Graduate Student / Postdoc Seminar

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Number Theoretic Instances in Nonlinear Dispersive PDE

Speaker: Zaher Hani

Location: Warren Weaver Hall 1302

Date: Friday, May 3, 2013, 1 p.m.

Synopsis:

A dispersive partial differential equation is one where different frequency components propagate at different velocities. On non-compact domains (like \(\mathbb{R}^d\)), dispersion is a mechanism for (conservative or non-dissipative) decay and ultimately stability. This is not the case on compact domains where conservation laws prohibit any form of decay, and different frequency components have no escape from "overlapping" with each other all the time. Consequently, the mathematical analysis of such equations is considerably more delicate and, more importantly, the asymptotic behavior is very rich.

A mathematically and physically important example of a compact domain is that of a box, in which case one can retain the power of Fourier analysis. Having the lattice \(\mathbb{Z}^d\) as Fourier space, the analysis ultimately requires input from number theory, a connection first explored by Bourgain in the early nineties. I will try to present some instances of this rather beautiful interaction between these two, seemingly distant, fields of mathematics, both at the level of linear and nonlinear problems.