简介

My research interests are in differential geometry in finite and infinite dimensions, particularly with applications to/from mathematical physics. Almost all this work uses Laplacian-type operators sooner or later. My current work focuses on characteristic classes for infinite dimensional bundles, with collaborations with Andres Larrain-Hubach, Yoshi Maeda, Sylvie Paycha, Simon Scott, and Fabian Torres-Ardila. Older work includes the functional/zeta determinant of Laplacians, which is a key element of quantum field theory (or non-theory), and (with K. D. Elworthy and Xue-Mei Li) applications of Brownian motion to differential geometry. This has given a series of results of the type: topological condition A on a compact manifold implies that metrics of type B cannot exist on the manifold. In particular, these theorems extend the classical Bochner and Myers type theorems. Heat operators associated to Laplacians figure heavily in this work; after all, Brownian motion is supposed to model heat flow as an example of infinite dimensional Riemannian geometry. More recently, in the work on primary and secondary characteristic classes on infinite dimensional manifolds such as loop spaces, Laplacians enter in the curvature of connections on these manifolds