Microlocal Sheaves, Symplectic Geometry and Applications in Representation Theory

微局域滑轮、辛几何及其在表示论中的应用

• 批准号: 1854232
• 负责人: Xin Jin
• 金额: $ 7.01万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1854232&HistoricalAwards=false
• 关键词: Microlocal; Sheaves; Symplectic; Geometry; Applications

Symplectic geometry is a mathematical framework for the study of classical and quantum mechanics. It centers around questions about the structures and symmetries of a symplectic manifold--an even-dimensional space that locally looks like the phase space containing the position and momentum of a moving particle. Modern physics has led to the discoveries of many important and sophisticated invariants of symplectic manifolds, and has predicted a number of deep connections to different fields of mathematics. Current definitions and approaches to these invariants are based on the analysis of a special kind of mapping of a surface to a symplectic manifold, known as the theory of holomorphic curves. The primary goal of the project is to develop an alternative approach to some of these invariants, which is more accessible and which will form the foundations of effective calculations. The approach is based on the microlocal sheaf theory, which was invented as an algebraic and topological method to study differential equations. The project will have many applications in the field of representation theory, a rich subject focusing on the study of symmetries appearing in mathematics and physics. The PI will also continue to organize seminars on related topics, disseminate her results through academic events, and provide research opportunities for undergraduate students.More specifically, the PI will use microlocal sheaf theory to quantize Lagrangian submanifolds in an exact symplectic manifold, and will give the definition of a microlocal sheaf category of a symplectic manifold, which is expected to be equivalent to the important symplectic invariant, known as the Fukaya category.  The approach is purely topological, and it allows the coefficient ring to be a ring spectrum, which opens up interesting connections to stable homotopy theory. The PI will develop a parallel story in the complex setting of holomorphic Lagrangians in an exact holomorphic symplectic manifold, which exhibits new and richer structures and which will have important applications in geometric representation theory. The PI will use this to quantize holomorphic Lagrangians in symplectic resolutions, a class of holomorphic symplectic manifolds in the center of modern representation theory, with the goal of understanding the mysterious phenomena of symplectic duality. Other applications to representation theory involve calculations in the Hecke category using sheaf quantizations of the braid group action as symplectomorphisms and a proposal to realize the nonabelian Hodge theory using microlocal perverse sheaves.

Symple毒物的几何形状是研究经典和量子力学的数学框架。它围绕着关于对称歧管的结构和对称性的问题，即局部看起来像是包含移动粒子位置和动量的相位空间。现代物理学导致了许多重要和复杂的对称流形的发现，并预测了与数学不同领域的许多深厚联系。这些不变性的当前定义和方法是基于对表面映射到对称歧管的特殊映射的分析，称为霍明态曲线理论。该项目的主要目的是为其中一些不变式开发一种替代方法，该方法更容易访问，并且将构成有效计算的基础。该方法是基于微局部捆的理论，该理论是作为研究微分方程的代数和拓扑方法发明的。该项目将在代表理论领域中有许多应用，这是一个重点，侧重于数学和物理学中出现的对称性的研究。 PI还将继续组织有关相关主题的半手，通过学术活动来传播她的结果，并为本科生提供研究机会。更具体地说，PI将使用微观的链条理论来量化Lagrangian submanifolds，以确切的对称流形量化Lagrangian Submanifolds，并将其定义与预期的Syment syment syment的定义相等，以等于等于，这是等于等于的。福卡亚类别。该方法纯粹是拓扑结构，它允许核心环成为环光谱，从而打开了与稳定同型理论的有趣联系。 PI将在精确的全态对称歧管中在复杂的Holomorthic Lagrangians的复杂环境中发展一个平行的故事，该歧管表现出新的，更丰富的结构，并且在几何表示理论中具有重要的应用。 PI将使用它来量化对称分辨率的全态Lagrangians，这是现代代表理论中心的一类全体形态对称歧管，其目的是了解符号二元性的神秘现象。代表理论的其他应用涉及使用编织组作用的捆绑量化作为符号切除型的分层计算，以及使用微局部变形束带实现非阿比尔霍奇理论的建议。

项目成果

Hyperbolicity of asymmetric lemon billiards

不对称柠檬台球的双曲性

• DOI: 10.1088/1361-6544/abaff2
• 发表时间: 2021
• 期刊: Nonlinearity
• 影响因子: 1.7
• 作者: Jin, Xin;Zhang, Pengfei
• 通讯作者: Zhang, Pengfei

Symplectomorphisms of T^*(G_C/B) and the braid group I: A homotopy equivalence for G_C=SL_3(C)

T^*(G_C/B) 和辫群 I 的辛同态：G_C=SL_3(C) 的同伦等价

• 发表时间: 2019
• 期刊: Journal of symplectic geometry
• 影响因子: 0.7
• 作者: Jin, Xin

Representing the Big tilting sheaves as holomorphic Morse Branes

将大倾斜滑轮表示为全纯莫尔斯布拉内斯

• DOI: 10.1016/j.aim.2019.01.035
• 发表时间: 2019
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Jin, Xin