Abstracts

Week of February 7, 2021

February 8, 2021
Group Actions and Dynamics Effective discreteness radius of stabilizers for stationary actions 4:00pm -
Zoom

I will discuss a quantitative variant of the classical Kazhdan-Margulis theorem generalized to stationary actions of semisimple groups over local fields. More precisely, the probability that the stabilizer of a random point admits a non-trivial intersection with a small r-neighborhood of the identity is at most a constant times r^d, for some explicit d > 0 depending only on the semisimple group in question. Our proof involves some of the original ideas of Kazhdan and Margulis, combined with methods of Margulis functions as well as (C,alpha)-good functions on varieties. As an application, we present a new unified proof of the fact that all lattices in these groups are weakly cocompact, i.e admit a spectral gap. The talk is based on a preprint joint with Gelander and Margulis.
Geometry, Symmetry and Physics Brane quantization 4:30pm -
https://yale.zoom.us/j/92811265790 (Password is the same as last semester)

I will discuss some ongoing work with E.Witten, exploring features of
quantization from the perspective of the A-model. I will briefly discuss
the relationship between this work and analytic Geometric Langlands.
February 10, 2021
Geometry & Topology Random covers of closed hyperbolic surfaces: spectral gaps and asymptotic statistics 10:15am -

On a compact hyperbolic surface, the Laplacian has a spectral gap between 0 and the next smallest eigenvalue if and only if the surface is connected. The size of the spectral gap measures how "highly connected" the surface is. We study the spectral gap of a random covering space of a fixed surface, and show that for every ε>0 , with high probability as the degree of the cover tends to ∞, the smallest new eigenvalue is at least 3/16-ε.

Our main tool is a new method to analyze random permutations "sampled by surface groups". I intend to give some background to the result and discuss some ideas from the proof.

This is based on joint works with Michael Magee and Frederic Naud.
February 11, 2021
Analysis Continuum and discrete models of waves in 2D materials 4:15pm -

Abstract:

We discuss continuum Schroedinger operators which are basic models of 2D-materials, like graphene; in its bulk form or deformed by edges (sharp terminations or domain walls). For non-magnetic and strongly non-magnetic systems we discuss the relationship to effective tight binding (discrete) Hamiltonians through a result on strong resolvent convergence. An application of this convergence is a result on the equality of topological (Fredholm) indices associated with continuum and discrete models (for bulk and edge systems). Finally, we discuss the construction of edge states in continuum systems with domain walls. Away from the tight binding regime there are resonant phenomena, and we conjecture that there are meta-stable (finite lifetime, but long-lived) edge states which slowly diffract into the bulk.