Abstracts

Week of April 18, 2021

April 19, 2021
Group Actions and Dynamics Hausdorff dimension of limit sets and its second derivative, some higher rank situations 10:15am -
Zoom
Let G be a semi-simple real algebraic group, $G_\mathbb R$ its real points and $G_\mathbb C$ its complex points. Let $\Gamma$ be a discrete subgroup of the real points $G_\mathbb R.$ The purpose of the talk is to explain a general procedure relating (in some situations) the second variation of the Hausdorff dimension of the limit set of $\Gamma,$ when deforming $\Gamma$ inside the complex group $G_\mathbb C,$ with natural geometries (known as pressure forms) of the real characters $X(\Gamma,G_\mathbb R).$ 
 
This is joint work with M. Bridgeman, B. Pozzetti and A. Wienhard
Applied Mathematics Graph Convolutional Neural Networks: The Mystery of Generalization 1:00pm -
Zoom Meeting ID: 97670014308

Abstract:  The tremendous importance of graph structured data due to recommender systems or social networks led to the introduction of graph convolutional neural networks (GCN). Those split into spatial and spectral GCNs, where in the later case filters are defined as elementwise multiplication in the frequency domain of a graph. Since often the dataset consists of signals defined on many different graphs, the trained network should generalize to signals on graphs unseen in the training set. One instance of this problem is the transferability of a GCN, which refers to the condition that a single filter or the entire network have similar repercussions on both graphs, if two graphs describe the same phenomenon. However, for a long time it was believed that spectral filters are not transferable.
In this talk by modelling graphs mimicking the same phenomenon in a very general sense, also taking the novel graphon approach into account, we will debunk this misconception. In general, we will show that spectral GCNs are transferable, both theoretically and numerically. This is joint work with R. Levie, S. Maskey, W. Huang, L. Bucci, and M. Bronstein.

email tatianna.curtis@yale.edu for info.
Geometry, Symmetry and Physics Harmonic Analysis on GLn over Finite Fields. 4:30pm -
https://yale.zoom.us/j/92811265790 (Password is the same as last semester)

There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio:

Trace(ρ(g)) / dim(ρ),

for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.

Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank. Rank suggests a new organization of representations based on the very few “Small” ones. This stands in contrast to Harish-Chandra’s “philosophy of cusp forms”, which is (since the 60s) the main organization principle, and is based on the (huge collection) of “Large” representations. This talk will discuss the notion of rank for the group GLn over finite fields, illustrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks. This is joint work with Roger Howe (Yale). The numerics for this work was carried with Steve Goldstein (Madison) and John Cannon (Sydney).
April 20, 2021
Algebra and Number Theory Seminar Doubling integrals for Brylinski-Deligne extensions of classical groups 9:00am -
Zoom

In the 1980s, Piatetski-Shapiro and Rallis discovered a family of Rankin-Selberg integrals for the classical groups that did not rely on Whittaker models. This is the so-called doubling method. It grew out of Rallis' work on the inner products of theta lifts -- the Rallis inner product formula.

Recently, a family of global integrals that represent the tensor product L-functions for classical groups (joint with Friedberg, Ginzburg, and Kaplan) and the tensor product L-functions for covers of symplectic groups (Kaplan) was discovered. These can be viewed as generalizations of the doubling method. In this talk, we explain how to develop the doubling integrals for Brylinski-Deligne extensions of connected classical groups. This gives a family of Eulerian global integrals for this class of non-linear extensions.
Geometry & Topology Branching properties of random trees in the boundary of Outer space 4:00pm -
https://yale.zoom.us/j/96501374645

It is well known that a pseudo-Anosov homeomorphism of a closed hyperbolic surface admits two invariant transverse measured laminations. The singularity properties of those laminations record important dynamical characteristics of the homeomorphism. More generally, the same is true for arbitrary points of the of the Teichmuller space. Recently, the study of the properties of “random” pseudo-Anosovs as well as of “random” points in the Thurston boundary, and the properties of the corresponding measured laminations, became an important topic in geometric topology. We undertake a similar study for free group automorphisms and the boundary of the Culler-Vogtmann Outer space. In that contexts pseudo-Anosovs are replaced by the so-called “fully irreducible” automorphisms. Points of the boundary of the Outer space are R-trees, equipped with (usually highly mixing) isometric free group actions that in various senses generalize measured laminations on surfaces. We discuss the branching properties of "random" automorphisms and "random" (in the sense of harmonic measures) trees in the boundary of the Outer space. The talk is based on joint papers with Catherine Pfaff, Joseph Maher and Samuel Taylor.