9504900 Pardon This project will attempt to define characteristic classes and then prove (by the heat equation method) a Riemann-Roch theorem for the moduli space of rank-two, degree-zero vector bundles on a Riemann surface. It will also attempt to find an analytic understanding of the difference between Neumann and Dirichlet boundary conditions for differential forms on the smooth part of a complex projective variety. Finally, it is planned that prior results of the investigator and Mark Stern concerning pure Hodge structures on the intersection cohomology of a projective variety will be extended without recourse to the theory of D-modules. There are many quantitative approaches to the study of curved objects, parts of which are not smooth or even, but instead have corners or cusps. (Imagine a crumpled piece of metal.) For instance, one may ask how these cusps affect the diffusion of heat within the object; or, conversely, whether observed properties of the diffusion determine the nature of the cusps. For very complicated smooth objects (even those of more than three dimensions), the effect of the curvature of the object on heat diffusion within it is fairly well understood. One goal of this project is to extend to objects with cusps, the known analysis of heat diffusion in smooth objects. ***
