Quantum Invariants, Enhanced Moduli, and Integrable Systems

该奖项支持代数几何领域的研究。该研究为以适合在广泛应用中实际使用的方式理解基本问题的基本结构奠定了基础。这项工作的结果旨在与可积系统理论、弦理论、规范理论和拓扑绝缘体研究中的问题直接相关。 此外，该研究项目旨在对场论和量化进行具体应用。 该项目还旨在建立一个新的几何技术库，适用于代数循环、辛拓扑和高能物理理论。这将通过为数学和物理学的研究生和博士后提供几何和弦理论接口的研究机会，以及开发关于形式局部化和移位量子化以及卡拉比-丘可积系统和高级唐纳森-托马斯理论的课程来实现。该项目整合了派生几何和量子场论的思想，以揭示模问题隐藏的复杂性，提取新的簇枚举不变量，并研究完全可积系统。这些问题的解决将巩固和揭开几何、辛拓扑和场论中现有的几种量化方案的神秘面纱。将研究三个方向。第一个目标是描述那些承认实现作为势的关键轨迹的模空间。表征需要发展派生几何中的移辛和泊松结构的形式主义以及各向同性叶理理论。在第二个项目中，将研究一种新方法，通过构建显式拉格朗日叶状结构并计算相关的势函数来构建非阿贝尔上同调的动机方向数据。该项目旨在构造更高的 Chern-Simons 泛函，并为从高维量子场论中提取枚举不变量奠定基础。最终项目寻找卡拉比-丘可积系统，以实现 ADE 结构群的驯化或野生亚形希钦纤维。

This award supports research in the field of algebraic geometry. The research sets the stage for understanding the basic structure of fundamental problems in a way suitable for pragmatic use in a broad spectrum of applications. The results of the work are intended to be immediately relevant to questions in the theory of integrable systems, string theory, gauge theory, and the study of topological insulators. In addition, the research project aims to have concrete applications to field theory and quantization. The project also aims to build a new arsenal of geometric techniques applicable to the theory of algebraic cycles, symplectic topology, and high energy physics. This will be achieved by providing research opportunities on the interface of geometry and string theory for graduate students and postdoctoral associates in mathematics and physics, and by development of courses on formal localization and shifted quantization and on Calabi-Yau integrable systems and higher Donaldson-Thomas theory. The project integrates ideas from derived geometry and quantum field theory to unravel the hidden complexity of moduli problems, to extract new enumerative invariants of varieties, and to study completely integrable systems. The resolution of these questions will consolidate and demystify several existing quantization schemes in geometry, symplectic topology, and field theory. Three directions will be studied. The first aims to characterize those moduli spaces that admit a realization as the critical locus of a potential. The characterization requires the development of the formalism of shifted symplectic and Poisson structures in derived geometry and the theory of isotropic foliations. In the second project a new method will be investigated for constructing motivic orientation data on non-abelian cohomology by building explicit Lagrangian foliations and computing the associated potential functions. The project aims to construct higher Chern-Simons functionals, and sets the stage for extracting enumerative invariants from higher dimensional quantum field theory. The final project searches for Calabi-Yau integrable systems that realize the tame or wild meromorphic Hitchin fibrations for ADE structure groups.