Welcome to Steve Rosenberg's Home Page

Office: MCS 248
Office Hours: Wednesday 11-12, 1-3
Email: sr(at)math.bu.edu

Fall 2017 Courses

MA242 Linear Algebra

Click here for the course syllabus.

Check the class Blackboard site at learn.bu.edu for class handouts, quiz answers, review sheets and test answers.

MA725 Differential Geometry

Click here for the course syllabus.

Click here for a pdf file of the text.


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 to analysis of networks (with Cedric Geneset, Prakash Balanchandran, and Eric Kolaczyk), applications of differential geometry to machine learning (with Qinxun Bai, Stan Sclaroff, and Zheng Wu), and on applications of Kaehler geometry to quantum mechanics (with Michael Kolodrubetz, Anatoli Polkovnikov, Mischi Tomka, and Tiago Souza).

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. This book is aimed at graduate students who have had a basic course in manifolds through integration of forms. The goal is to get students to appreciate current areas of research in global geometry. 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. You can preview the introduction. 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 print it out, but consider making a donation to a good cause in lieu of buying the text. I would like to thank Cambridge University Press for allowing me to make the text available online, in contrast to the attitude of other math text publishers.

For an antiquated streaming video of a lecture at MSRI in 1998, click here and select the "start 56kps video" option in the top left corner. Good luck.

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Visitors since September 2, 1999

Creation Date: March 29,1996