Welcome to Steve Rosenberg's Home Page
Office: MCS 248
Office Hours: Wednesday 11-1, 2-3 or by appointment
Phone: (617) 353-9556; (617) 353-8100 (fax)
Email: sr(at)math.bu.edu
Spring 2008 Courses: MA242 Linear Algebra
Click here for
the course syllabus, which
contains all the course information and homework assignments.
Click here
for in-class demos.
MA822 Topics in Geometry and Topology
Click here for
the course syllabus.
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. Topics include 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 Yoshiaki Maeda, Sylvie Paycha 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.
My book, "The Laplacian on a Riemannian
Manifold," now in its second (corrected) printing, is also available,
just not for free.
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 $75.00 for hardcover (ISBN 0 521 46300 9)
and $29.99 for paperback (ISBN 0 521 46831 0). (The hardcover is "temporarily
unavailable.") 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.
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.
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Creation Date: March 29,1996
Last Serious Update: November 8,
2005