Algebraic Geometry of Hitchin Integrable Systems and Beyond

希钦可积系统及其他代数几何

• 批准号: 2301474
• 负责人: Junliang Shen
• 金额: $ 33万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2301474&HistoricalAwards=false
• 关键词: Algebraic; Geometry; Hitchin; Integrable; Systems

This research project focuses on algebraic geometry and moduli spaces. Algebraic geometry is the study of varieties, which are in turn the sets of solutions of polynomial equations. Moduli spaces are parameter spaces of varieties, which concern the behavior of varieties as the defining polynomials vary. In the last few decades, fundamental connections have been found relating moduli spaces in algebraic geometry to other fields including representation theory, topology, and quantum field theory in mathematical physics. This project aims to study several classes of moduli spaces which lie at the crossroads of central areas in mathematics and physics. The investigator will develop new tools concerning these varieties, attack long standing questions, and explore new connections. These projects will increase communication between the communities of enumerative geometry, topology, Hodge theory, and mathematical physics. The new developments will generate more activities and offer questions for graduate students and postdocs who are interested in these areas. Graduate students will be supported by this award. The research of the investigator will center around three projects: (1) to study the P=W phenomenon and the topological mirror symmetry for general reductive groups. This will bridge more systematically symmetries of groups in representation theory and symmetries of moduli spaces in algebraic geometry; moreover, a local version of P=W will be explored concerning several conjectures relating algebraic geometry of singularities to knot invariants in topology; (2) to study Hodge theory of Lagrangian fibrations. This will connect the general theory of perverse sheaves and Hodge modules to concrete and interesting examples of integrable systems and symplectic varieties; (3) to study perverse sheaves in enumerative geometry. This concerns relating Gromov-Witten and Donaldson-Thomas invariants to the more mysterious work of Gopakumar and Vafa. This direction will provide new perspectives in understanding the connections between algebraic geometry and quantum physics. To achieve these goals, the investigator together with his collaborators and students, will develop a set of tools including support theorems associated with the decomposition theorem, vanishing cycles techniques, localization methods, techniques in algebraic geometry of positive characteristics, and symmetries in hyper-Kähler geometries.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.

本研究项目的重点是代数几何和模空间。代数几何是对变量的研究，而变量又是多项式方程的解的集合。模空间是变量的参数空间，它关注变量在定义多项式变化时的行为。在过去的几十年里，已经发现了代数几何中的模空间与数学物理中的表示理论、拓扑和量子场论等其他领域之间的基本联系。本项目旨在研究几类模空间，它们位于数学和物理中心领域的十字路口。研究人员将开发有关这些品种的新工具，解决长期存在的问题，并探索新的联系。这些项目将增加枚举几何、拓扑、霍奇理论和数学物理团体之间的交流。新的发展将产生更多的活动，并为对这些领域感兴趣的研究生和博士后提供问题。研究生将获得该奖项的支持。研究者的研究将围绕三个项目展开：(1)研究一般约化群的P=W现象和拓扑镜像对称性。这将更系统地连接表示理论中群的对称性和代数几何中模空间的对称性；此外，我们将探讨P=W的一个局部版本，它涉及到一些关于拓扑中奇点的代数几何与结不变量的猜想；(2)研究霍奇拉格朗日颤振理论。这将把反常束和Hodge模块的一般理论与可积系统和辛变的具体而有趣的例子联系起来；(3)研究列举几何中的逆捆。这涉及到将Gromov-Witten和Donaldson-Thomas不变量与Gopakumar和Vafa更神秘的工作联系起来。这个方向将为理解代数几何和量子物理之间的联系提供新的视角。为了实现这些目标，研究者将与他的合作者和学生一起开发一套工具，包括与分解定理相关的支持定理、消失循环技术、局部化方法、代数几何中的正特征技术和hyper-Kähler几何中的对称性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Perverse filtrations, Chern filtrations, and refined BPS invariants for local P2

• DOI: 10.1016/j.aim.2023.109294
• 发表时间: 2023-11
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Y. Kononov;Weite Pi;Junliang Shen