The research supported by this grant lies at the crossroads of mathematics and physics. It involves a mix of pursuits, including the development of new tools and the solution of open problems. A main theme is finding new approaches to the geometry of infinite-dimensional spaces and nonlinear differential equations. For example, a primary aim of the project is to describe the dynamics of quantum particles in terms of a short list of combinatorial building blocks. This promises a new language to capture intricate phenomena through an elementary syntax. The methods are inspired by singularity theory, where symmetry-breaking often reveals hidden structure. In addition to original research, a broad goal of the project is the education of students in the new frontiers of rapidly developing fields. There will also be ample opportunities for outreach across fields and for increased public engagement with mathematics.
The research tackles symplectic manifolds, the modern descendant of classical phase spaces, and their quantum invariants. More specifically, the projects focus on symplectic manifolds arising in algebraic geometry (Kaehler manifolds) and gauge theory (moduli of bundles and connections). Specific directions focus on Lagrangian singularities and skeleta of Weinstein manifolds, microlocal sheaves in mirror symmetry, and the Betti Geometric Langlands correspondence. The main goals include a combinatorial approach to symplectic geometry, a strengthening of the applicability of microlocal sheaves to homological mirror symmetry, and a Verlinde formula for automorphic categories. The methods span a range of modern techniques in symplectic geometry, algebraic topology, and gauge theory. They also connect with central pursuits in supersymmetric gauge theory, specifically the study of phase spaces of gauge fields, their A-models, and higher structures coming from four-dimensional topological field theory.
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.