Quantum Invariants, Enhanced Moduli, and Integrable Systems

该奖项支持代数几何领域的研究。该研究为理解基本问题的基本结构奠定了基础，适合在广泛的应用中实用。这项工作的结果是为了立即相关的问题，在理论的可积系统，弦理论，规范理论和研究的拓扑绝缘体。 此外，该研究项目的目的是有具体的应用领域的理论和量化。 该项目还旨在建立一个新的武器库的几何技术适用于理论的代数循环，辛拓扑结构和高能物理。这将通过为数学和物理学的研究生和博士后研究生提供几何学和弦理论接口的研究机会，以及通过开发关于形式局部化和移位量子化以及卡-丘可积系统和更高的唐纳森-托马斯理论的课程来实现。该项目整合了衍生几何学和量子场论的思想，以揭示模问题的隐藏复杂性，提取新的变量的枚举不变量，并研究完全可积系统。这些问题的解决将巩固和揭开几个现有的量子化方案在几何，辛拓扑，场论。将研究三个方向。第一个目的是表征这些模空间，承认实现的关键轨迹的潜力。的表征需要的形式主义的发展转移辛和泊松结构在派生几何和各向同性叶理的理论。在第二个项目中，将研究一种新的方法，通过建立显式拉格朗日叶理和计算相关的势函数来构建非阿贝尔上同调的motivic方向数据。该项目旨在构建更高的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。最后一个项目是寻找能够实现ADE结构群的驯服或野生亚纯希钦纤维化的Calabi-Yau可积系统。

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.