Quantized Lagrangian submanifolds of moduli spaces and representation theory

尽管其相对较新的发展，集群代数的理论已被证明是一个强大的和通用的工具，在广泛的现代数学和物理领域，并已在这些学科之间的桥梁建设工具。这个项目的重点是探索新的结构表示论，量子拓扑和枚举几何从集群代数的角度来看。在这些领域的许多问题中，识别潜在的集群结构揭示了隐藏的组合结构和对称性，从而导致深层结果的明确的，建设性的证明。该研究计划将密切参与早期职业研究人员，计划通过组织针对相邻研究领域的研究生和博士后的小型学校来传播必要的背景思想和前沿成果。更具体地说，该项目侧重于表面上局部系统的模空间的量子几何，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希尔伯特空间中产生一个与曲面相关的正则向量，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维流形的几何密切相关。PI将系统地研究这个量子化问题，沿着新结构的基础上的模空间的本地系统的基础上，他们与代表性理论的连接。待探索的新方向包括可积系统的构建，提供Teichmueller空间上经典Fenchel-Nielsen Hamilton的更高Teichmueller理论类似物，以及理解局部系统的模空间的簇结构的行为，在穿孔处具有非通用单值数据，该奖项反映了NSF的法定使命，并被认为是值得的。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，

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.