Lagrangian Skeleta in Symplectic Geometry and Representation Theory

点击翻译按钮获取中文摘要

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 understanding complicated systems in terms of simple building blocks. For example, a primary aim is to describe global phase spaces in terms of a concrete list of local combinatorial models. This offers 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 centers around symplectic manifolds, the modern descendants of classical phase spaces, and their quantum invariants. More specifically, the projects focus on symplectic manifolds arising in algebraic geometry (Kahler 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 Weinstein manifolds, foundations of microlocal sheaves in homological mirror symmetry, and a Verlinde formula for automorphic categories. The methods span a range of techniques in symplectic geometry, algebraic topology, and gauge theory. They connect with central pursuits in supersymmetric gauge theory, in particular 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.