CAREER: Lattice Point Distribution and Homogeneous Dynamics

Boston College, Chestnut Hill MA

Investigators

Abstract

This project will use analytic and geometric tools to study classical problems in number theory. The basic arithmetic problems in question are finding, counting, and understanding the distribution of integer (or rational) solutions to algebraic equations that arise naturally in mathematics. The simplicity and intrinsic beauty of these problems, and the disproportionate depth and effort of their resolution, has inspired their study since ancient Greece. In many cases, the symmetries of the algebraic equations in question have a rich geometric and analytic structure. Using this structure it is possible to translate the arithmetic problems into geometric and dynamic problems on spaces of symmetries. Understanding how the arithmetic features of a problem manifest in the geometry of the corresponding space creates a link between arithmetic and geometric phenomena, and advances knowledge in both fields. The research in this proposal is complemented by educational and outreach activities, including the creation of a summer research workshop and a graduate seminar on analytic number theory. The PI will study two types of problems in homogenous dynamics, both originating from arithmetic. The first type of problems are shrinking target problems for unipotent flows on homogenous spaces. Shrinking target problems for diagonalizable group actions on homogenous spaces are very well understood. The corresponding problems for unipotent flows (as well as other slow mixing actions) are not yet understood, except for some special arithmetic cases. The PI will use methods from spectral theory and analytic number theory, in combination with methods of homogenous dynamics and ergodic theory, in order to analyze shrinking target problems for unipotent flows and their applications to the classical field of metric Diophantine approximations. The second type of problems regards the distribution of translates of closed orbits on homogenous spaces. The study of the distribution of translates of closed subgroup-orbits is an interesting problem with many applications. When the orbits are compact, or of finite measure, there are a number of techniques to study the limiting distribution of their translates, and these can be applied to study the classical problem of distribution of integer points in algebraic varieties; they are also amenable to more analytic applications in the study of L-functions of automorphic forms. The PI will extend such results to orbits of infinite measure, with an emphasis on cases having interesting arithmetic applications.