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Click here for the course syllabus.
My research interests in pure math 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 articles 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 Prakash Balanchandran, Cedric Geneset, Eric Kolaczyk, Nathaniel Josephs, 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 articles.
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 $165.00 for hardcover (ISBN 0 521 46300 9) and $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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Creation Date: March 29,1996