Moduli Spaces in Representation Theory and Symplectic Algebraic Geometry

对各种问题的数学研究，从粒子动力学到空间对称性，再到量子物理，都依赖于一种特殊类型的几何形式，即所谓的辛几何。在所有这些不同的问题中出现的辛几何的结构具有某些共同的特征，这些特征将在本项目中被进一步分离、探索和分类。然后，该项目将开发通用技术，以利用这些共同特征来计算辛几何中的基本量。由此产生的技术和计算将被应用于解决几何学、对称性的数学研究和数学物理中的一些突出问题。辛代数簇为几何表示理论中的基本构造提供了自然的几何背景。这些变化在超对称量子场论的研究中自然出现，如3DN=4理论中的真空模空间。这两种现象是深度相关的，因为量子场论为几何表示理论提供了洞察力，而几何表示理论通过回答物理中的重要问题来回报。该项目在一个共同的框架中定位了不同的辛代数变体-作为非对易泊松几何中的模空间-为它们的探索开发了通用工具，并产生了在表象理论、拓扑学、几何学和超对称量子场论中的应用。特别是，该项目将开发和应用新的方法来寻找与辛代数变体相关的拓扑和范畴不变量的生成器。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。

Mathematical studies of diverse problems, from the dynamics of particles, to spatial symmetries, to quantum physics, rely on the formulation of a particular type of geometry called symplectic geometry. The structures of symplectic geometry that arise in all these different problems possess certain common features that will be further isolated, explored, and catalogued in this project. The project will then develop general techniques to exploit those common features to calculate fundamental quantities in symplectic geometry. The resulting techniques and calculations will be applied to solve a number of outstanding problems in geometry, the mathematical study of symmetry, and mathematical physics. Symplectic algebraic varieties provide the natural geometric setting for basic constructions in geometric representation theory. Such varieties appear naturally in the study of supersymmetric quantum field theories, as moduli spaces of vacua in 3D N=4 theories. The two appearances are deeply inter-related, as quantum field theory provides insight to geometric representation theory, which representation theory repays by answering questions of importance in physics. The project locates diverse symplectic algebraic varieties in a common framework---as moduli spaces in noncommutative Poisson geometry---develops general tools for their exploration, and yields applications to representation theory, topology, geometry, and supersymmetric quantum field theories. In particular, the project will develop and apply new methods to find generators for topological and categorical invariants associated to symplectic algebraic varieties.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.