Lagrangian Skeleta in Symplectic Geometry and Representation Theory

这项资助所支持的研究处于数学和物理的交叉点。它涉及多种追求，包括开发新工具和解决悬而未决的问题。一个主要的主题是从简单的构建块的角度来理解复杂的系统。例如，一个主要的目标是描述全球相空间的一个具体的本地组合模型列表。这提供了一种新的语言，通过基本的语法来捕捉复杂的现象。这些方法受到奇点理论的启发，奇点破缺通常会揭示隐藏的结构。除了原创性研究，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。也将有大量的机会跨领域推广和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，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（卡勒流形）和规范理论（丛和连接的模量）中产生的辛流形。具体的方向集中在拉格朗日奇点和Weinstein流形，镜像对称的微局部层，和贝蒂几何朗兰兹对应。主要目标包括一个组合的方法温斯坦流形，基础的微观局部层同调镜像对称，和Verlinde公式自守类别。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。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.