Tuesday, April 20, 2021
04/20/2021 - 9:00am
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.
04/20/2021 - 10:15am
Abstract: What are normalized characters (aka character ratios) of finite groups good for?
We first discuss some applications, assuming good upper bounds on normalized charactersof simple groups. Then we will describe recent results that produce such character bounds.
04/20/2021 - 4:00pm
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.