Abstracts

Week of January 31, 2021

February 1, 2021
Group Actions and Dynamics A subspace theorem for manifolds 10:15am -
Zoom
In the late 90’s Kleinbock and Margulis solved a long-standing conjecture due to Sprindzuk regarding diophantine approximation on submanifolds of R^n. Their method used homogeneous dynamics via the so-called non-divergence estimates for unipotent flows on the space of lattices. In this talk I will explain how these ideas, combined with a certain understanding of the geometry at the heart of Schmidt’s subspace theorem, in particular the notion of Harder-Narasimhan filtration, leads to a metric version of the subspace theorem, where the linear forms are allowed to depend on a parameter. This subspace theorem for manifolds allows to quickly compute certain diophantine exponents, and it leads to several generalizations of the Kleinbock-Margulis results in a variety of contexts. Joint work with Nicolas de Saxcé.
Geometry, Symmetry and Physics Tate's thesis in the de Rham setting 4:30pm -
https://yale.zoom.us/j/92811265790 (Password is the same as last semester)

This is joint work with Justin Hilburn. We will explain a theorem showing that D-modules on the Tate vector space of Laurent series are equivalent to ind-coherent sheaves on the space of rank 1 de Rham local systems on the punctured disc equipped with a flat section. Time permitting, we will also describe an application of this result in the global setting. Our results may be understood as a geometric refinement of Tate's ideas in the setting of harmonic analysis. They also may be understood as a proof of a strong form of the 3d mirror symmetry conjectures in a special case.
February 2, 2021
Geometry & Topology Train track maps and CTs on graphs of groups 4:00pm -
https://yale.zoom.us/j/96501374645

Abstract: A train track map is a self-map of a graph with particularly nice properties. Train track maps and their cousins, relative train track maps, were developed by Bestvina and Handel in 1992 to prove the Scott conjecture: the fixed subgroup of an automorphism of a finite-rank free group has rank bounded by the rank of the free group. Since then, relative train track maps, particularly in their modern incarnation as CTs, have become perhaps the main tool in studying outer automorphisms of free groups. We will meet (relative) train track maps and describe a generalization of them to graphs of groups. As an application, we will see an index inequality implying a version of the Scott conjecture for automorphisms of free products.