Office: MCS 248

Office Hours: Monday 10-12, Wednesday 1-2

Email: sr(at)math.bu.edu

Click here for the course syllabus.

Click here for the course schedule.

Click here for the prep quiz to be handed in at Lecture 2 on September 5.

Click here for the first homework set, due Thursday, September 12.

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
mathematical physics.
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.
Older work
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
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 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), 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 Cedric Geneset, Prakash
Balanchandran,
Eric
Kolaczyk, Lizhen Li, Jackson Walters, Jie Xu), applications of differential geometry to machine learning
(with Qinxun Bai, Stan Sclaroff, and Zheng Wu), and on applications of Kaehler
geometry to quantum mechanics (with Michael Kolodrubetz,
Anatoli Polkovnikov, Mischi Tomka, and Tiago Souza).

Here is a list of available preprints/reprints.

- 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)

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 US$165.00 for hardcover (ISBN 0 521 46300 9) and US$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 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 *