Georgia Institute of Technology College of Sciences

I will describe a class of mathematical models of composites and polycrystals. The problems I will describe two research projects that are well suited for graduate student interested in learning more about this area of research.

The hyperplane conjecture of Kannan, Lovasz and Simonovits asserts that the isoperimetric constant of a logconcave measure (minimum surface to volume ratio over all subsets of measure at most half) is approximated by a halfspace to within an absolute constant factor. I will describe the motivation, implications and some developments around the conjecture and an approach to resolving it (which does not seem entirely ridiculous).

In this talk we will connect functional analysis and analytic function theory by studying the compact linear operators on Bergman spaces. In particular, we will show how it is possible to obtain a characterization of the compact operators in terms of more geometric information associated to the function spaces. We will also point to several interesting lines of inquiry that are connected to the problems in this talk. This talk will be self-contained and accessible to any mathematics graduate student.

Consider electrostatic plasmas described by Vlasov-Poisson with a fixed ion background. In 1946, Landau discovered the linear decay of electric field near a stable homogeneous state. This phenomena has been puzzling since the Vlasov-Poisson system is time reversible and non-dissipative. The nonlinear Landau damping was proved for analytic perturbations by Mouhot and Villani in 2009, but for general perturbations it is still largely open. I will discuss some recent results with C. Zeng on the failure of nonlinear daming in low regularity neighborhoods and a regularity threshold for the existence of nontrivial invariant structures near homogeneous states. A related problem to be discussed is nonlinear inviscid damping of Couette flow, for which the linear decay was first observed by Orr in 1907.