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Mathematics Colloquium


Date:  Tuesday, March 22, 2016

Title:  Long-time dynamics and turbulence of nonlinear waves. Part I

Abstract:  Nonlinear dispersive waves are partial differential equations that model numerous physical phenomena, ranging from plasma physics, ocean and atmospheric science, to general relativity. Over the past twenty years, the long-time behavior of small amplitude solutions to such equations on Euclidean space (R^n) became relatively well-understood. In contrast, the situation is much less understood on bounded domains, that feature a markedly different and rich set of behaviors. In particular, the dynamics in this setting is characterized by out-of-equilibrium behavior, in the sense that solutions typically do not exhibit long-time stability near equilibrium configurations. At the level of the physics underlying these problems, studying this out-of-equilibrium behavior leads to an interesting interplay between dynamics and statistical mechanics, in what is often known as wave turbulence theory. At the level of the mathematics, this study features a beautiful interaction between PDE methods, dynamical systems theory, probability theory, as well as a surprising and very elegant input from analytic number theory. In this first talk, we shall discuss all these aspects, and survey some recent advances in this direction of research.

Speaker:  Zaher Hani
Institution:  Georgia Tech


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