Michael Bersudsky
The Ohio State University
Title
Equidistribution of polynomially bounded o-minimal curves in homogenous spaces
Abstract
Given a real algebraic group G and an o-minimal structure on the real field, one can naturally define subsets of G that are definable in this structure. Peterzil and Starchenko recently showed that when G is the group of upper-triangular matrices and L is a lattice in G, the closure of the image in G/L of a definable set in G is the closed image of a potentially larger definable set in G. Moreover, they showed that if a curve in G is definable in a polynomially bounded o-minimal structure and its image in G/L is dense, then the curve is uniformly distributed in G/L. In this talk, I will present recent joint work with Nimish Shah and Hao Xing, where we extend these results for curves definable in a polynomially bounded o-minimal structure in a general real algebraic group G and a general lattice L in G, under a 'non-contraction' condition on the curves. This work builds upon Shah's earlier technique for polynomial curves in homogeneous spaces, ‘tangency at infinity’ property of o-minimal curves shown by Peterzil and Steinhorn, and Ratner's groundbreaking theorems. A key innovation in our analysis is a proof of a certain growth property for families of polynomially bounded o-minimal functions.
