Quantized Lagrangian submanifolds of moduli spaces and representation theory
模空间的量化拉格朗日子流形和表示理论
基本信息
- 批准号:2302624
- 负责人:
- 金额:$ 28万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard 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:
- 关键词:
项目摘要
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.
尽管簇代数理论的发展相对较新,但它已被证明是现代数学和物理的广泛领域中的一种强大而通用的工具,并在这些学科之间建立了桥梁。本项目致力于从簇代数的角度探索表象理论、量子拓扑和计数几何的新结构。在这些领域的许多问题中,识别潜在的团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显性、建设性的证明。这项研究计划将与早期职业研究人员密切合作,计划通过组织针对邻近研究领域的研究生和博士后的迷你学校来传播该项目的必要背景思想和前沿成果。更具体地说,该项目专注于曲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与曲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与三维流形的几何密切相关。PI将系统地研究这一量子化问题,根据它们与表示理论的联系,在局部系统的下层模空间上开发新的结构。有待探索的新方向包括:在Teichmueller空间上构造可积系统,提供经典Fichel-Nielsen哈密顿的更高Teichmueller理论类比,以及理解具有非普通单列数据的局部系统的模空间的簇结构的行为,这与双仿射Hecke代数及其高亏格类似物密切相关。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。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Gus Schrader其他文献
A cluster realization of $$U_q(\mathfrak {sl}_{\mathfrak {n}})$$ from quantum character varieties
- DOI:
10.1007/s00222-019-00857-6 - 发表时间:
2019-01-19 - 期刊:
- 影响因子:3.600
- 作者:
Gus Schrader;Alexander Shapiro - 通讯作者:
Alexander Shapiro
Gus Schrader的其他文献
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{{ truncateString('Gus Schrader', 18)}}的其他基金
Conference: A Meeting on Poisson Geometry
会议:泊松几何会议
- 批准号:
2410632 - 财政年份:2024
- 资助金额:
$ 28万 - 项目类别:
Standard Grant
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