Overview
The focus of Professor Stern's research is the study of analytic problems arising in geometry and physics.
In recent and ongoing work, Professor Stern has studied analytical, geometric, and topological questions arising in Yang-Mills theory. These include analyzing the moduli space of Yang Mills instantons on gravitational instantons, analyzing the asymptotic structure of instantons (proving a nonlinear analog of the inverse square law of electromagnetism), and analyzing the structure of singularities of instantons and of harmonic maps.
In addition, Professor Stern has recently studied questions arising in the interplay between geometric group theory and Lp and L2 cohomology. This work includes finding new bounds on L2 betti numbers of negatively curved manifolds, and new growth,
stability, and vanishing results for Lp and L2 cohomology of symmetric and locally symmetric spaces.
Current Appointments & Affiliations
Recent Publications
Instantons on multi-Taub-NUT Spaces III: Down Transform, Completeness, and Isometry
Journal Article Journal of differential geometry · January 2026 The index bundle of a family of Dirac operators associated to an instanton on a multi-Taub-NUT space forms a bow representation. We prove that the gauge equivalence classes of solutions of this bow representation are in one-to-one correspondence with the i ... Full text Link to item CiteON THE BETTI NUMBERS OF FINITE VOLUME REAL- AND COMPLEX-HYPERBOLIC MANIFOLDS
Journal Article Journal of Differential Geometry · June 1, 2025 We obtain strong upper bounds for the Betti numbers of compact complex-hyperbolic manifolds. We use the unitary holonomy to improve the results given by the most direct application of the techniques of [DS22]. We also provide effective upper bounds for Bet ... Full text CiteHarmonic Forms, Price Inequalities, and Benjamini–Schramm Convergence
Journal Article Journal of Geometric Analysis · January 1, 2025 We study Betti numbers of sequences of Riemannian manifolds which Benjamini–Schramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, under a negative Ricci curvature ... Full text CiteRecent Grants
The Geometry and Analysis of Yang-Mills Instantons.
Institutional SupportPrincipal Investigator · Awarded by Simons Foundation · 2023 - 2028Instanton Decay and Nonlinear Harmonic Forms
ResearchPrincipal Investigator · Awarded by Simons Foundation · 2015 - 2022Chern-Simons Invariance Instantons and Mass
ResearchPrincipal Investigator · Awarded by National Science Foundation · 2010 - 2014View All Grants