Click here for the course syllabus.
Click here for class handouts, homework sets and lecture notes.
My research interests are in
differential geometry in finite and infinite dimensions,
particularly with applications to/from
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.
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
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
cannot exist on the manifold. In particular, these theorems extend
the classical Bochner and Myers type theorems.
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), on Lax pairs and Feynman diagrams (with Gabriel Baditoiu), on applications of differential topology to analysis of networks (with Cedric Geneset, Prakash Balanchandran, and Eric Kolaczyk), and on applications of differential geometry to machine learning (with Qinxun Bai and Stan Sclaroff).
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 $128.00 for hardcover (ISBN 0 521 46300 9) and $51.00 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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Creation Date: March 29,1996