Algebraic Geometry of Hitchin Integrable Systems and Beyond

这项研究主要集中在代数几何和模空间。代数几何研究的是变元，而变元又是多项式方程的解的集合。模空间是簇的参数空间，它关注簇随着定义多项式的变化而变化的行为。在过去的几十年里，人们发现了代数几何中的模空间与数学物理中的表示论、拓扑学和量子场论等其他领域之间的基本联系。这个项目旨在研究位于数学和物理中心领域的十字路口的几类模空间。研究人员将开发与这些品种有关的新工具，解决长期存在的问题，并探索新的联系。这些项目将加强列举几何、拓扑学、霍奇理论和数学物理社区之间的交流。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出问题。研究生将获得该奖项的资助。研究人员将围绕三个项目展开研究：(1)研究一般约化群的P=W现象和拓扑镜对称。这将更系统地连接表示论中的群的对称性和代数几何中的模空间的对称性；此外，还将探索P=W的局部版本，涉及几个将奇点的代数几何与拓扑学中的纽结不变量联系起来的猜想；(2)研究拉格朗日纤维的Hodge理论。本课程将把一般的倒叶理论和Hodge模与可积系和辛簇的具体而有趣的例子联系起来；(3)学习计数几何中的倒叶。这关系到格罗莫夫-维腾和唐纳森-托马斯不变量与更神秘的戈帕库马尔和瓦法的工作之间的关系。这一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，研究人员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、消失圈技术、局部化方法、正特征代数几何技术以及超Kähler几何中的对称性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.