Singularities and Sheaves in Symplectic Geometry and Geometric Representation Theory

辛几何和几何表示理论中的奇点和滑轮

基本信息

批准号：
1802373
负责人：
David Nadler
金额：
$ 33万
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：
2018
资助国家：
美国
起止时间：
2018-06-01 至 2022-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802373&HistoricalAwards=false
关键词：
Singularities Sheaves Symplectic Geometry Geometric

项目摘要

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.

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。一个主要主题是找到无限维空间和非线性微分方程的几何形状的新方法。例如，该项目的主要目的是用组合构建块的简短列表来描述量子粒子的动力学。这有望通过基本语法捕获复杂现象的新语言。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。在跨领域的外展和公众参与数学也将有足够的机会。研究涉及象征性的歧管，古典相位空间的现代后代及其量子不变性。更具体地说，这些项目着重于代数几何形状（Kaehler歧管）和仪表理论（束和连接模量）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。主要目标包括一种组合方法的合并方法，增强了微局部滑轮对同源镜子对称性的适用性，以及用于自动形态类别的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现代技术。它们还与超对称规格理论中的中心追求联系，特别是对量规场的相位空间，其A模型的研究以及来自四维拓扑场理论的较高结构。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和广泛影响的评估来评估CRETERIA的评估。