The investigator plans to study three problems in the area of stochastic differential geometry. The first problem concerns the Gauss-Bonnet-Chern theorem. The investigator has proved a generalization of this theorem valid for an oriented even-dimensional Riemannian vector bundle over a closed compact manifold, equipped with a metric connection. The investigator would like to simplify his proof, replacing the use of the Splitting Principle, which constitutes a key step in his argument, with a more elementary and transparent stochastic argument. The second problem concerns the derivation of integration by parts formulae for the law of a diffusion process on a compact manifold. The investigator has recently found a new method for obtaining integration by parts formulae in the case where the diffusion is strictly elliptic. He plans to use his method to study the analogous problem for degenerate (non-elliptic) diffusions. He believes he will be able to prove a dichotomy theorem giving conditions very close to necessary and sufficient for the existence of integration by parts formulae, for a large class of vector fields in the degenerate case. In the third problem, the investigator in collaboration with Elton Hsu, will use ideas in two papers of Hsu to give a simplified stochastic proof of the Gauss-Bonnet-Chern theorem for manifolds with boundary. They will use the expertise they gain from this work to study the index theorem for the Dirac operator on spin manifolds with boundary. The proposal combines two areas of mathematics, stochastic analysis, the study of randomness, and differential geometry, the study of shape. Both areas have close connections with the physical world. For example, stochastic differential equations, a central theme of the proposal, are widely used to model physical systems subject to the influence of random noise. Examples include the flow of heat in a material, weather systems, the trajectory of a spacecraft, and the pricing of stock options. Curved spaces are often the natural setting for these phenomena, e.g. if one wishes to study the heat at various points on a cylindrical pipe. A deeper understanding of the mathematics underlying physical systems is often crucial in developing good mathematical models of these systems. Thus, although it is of a theoretical nature, the research outlined in this proposal may prove useful in many areas of applied science and technology.
