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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。一个主要的主题是寻找新的方法来几何的无限维空间和非线性微分方程。例如，该项目的主要目标是用一系列组合构建块来描述量子粒子的动力学。这预示着一种新的语言可以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。也将有足够的机会跨领域推广和增加公众对数学的参与。研究处理辛流形，经典相空间的现代后代，及其量子不变量。更具体地说，这些项目集中在代数几何（Kaehler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辛几何，加强的适用性microlocal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现代技术。他们还与超对称规范理论的核心追求有关，特别是规范场的相空间，它们的A模型和来自四维拓扑场论的更高结构的研究。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而被认为值得支持。