Representation Theory and Symplectic Geometry Inspired by Topological Field Theory

拓扑场论启发的表示论和辛几何

• 批准号: 2401178
• 负责人: David Nadler
• 金额: $ 27万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401178&HistoricalAwards=false
• 关键词: Representation; Theory; Symplectic; Geometry; Inspired

Geometric representation theory and symplectic geometry are two subjects of central interest in current mathematics. They draw original inspiration from mathematical physics, often in the form of quantum field theory and specifically the study of its symmetries. This has been an historically fruitful direction guided by dualities that generalize Fourier theory. The research in this project involves a mix of pursuits, including the development of new tools and the solution of open problems. A common theme throughout is finding ways to think about intricate geometric systems in elementary combinatorial terms. The research also offers opportunities for students entering these subjects to make significant contributions by applying recent tools and exploring new approaches. Additional activities include educational and expository writing on related topics, new interactions between researchers in mathematics and physics, and continued investment in public engagement with mathematics.The specific projects take on central challenges in supersymmetric gauge theory, specifically about phase spaces of gauge fields, their two-dimensional sigma-models, and higher structures on their branes coming from four-dimensional field theory. The main themes are the cocenter of the affine Hecke category and elliptic character sheaves, local Langlands equivalences and relative Langlands duality, and the topology of Lagrangian skeleta of Weinstein manifolds. The primary goals of the project include an identification of the cocenter of the affine Hecke category with elliptic character sheaves as an instance of automorphic gluing, the application of cyclic symmetries of Langlands parameter spaces to categorical forms of the Langlands classification, and a comparison of polarized Weinstein manifolds with arboreal spaces.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.

几何表示理论和符号几何形状是当前数学中核心兴趣的两个主题。他们从数学物理学中汲取了原始灵感，通常以量子场理论的形式，特别是对其对称性的研究。这是一个历史上富有成果的方向，这些方向是二元性的二元性，这些方向概括了傅立叶理论。该项目的研究涉及各种追求，包括开发新工具和解决开放问题的解决方案。整个过程中的一个共同主题是寻找以基本组合术语来思考复杂的几何系统的方法。这项研究还为进入这些学科的学生提供了机会，通过应用最新工具和探索新方法来做出重大贡献。其他活动包括有关相关主题的教育和说明性写作，数学和物理学研究人员之间的新互动，以及在公众参与数学方面的持续投资。具体项目在超对称规格理论中面临着核心挑战，特别是关于仪表领域的相位空间，他们的二维Sigma模型以及对布兰斯的二维式结构以及来自四维理论的更高结构。主要主题是Aggine Hecke类别和椭圆形的束带，本地Langlands等价和相对Langlands二重性的对焦，以及Weinstein歧管Lagrangian Skeleta的拓扑结构。该项目的主要目标包括识别具有椭圆形特征的仿射类别的合伙人作为自动胶合的实例值得通过基金会的智力优点和更广泛的影响审查标准来通过评估来支持。