Tuesday, October 8, 2019
10/08/2019 - 4:00pm
Abstract: Understanding the statistical properties of Laplacian eigenfunctions in general and their nodal sets in particular, have an important role in the field of spectral geometry, and interest both mathematicians and physicists. A quantum graph is a system of a metric graph with a self-adjoint Schrodinger operator. It was proven for quantum graphs that the number of points on which each eigenfunction vanish (also known as the nodal count) is bounded away from the spectral position of the eigenvalue by the first Betti number of the graph. A remarkable result by Berkolaiko and Weyand showed that the nodal surplus is equal to a magnetic stability index of the corresponding eigenvalue. A similar result for discrete graphs holds as well proved first by Berkoliako and later by Colin deVerdiere.
A numerical study indicates that this property might be universal and led us to state the following conjecture. For every sequence of graphs with an increasing number of cycles, the corresponding sequence of properly normalized nodal count distributions will converge to a normal distribution.
In my talk, I will present our latest results extending the number of families of graphs for which we can prove the conjecture.
10/08/2019 - 4:15pm
Abstract: Typically local converse theorems are done using the Rankin-Selberg gamma factors, but in this talk we will follow a conjecture of Ramakrishnan to present a new kind of local converse theorems using twisted exterior power gamma factors. We then prove a local converse theorem for simple supercuspidal representations using twisted exterior square gamma factors. This is joint work with Elad Zelingher.
10/08/2019 - 5:20pm
For a (finitely generated) subgroup of a (finitely generated) group, we will consider two properties of this subgroup: the separability of this group and its subgroup distortion. The separability of a subgroup measures whether the property of an element not lying in this subgroup is visible by taking some finite quotient. We will give a characterization on whether a subgroup of a 3-manifold group is separable. The subgroup distortion compares the intrinsic and extrinsic geometry of a subgroup. For an arbitrary subgroup of a 3-manifold group, we prove that the subgroup distortion can only be linear, quadratic, exponential and double exponential. It turns out the subgroup separability and subgroup distortion are closed related for subgroups of 3-manifold groups. The subgroup distortion part is joint work with Hoang Nguyen.