Welcome to Steve Rosenberg's Home Page

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

Research

My research interests in pure math are in differential geometry in finite and infinite dimensions. Recently, I've been looking at applications of differential geometry to statistics and machine learning, exotic characteristic classes and isometry groups of manifolds, positive scalar curvature metrics, and the Riemannian geometry of moduli spaces of pseudoholomorphic curves.













Here is a list of available articles. Most are with co-authors.

• The Geometry of Loop Spaces IV: Closed Sasakian Manifolds (2024)
• A Codimension Two Approach to the S¹-Stability Conjecture (2024)
• The Conformal Laplacian and Positive Scalar Curvature Metrics on Manifolds with Boundary (2023)
• Generalized Tangent Kernel: A Unified Geometric Foundation for Natural Gradient and Standard Gradient (2022)
• Solving the Yamabe Problem by an Iterative Method on a Small Riemannian Domain (2021)
• The Geometry of Loop Spaces III: Diffeomorphisms of Contact Manifolds (2020)
• Bayesian classification, anomaly detection, and survival analysis using network inputs with application to the microbiome (2020)
• Central Limit Theorems on Compact Metric Spaces (2020)
• Discretized Gradient Flow for Manifold Learning in the Space of Embeddings (2019)
• Averages of Unlabeled Networks: Geometric Characterization and Asymptotic Behavior (2017)
• Geodesic Paths for Quantum Many-Body Systems (2016)
• Enabling Adiabatic Passages Between Disjoint Regions in Parameter Space through Topological Transitions (2015)
• A differential geometric approach to classification (2015)
• Hypothesis testing for network data in functional neuroimaging (2014)
• The geometry of loop spaces I: H^s-Riemannian metrics (2014)
• The geometry of loop spaces II: Characteristic classes (2014); The geometry of loop spaces II: Corrections (2024)
• Traces and characteristic classes in infinite dimensions (2014)
• Equivariant, string, and leading order characteristic classes associated to fibrations (2013)
• Chern-Weil theory for certain infinite-dimensional Lie groups (2013)
• Riemannian geometry on loop spaces (2010) (the Mathematica file ComputationsChernSimonsS2xS3.pdf of computations in this paper)
• Characteristic classes and zeroth order pseudodifferential operators (2010)
• Feynman diagrams and Lax pair equations (2006)
• Conformal anomalies via canonical traces (2005)
• Infinite dimensional Chern-Simons theory (2004)
• Chern-Weil constructions on PDO bundles (2003)
• Characteristic classes and traces on loop spaces (2003)
• Curvature on determinant bundles and first Chern forms (2000)
• Gauge Theory Techniques in Quantum Cohomology (2000)
• Mathai-Quillen forms and Lefschetz theory (1998)
• Nonlocal invariants in index theory -- a survey (1997)
• L² and bounded harmonic forms on universal covers (1996)
• Minimal submanifolds of metrics (1995)
• Homotopy and homology vanishing theorems and the stability of stochastic flows (1995)
• Bounds on the fundamental group of a manifold with almost nonnegative Ricci curvature (1994)
• The Witten Laplacian on negatively curved simply connected manifolds (1993)
• Anomalies associated to the polar decomposition of GL(n,C) (1992)
• Manifolds with wells of negative curvature (1991)
• The determinant of a conformally covariant operator (1987)
• The variation of the de Rham zeta function (1987)
• Invariants of conformal Laplacians (1985)
• On the Gauss-Bonnet theorem for complete manifolds (1985)
• Harmonic forms and L² cohomology on manifolds with cylinders (1985)
• Gauss-Bonnet theorems for noncompact surfaces (1982)

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.

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