Quantized Lagrangian submanifolds of moduli spaces and representation theory

尽管它的发展相对较晚，但聚类代数理论已被证明是一种强大而通用的工具，可以跨越现代数学和物理学的广泛领域，并有助于在这些学科之间建立桥梁。本项目主要从集群代数的角度探索表征理论、量子拓扑和枚举几何的新结构。在这些领域的许多问题中，识别潜在的簇结构揭示了隐藏的组合结构和对称性，从而导致对深层结果的明确的、建设性的证明。该研究项目将密切涉及早期职业研究人员，并计划通过组织针对邻近研究领域的研究生和博士后的小型学校，传播项目的必要背景思想和前沿成果。更具体地说，本项目重点研究了曲面上局部系统模空间的量子几何，以及这些辛模空间的拉格朗日子流形的量子化问题。构造这样的量子化相当于在与表面相关的希尔伯特空间中产生一个正则向量，并且根据拓扑量子场论的哲学，这些量子化的拉格朗日量与三流形的几何结构密切相关。PI将系统地研究这个量化问题，在局部系统的基础模空间上基于它们与表示理论的联系开发新的结构。新的研究方向包括在Teichmueller空间上构造与经典fenchell - nielsen hamilton量具有更高的Teichmueller理论类似的可积系统，以及在点处具有非一般单数据的局部系统的簇结构的行为，这与双仿射Hecke代数及其高属类似的理论密切相关。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。

Despite its relatively recent development, the theory of cluster algebras has proven to be a powerful and versatile tool across a broad range of areas of modern mathematics and physics, and has been instrumental in building bridges between these disciplines. This project focuses on exploring new structures in representation theory, quantum topology and enumerative geometry from the cluster-algebraic perspective. In many problems in these areas, identifying an underlying cluster structure reveals hidden combinatorial structures and symmetries, thereby leading to explicit, constructive proofs of deep results. This research program will closely involve early career researchers, with plans to disseminate both the necessary background ideas and cutting edge results from the project through the organization of mini-schools aimed at graduate students and postdocs in adjacent areas of research.More specifically, this project focuses on the quantum geometry of moduli spaces of local systems on surfaces, and the problem of quantizing Lagrangian submanifolds of these symplectic moduli spaces. Constructing such a quantization amounts to producing a canonical vector in the Hilbert space associated to the surface, and in accordance with the philosophy of topological quantum field theory, these quantized Lagrangians are closely related to the geometry of three-manifolds. The PI will systematically study this quantization problem, developing along the way new structures on the underlying moduli spaces of local systems based on their connection with representation theory. New directions to be explored include the construction of integrable systems providing higher Teichmueller-theoretic analogs of the classical Fenchel-Nielsen Hamiltonians on Teichmueller spaces, as well understanding the behavior of the cluster structure for moduli spaces of local systems with non-generic monodromy data at punctures, which is intimately connected with the theory of double affine Hecke algebras and their higher genus analogs.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.