Office: MCS 276

Office Hours: On leave this semester

Email: (insert my two initials)@math.bu.edu

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.

- Invariants of conformal Laplacians (1985)
- Manifolds with wells of negative curvature (1991)
- Bounds on the fundamental group of a manifold with almost nonnegative Ricci curvature (1994)
- Minimal submanifolds of metrics (1995)
- Homotopy and homology vanishing theorems and the stability of stochastic flows (1995)
- L2 and bounded harmonic forms on universal covers (1996)
- Nonlocal invariants in index theory -- a survey (1997)
- Mathai-Quillen forms and Lefschetz theory (1998)
- Curvature on determinant bundles and first Chern forms (2000)
- Gauge Theory Techniques in Quantum Cohomology (2000)
- Chern-Weil constructions on PDO bundles (2003)
- Characteristic classes and traces on loop spaces (2003)
- Infinite dimensional Chern-Simons theory (2004 - now partially incorporated into the last article below)
- Conformal anomalies via canonical traces (2005)
- Feynman diagrams and Lax pair equations (2006)
- Characteristic classes and zeroth order pseudodifferential operators (2010)
- Riemannian geometry on loop spaces (2010) ( the Mathematica file ComputationsChernSimonsS2xS3.pdf of computations in this paper)
- Chern-Weil theory for certain infinite-dimensional Lie groups (2013)
- Equivariant, string, and leading order characteristic classes associated to fibrations (2013)
- Traces and characteristic classes in infinite dimensions (2014)
- The geometry of loop spaces I: $H^s$-Riemannian metrics (2014)
- The geometry of loop spaces II: Characteristic classes (2014)
- Hypothesis testing for network data in functional neuroimaging (2014)
- A differential geometric approach to classification (2015)
- Enabling Adiabatic Passages Between Disjoint Regions in Parameter Space through Topological Transitions (2015)
- Geodesic Paths for Quantum Many-Body Systems (2016)
- Averages of Unlabeled Networks: Geometric Characterization and Asymptotic Behavior (2017)
- Discretized Gradient Flow for Manifold Learning in the Space of Embeddings (2019)
- Central Limit Theorems on Compact Metric Spaces (2020)
- Bayesian classification, anomaly detection, and survival analysis using network inputs with application to the microbiome (2020)
- The Geometry of Loop Spaces III: Diffeomorphisms of Contact Manifolds (2020)
- Solving the Yamabe Problem by an Iterative Method on a Small Riemannian Domain (2021)

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 EducationReturn to Math/Stats home page

Creation Date:* March 29,1996 *