Welcome to Steve Rosenberg's Home Page


Office: MCS 278 -- but not during Fall 2020
Office Hours: Tuesday/Thursday 1-2:30
Email: sr(at)math.bu.edu


Fall 2020 Courses

MA242 C1 Linear Algebra

Click here for the course syllabus.

Visit the Blackboard course site for more information.

MA721 Differential Toopology I

Click here for the course syllabus.


Research

My research interests in pure math 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.

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 length spaces (with Deane Yang), on quantum cohomology (with Mihaela Vajiac), and on Lax pairs and Feynman diagrams (with Gabriel Baditoiu). More recent work in applied math includes applications of differential topology and geometry to analysis of networks (papers with Cedric Geneset, Prakash Balanchandran, Eric Kolaczyk, Lizhen Li, Jackson Walters, Jie Xu), applications of differential geometry to machine learning (with Qinxun Bai, Stan Sclaroff, Zheng Wu; Dara Gold), applications of Kaehler geometry to quantum mechanics (with Michael Kolodrubetz, Anatoli Polkovnikov, Mischi Tomka, and Tiago Souza), and Central Limit Theorems on metric spaces (with Jie Xu).

Here is a list of available preprints/reprints.

My book, "The Laplacian on a Riemannian Manifold," now in its second (corrected) printing, is also available. The book covers Hodge theory, basics of differential geometry, heat flow on functions and forms, the heat equation/supersymmetric proof of the Chern-Gauss-Bonnet theorem, an overview of the Atiyah-Singer Index Theorem, the zeta function for Laplacians and analytic torsion. There are lots of exercises. The price is US$165.00 for hardcover (ISBN 0 521 46300 9) and US$62.00 for paperback (ISBN 0 521 46831 0). You can order copies (no upper limit) from the publisher, Cambridge University Press. Alternatively, you can view the book here. Feel free to download it, but consider making a donation to a good cause in lieu of buying the text. I would like to thank CUP for allowing me to make the text available online, in contrast to some other math text publishers.

Link to Higher Education

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Visitors since September 2, 1999

Creation Date: March 29,1996