Interactions Between Homogeneous Dynamics and Number Theory

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

ABSTRACT The problems that are to be addressed in this project involve, on one hand, dynamics of flows on homogeneous spaces of Lie groups, and, on the other hand, the multi-dimensional theory of Diophantine approximations. Various connections between these two fields have been found in the last two decades, which significantly stimulated progress in both fields. During recent years, the proposer's research has been centered on developing new links between homogeneous dynamics and number theory, which has resulted in solving many important problems, as well as in creating new directions for further research. The investigator is to continue his work on bounded trajectories, growth rate of orbits, Diophantine approximation with weights and Khintchine-type theorems on manifolds. This project deals with algebraic dynamical systems and their applications to number theory. Many problems concerning simultaneous approximation of real numbers by rational numbers can be cast in terms of the behavior of certain orbits. Dynamical systems in the present context deal with how points in a system move over time, given a set of differential equations (or laws of nature) governing the system. It turns out that various numerical approximations used in the theory of integer equations can be better calculated once they are phrased in dynamical systems language.