In recent years, it has become clear that many interesting problems, in particular problems in arithmetic, quantum chaos and the theory of L-functions, may be profitably reduced to questions concerning equidistribution of points or measures on homogeneous spaces. These questions regarding equidistribution can be approached from many angles. Two theories which have proved to be particularly well-suited to study such questions are the spectral theory of automorphic forms, which is closely related to the theory of L-functions, and the theory of dynamical systems, particularly the study of unipotent and more general flows on these homogeneous spaces. Recently there has been considerable progress involving tools such as special value formulae for L-functions, and (partial) classification results for measures invariant under higher rank torus actions. Particularly exciting is the possibility, already realized in some instances, of combining these techniques. The purpose of the proposed FRG is to investigate further this circle of ideas, which we believe has the potential to impact many other problems related to the above. The result of these investigations will be a deeper understanding of the dynamics of group actions on homogeneous spaces, of the analytic theory of automorphic forms, and the (sometimes unexpected) applications to problems of arithmetic nature.
The present project is concerned with a surprising link between two classical fields of mathematics of quite disparate origin: number theory and dynamics. The study of number theory began thousands of years ago, motivated, in significant part, by questions about prime numbers. On the other hand, ergodic theory and dynamics are mathematical fields of more recent provenance, which arose from studying the long-term evolution of complicated deterministic processes -- such as planetary motion. It is a striking fact (which has only recently begun to be heavily exploited) that, in certain contexts, ideas from ergodic theory interact very deeply with classical problems in number theory. This project will enhance our understanding of this inter-relation and how we can combine knowledge from both of these fruitful disciplines effectively.
