Applying unipotent flows to number theoretic problems

Prof. Nimish Shah (The Ohio State University)

June 17, 10:00 to 12:00, HG G 19.1
June 20, 10:00 to 11:00, HG G 19.1
June 22, 10:00 to 11:00, HG G 19.1

Abstract

Dynamics of unipotent flows on the space of unimodular lattices or more general homogeneous spaces show up in various kinds of problems in number theory and geometry. The algebraic and arithmetic rigidity of unipotent flows as observed and established by Dani, Margulis, and Ratner have been very successfully generalized and applied to these problems in recent years. In these lectures we want to (i) show by examples, how unipotent flows enter into picture in different problems (Oppenheim conjecture, lattice point counting, equidistribution of Hecke orbits), (ii) describe the building blocks of the theory (Ratner‘s classification theorem, Dani-Margulis non-divergence), (iii) explain some basic techniques (linearization, linear dynamics), and (iv) look at a few specific instances of problems resolved by putting all these together.