Lagrangian Skeleta in Symplectic Geometry and 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。一个主要的主题是用简单的构建模块来理解复杂的系统。例如，主要目标是根据局部组合模型的具体列表来描述全局相空间。这提供了一种新的语言，可以通过基本语法捕捉复杂的现象。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。研究中心围绕辛流形，经典相空间的现代后代，和他们的量子不变量。更具体地说，这些项目侧重于代数几何中的辛流形（Kahler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Weinstein流形的组合方法，同调镜像对称中微局部束的基础，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技术。它们与超对称规范理论的中心追求有关，特别是来自四维拓扑场论的高级结构。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 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.