Welcome to Steve Rosenberg's Home Page
Office: MCS 248
Office Hours: Monday, Wednesday, Thursday 3-4
Email: sr(at)math.bu.edu
Research
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 a series of papers with Andres Larrain-Hubach, Yoshiaki Maeda, Sylvie Paycha, Simon Scott and Fabian Torres-Ardila, we've studied primary and secondary characteristic classes on infinite dimensional manifolds such as loop spaces; here the Laplacians enter in the curvature of connections on these manifolds
Other work: Yoshiaki Maeda, Philippe Tondeur and I have worked on the geometry of the gauge orbits in the space of connections, and on the geometry of the orbits of the diffeomorphism group in the space of metrics on a manifold. Mihail Fromosu and I have studied Mathai-Quillen forms, which have formal applications in QFT and rigorous applications in differential geometry. There are also preprints on quantum cohomology (with Mihaela Vajiac), and on Lax pairs and Feynman diagrams (with Gabriel Baditoiu). Here is a list of available preprints/reprints.
