustrative applications. The second and third lectures will be devoted to a more technical presentation of RH methods, Painlev'e equations and RMT. Applications of integrable methods to numerical analysis will also be discussed.
Percy Deift is Professor of Mathematics at the Courant Institute, NYU. He obtained a Master's degree in Chemical Engineering from the University of Natal, Durban, South Africa in 1970 and a Master's degree in Physics from Rhodes University, Grahamstown, South Africa in 1971. He obtained his PhD in Mathematical Physics under Barry Simon from Princeton University in 1977. He has been on the faculty at the Courant Institute since 1976. His principal interests lie in spectral theory, inverse spectral theory, integrable systems and random matrix theory. He is a joint winner of the George Polya Prize (1998), he was a Guggenheim Fellow in 1999-2000, and he has given plenary talks at the International Congress on Mathematical Physics (ICMP2006) and at the International Congress of Mathematicians (ICM2006). In 2009 he gave the Gibbs Memorial Lecture. He was elected to the American Academy of Arts and Sciences in 2003 and to the National Academy of Sciences in 2009.
The Thomas Wolff lectures, sponsored by donations from his widow and his parents, memorialize Caltechs 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.