Automorphic Forms and Applications
The 2002 IAS/ParkCity Mathematics Institute conference on “Automorphic form and their applications” took place in Park City, Utah. The lectures were divided into six general areas. Lectures by A. Borel and J. Bernstein on “Basic theory of Eisenstein series” addressed general aspects of the theory of automorphic forms on reductive groups over adèles (the lectures by Bernstein are not included in the proceedings). Lectures on “Converse Theorems and Langlands-Shahidi method” were delivered by J. Cogdell and F. Shahidi. The lectures by L. Clozel, W. Li and A. Valette covered the topic of the (now proven) Ramanujan conjectures for definite quaternion algebras and their applications to Ramanujan graphs and spectral gaps. The lectures by P. Michel on analytic number theory were devoted to the technique of studying averages over “families of automorphic L-functions”.
The lectures by A. Terras focused on a topic of current interest called arithmetic quantum chaos. For example, the geodesic flow on the unit sphere tangent bundle of an arithmetic hyperbolic surface leads to an interesting area for various conjectures in number theory. The last lecture series by A. Eshin described a very powerful tool that has emerged in recent years in connection with problems of equi-distribution of measures associated with arithmetic and spectral geometry in ergodic theory, e.g. the Ratner theorem.