CAREER: Curvature, Topology, and Geometric Partial Differential Equations, with new tools from Applied Mathematics

Research Foundation Of The City University Of New York (Lehman), Bronx NY

Investigators

Abstract

This award is funded in whole or in part under the American Rescue Plan Act of 202 (Public Law 117-2). This project will harness methods associated with applied mathematics to answer fundamental questions in geometry regarding how the curvature of a multi-dimensional object constrains its global shape, and how rigid or malleable geometric configurations can be under various intrinsic or extrinsic conditions. The anticipated findings of this agenda should be applicable to mathematical models used to predict the stability and optimal shape of a wide range of small and large parts of our physical world: from cell membranes, fluid droplets, and airplane wings, to the interface between different layers of the Earth's atmosphere, the event horizon of a black hole, and even the entire universe. Pedagogical efforts will engage graduate and undergraduate students in the discovery process. The latter will be supported by the creation of a 3D printing and visualization lab at CUNY Lehman College, which will be the first of its kind in any public institution of higher education in the Bronx borough of New York City, enabling new forms of inquiry-based instruction grounded on experiential learning. This facility will also be used to host events in partnership with CUNY Bronx Community College, to attract more students to Mathematics and help address the current overall shortage of workers with STEM qualifications. The lines of investigation in this project can be separated in two main categories, involving novel applications of either convex algebraic geometry or bifurcation theory to geometric analysis. In the first category, new topological obstructions to curvature conditions on closed manifolds will be sought through strategies that combine recently developed convex optimization tools, such as semidefinite programming, and classical local-to-global methods, including Chern-Weil theory, Index theory for twisted Dirac operators, and the Bochner technique. In particular, extremal values of polynomials on spectrahedral shadows of curvature operators will be used to bound characteristic numbers of certain manifolds with nonnegative or nonpositive sectional curvature, or special holonomy. These bounds are expected to shed new light on the Hopf Questions about existence of positively curved metrics in products of spheres, and the sign of the Euler characteristic in nonnegative or nonpositive curvature, as well as on the Stolz conjecture on the Witten genus of string manifolds with positive Ricci curvature. In the second category, global results from bifurcation theory will be used to analyze issues regarding symmetry, stability, rigidity, and multiplicity of minimal and constant mean curvature hypersurfaces, Einstein metrics, and solutions to other partial differential equations that arise in conformal or complex geometry, such as the Yamabe problem and its many variants. This bifurcation-theoretic approach provides several advantages which complement existing variational methods, including a finer control on the topology and regularity of the solutions produced. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.