Singularities and Sheaves in Symplectic Geometry and Geometric Representation Theory

这项经费支持的研究处于数学和物理学的交叉领域。它涉及多种追求，包括新工具的开发和开放性问题的解决。一个主要的主题是寻找新的方法来无限维空间和非线性微分方程的几何。例如，这个项目的一个主要目标是用一个简短的组合构建块列表来描述量子粒子的动力学。这意味着一种新的语言可以通过基本的语法来捕捉复杂的现象。这些方法受到了奇点理论的启发，在奇点理论中，对称破缺通常会揭示隐藏的结构。除了原创性研究外，该项目的一个广泛目标是在快速发展领域的新前沿教育学生。此外，还将有充分的机会进行跨领域的推广，并增加公众对数学的参与。该研究涉及辛流形，经典相空间的现代后代，以及它们的量子不变量。更具体地说，该项目侧重于代数几何中的辛流形（Kaehler流形）和规范理论（束和连接的模）。具体方向集中在拉格朗日奇点和温斯坦流形的骨架，镜像对称中的微局部束，以及贝蒂几何朗兰兹对应。主要目标包括辛几何的组合方法，加强微局部束对同调镜像对称的适用性，以及自同构范畴的Verlinde公式。这些方法涵盖了辛几何、代数拓扑和规范理论中的一系列现代技术。它们也与超对称规范理论的核心研究有关，特别是规范场的相空间、它们的a模型和来自四维拓扑场理论的更高结构的研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.