Singularities and Sheaves in Symplectic Geometry and Geometric Representation Theory

这笔拨款支持的研究处于数学和物理的十字路口。它涉及多种追求，包括开发新工具和解决悬而未决的问题。一个主要的主题是寻找无限维空间几何和非线性微分方程的新方法。例如，该项目的一个主要目标是用一小串组合积木来描述量子粒子的动力学。这承诺了一种新的语言，可以通过基本的语法来捕捉复杂的现象。这些方法的灵感来自奇点理论，在奇点理论中，对称破缺往往揭示了隐藏的结构。除了原创研究，该项目的一个广泛目标是在快速发展的领域的新前沿教育学生。还将有大量的机会扩展到各个领域，并增加公众对数学的参与。这项研究涉及辛流形，经典相空间的现代后裔，以及它们的量子不变量。更具体地说，这些项目集中在代数几何(Kaehler流形)和规范理论(丛和联系的模数)中产生的辛流形。具体方向集中在Weinstein流形的拉格朗日奇点和骨架，镜像对称的微局部层，以及Betti几何朗兰兹对应。主要目标包括辛几何的组合方法，加强微局部层对同调镜像对称性的适用性，以及自同构范畴的Verlinde公式。这些方法涵盖了辛几何、代数拓扑学和规范理论中的一系列现代技术。他们还与超对称规范理论的核心追求联系在一起，特别是对规范场的相空间及其A模型的研究，以及来自四维拓扑场理论的更高结构的研究。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.