Professor Sarnak's research will be in the spectral theory of automorphic forms with applications to number theory, combinatorics and mathematical physics. Two of the unsolved problems in this area which he plans to work on are (1) The existence of Maass cusp forms for nonuniform lattices and (2) The "hearing the shape of a drum" problem, i.e. what can be said about an isospectral set of manifolds. Professor Osgood will work on several problems in geometric function theory, some in joint work with Sarnak. This work will concern the isospectral problem for surfaces with a generalization of the classical Schwarzian derivative to higher dimensions. This research will involve a combination of geometry and analysis with applications to other areas of mathematics. It turns out that the theory of automorphic forms, a subject arising originally from number theory often as generating functions for arithmetic functions, has many applications to other areas. This theory is one of the main focuses of this research.
