Gregory Margulis: Homogeneous dynamics and number theory

Homogeneous dynamics is another name for the theory of flows on homogeneous spaces, or homogeneous flows. The study of homogenous flows has been attracting considerable attention for the last 40-50 years. During the last three decades, it has been realized that some problems in number theory and, in particular, in Diophantine approximation can be solved using methods from the theory of homogeneous flows. The purpose of the first lecture is to give examples of interactions between number theory and homogenous dynamics; mostly only formulations will be given, but there will be also very brief description of some proofs. The second and third lectures will be the continuation, on a more technical level, of the first lecture. Topics to be (very briefly) covered include: (1) recurrence to compact sets and Diophantine approximation on manifolds; (2) orbit closures, the Oppenheim conjecture and the Littlewood conjecture; (3) classification of ergodic invariant measures and equidistribution; (4) quantitative Oppenheim conjecture and counting of integral points on homogenous varieties; (5) effective results in homogeneous dynamics.