My research interests in pure math are in
differential geometry in finite and infinite dimensions,
particularly with applications to/from
mathematical physics. Recently, I've been looking at applications of
differential geometry to statistics and machine learning, exotic
characteristic classes and diffeomorphism groups of manifolds, and
the Riemannian geometry of moduli spaces of pseudoholomorphic curves.
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 $138 for hardcover, $61 for paperback, and $49 for eBook. 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 Cambridge University Press for allowing me to make the text available online, in contrast to some other math text publishers.
Link to Higher Education