In this project, the PI will study the actions of Lie groups and their discrete subgroups from ergodic-theoretic point of view. Note that only few results about statistical properties of orbits of large nonamenable groups are known. A part of this project addresses the question about the distribution of orbits of lattices in Lie groups acting on homogeneous spaces. The methods, which utilize the equidistribution properties of unipotent flows (due to Ratner and others), apply to a large variety of natural actions of lattices. Since the methods based on the Ratner theory do not provide effective estimates on the rate of convergence, it is intended to develop a different approach that gives effective error terms. The expected results of the proposal have several potential applications to number theory. In particular, we plan to investigate distribution of values at integer points of systems consisting of linear and quadratic forms. Another direction of research is the study of mixing properties of large groups of automorphisms of nilmanifolds.
The main objective of this project is to investigate distribution of orbits of large discrete and continuous groups of motions on various spaces. Questions about asymptotic distribution of families of mathematical objects appear in many different areas of mathematics: ergodic theory, number theory (e.g., distribution of prime numbers), geometry (e.g., distribution of closed geodesics on a compact surface), PDE (e.g., distribution of eigenvalues of the Laplace operator), and others. When a family of the objects exhibits very complicated behavior, statistical properties of this family provide important insights into its structure. In this project, we use methods of the theory of dynamical systems to derive results in number theory on the distribution of integer points. Our study uncovers promising interplay between ergodic theory and number theory simultaneously enriching both of these areas of research. This subject and its connections with many other branches of mathematics provide an excellent introduction for graduate as well as undergraduate students to active research and can be used for presentations accessible to audiences of different levels.
