Office: MCS 278 -- but not during Fall 2020

Office Hours: Tuesday/Thursday 1-2:30

Email: sr(at)math.bu.edu

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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, 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 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)
- 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)

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 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 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.

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Creation Date:* March 29,1996 *