RUI: Problems in Stochastic Geometry

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.