Ergodic-Theoretic Properties of Actions of Discrete Groups

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In this project, the PI will study the actions of Lie groups and their discretesubgroups from ergodic-theoretic point of view. Note that only few resultsabout statistical properties of orbits of large nonamenable groups are known.A part of this project addresses the question about the distribution oforbits of lattices in Lie groups acting on homogeneous spaces. The methods,which utilize the equidistribution properties of unipotent flows (due to Ratnerand others), apply to a large variety of natural actions of lattices. Sincethe methods based on the Ratner theory do not provide effective estimateson the rate of convergence, it is intended to develop a different approachthat gives effective error terms. The expected results of the proposal haveseveral potential applications to number theory. In particular, we plan toinvestigate distribution of values at integer points of systems consistingof linear and quadratic forms. Another direction of research is the studyof mixing properties of large groups of automorphisms of nilmanifolds.The main objective of this project is to investigate distribution of orbitsof large discrete and continuous groups of motions on various spaces.Questions about asymptotic distribution of families of mathematical objectsappear in many different areas of mathematics: ergodic theory, number theory(e.g., distribution of prime numbers), geometry (e.g., distribution of closedgeodesics on a compact surface), PDE (e.g., distribution of eigenvalues of theLaplace operator), and others. When a family of the objects exhibits verycomplicated behavior, statistical properties of this family provide importantinsights into its structure. In this project, we use methods of the theory ofdynamical systems to derive results in number theory on the distribution ofinteger points. Our study uncovers promising interplay between ergodic theoryand number theory simultaneously enriching both of these areas of research.This subject and its connections with many other branches of mathematicsprovide an excellent introduction for graduate as well as undergraduatestudents to active research and can be used for presentations accessible toaudiences of different levels.