The principal investigators will continue work on various areas of differential geometry including large time heat flow on Riemannian manifolds and Teichmuller space. The work on heat flow will use method from probability to attempt to describe the relationship between heat diffusion and geometric quantities. One of the investigators will also study discrete versions of some problems in spectral theory. The research on Teichmuller theory will focus on applications of the theory to dynamical systems. The principal investigators will also attempt to develop a Teichmuller theory for quadratic and polynomial-like maps. This award will support research in the general area of differential geometry and global analysis. Differential geometry is the study of the relationship between the geometry of a space and analytic concepts defined on the space. Global analysis is the study of the overall geometric and topological properties of a space by piecing together local information. Applications of these areas of mathematics in other sciences include the structure of complicated molecules, liquid-gas boundaries, and the large scale structure of the universe.
