Back to Departmental Colloquium: Spring 2011
Departmental Colloquium
Date: Friday, Apr 1, 2011
Time: 4:15PM
Location: JWB 335
Gopal Prasad
University of Michigan
Title |
Number theoretic techniques in the study of Lie groups and locally symmetric spaces. |
Abstract |
In this talk I will describe recent joint work with Andrei Rapinchuk in which we have used number theoretic techniques to prove the existence of elements with interesting properties in any Zariski-dense subgroup of a real semi-simple Lie group. These elements have been used in several different contexts. In our recent work we have used them to decide when two “weakly commensurable” arithmetic subgroups are actually commensurable. We have used known results, and a well-known conjecture, in transcendental number theory to show that for the symmetric space X of a noncompact absolutely simple real Lie group G, if the quotients X/\Gamma_1 and X/\Gamma_2, where \Gamma_1 and \Gamma_2 are lattices in G and at least one of them is arithmetic, are either isolength, or they are compact and isospectral, then the subgroups \Gamma_1 and \Gamma_2 are weakly commensurable. Therefore, our results on weakly commensurable arithmetic groups have important consequences for the Riemannian Geometry of locally symmetric spaces. Our work has led us to investigate local-global principles for embedding of fields with involution into a central simple algebra with involution. The failure of such a local-global principle for certain central simple algebras of degree 4n, given with an orthogonal involution, implies interesting results, for example, for compact arithmetic hyperbolic spaces of dimension 4n-1. |
Number Theory
Representation Theory
Geometry and Topology