Collaborative Research: Arithmetic and Equidistribution on Homogeneous Spaces

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In recent years, it has become clear that many interesting problems,in particular problems in arithmetic, quantum chaos and the theory ofL-functions, may be profitably reduced to questions concerningequidistribution of points or measures on homogeneous spaces. Thesequestions regarding equidistribution can be approached from manyangles. Two theories which have proved to be particularly well-suitedto study such questions are the spectral theory of automorphic forms,which is closely related to the theory of L-functions, and the theoryof dynamical systems, particularly the study of unipotent and moregeneral flows on these homogeneous spaces. Recently there has beenconsiderable progress involving tools such as special value formulaefor L-functions, and (partial) classification results for measuresinvariant under higher rank torus actions. Particularly exciting isthe possibility, already realized in some instances, of combiningthese techniques. The purpose of the proposed FRG is to investigatefurther this circle of ideas, which we believe has the potential toimpact many other problems related to the above. The result of theseinvestigations will be a deeper understanding of the dynamics of groupactions on homogeneous spaces, of the analytic theory of automorphicforms, and the (sometimes unexpected) applications to problems ofarithmetic nature.The present project is concerned with a surprising link between twoclassical fields of mathematics of quite disparate origin: numbertheory and dynamics. The study of number theory began thousands ofyears ago, motivated, in significant part, by questions about primenumbers. On the other hand, ergodic theory and dynamics aremathematical fields of more recent provenance, which arose fromstudying the long-term evolution of complicated deterministicprocesses -- such as planetary motion. It is a striking fact (whichhas only recently begun to be heavily exploited) that, in certaincontexts, ideas from ergodic theory interact very deeply with classicalproblems in number theory. This project will enhance ourunderstanding of this inter-relation and how we can combine knowledgefrom both of these fruitful disciplines effectively.