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
ON 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 CiteLp-cohomology and the geometry of p-harmonic forms
Chapter · January 1, 2025 In this note we describe basic geometric properties of p-harmonic forms and p-coclosed forms and use them to reprove vanishing theorems of Pansu and new injectivity theorems for the Lp-cohomology of simply connected, pinched negatively curved ma ... 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