Title:Homogeneous dynamics for Number Theory (Pt 1)
Speaker: Nimish Shah (OSU)
Abstract: Many problems of number theoretic nature, like finding and counting integral solutions of various types of algebraic equations or inequalities, Diophantine approximation, special points on Shimura varieties etc., involve two large groups of symmetries. Roughly speaking, one of them would be an algebraic group associated to the variety and another would be a discrete subgroup associated to the integral points. Such number theoretic problems become much harder to resolve using the classical methods of Hardy-Littlewood circle method or Harmonic analysis, when the number of freedom in variables is small. In many such situations, the study of ergodic properties of subgroup actions on quotient spaces of Lie groups (in short, homogeneous dynamics) provides satisfactory solutions to the problems.
Our goal of these lectures is to introduce a small sample of such problems, and indicate how they can be converted to questions about closures or limiting distributions of orbits of subgroups on homogeneous spaces, typically SL(n,R)/SL(n,Z). We will also indicate what kind of modern tools that have been developed to understand this dynamics.
Note: This is part of the Invitations to Mathematics lecture series given each year in Autumn and Spring semesters. Pre-candidacy students can sign up for this lecture series by registering for one or two credit hours of Math 6193, Call/Class # 17170 (with Prof H. Moscovici).
