Lagrangian Skeleta in Symplectic Geometry and Representation Theory

辛几何与表示论中的拉格朗日骨架

• 批准号: 2101466
• 负责人: David Nadler
• 金额: $ 40.5万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101466&HistoricalAwards=false
• 关键词: Lagrangian; Skeleta; Symplectic; Geometry; Representation

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.

这项赠款支持的研究在于数学和物理学的十字路口。它涉及各种追求，包括开发新工具和解决开放问题的解决方案。一个主要主题是从简单的构建块来理解复杂的系统。例如，主要目的是用局部组合模型的具体列表来描述全球相位空间。这提供了一种新的语言，可以通过基本语法捕获复杂的现象。这些方法的灵感来自奇异理论，在这种理论中，对称性破裂通常揭示了隐藏的结构。除原始研究外，该项目的一个广泛目标是在快速发展领域的新领域对学生的教育。在整个领域的外展和公众参与数学的参与也将有足够的机会。 研究以符号歧管，古典相位空间的现代后代及其量子不变性为中心。更具体地说，这些项目着重于代数几何形状（Kahler歧管）和仪表理论（捆绑和连接模量）中产生的符合歧管。特定方向着重于温斯坦歧管的拉格朗日奇异性和骨骼，镜子对称性中的微局部滑轮和贝蒂几何兰兰兹对应关系。主要目标包括用于Weinstein歧管的组合方法，同源镜对称性中的微局部滑轮的基础以及用于自代化类别的Verlinde公式。这些方法涵盖了符号几何，代数拓扑和量规理论中的一系列技术。他们与超对称规格理论中的中心追求联系，特别是来自四维拓扑领域理论的更高结构。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛影响的评估标准来通过评估来获得支持的。