Enhanced moduli, Hodge theory, and quantization

这是对代数几何领域的一项研究。该项目融合了几何学和量子场论的技术，以揭开代数奇点隐藏的复杂性，提取新的变种计数不变量，并在表示理论中构建新的对偶。将研究五个问题。第一个目的是构造模空间和辛奇点的新的代数不变量。这些不变量的建立和计算需要非对易几何中移位辛结构形式的理论发展。在第二个项目中，提出了一种通过移位量化Calabi-Yau型微分分次范畴中对象的模来构造Motivic计数不变量的新方法。第三个项目分析了非对易Hodge理论控制Landau-Ginzburg模型形变的方式，并提出了一个非常一般的无阻定理。第四个项目给出了一种策略，将经典的极限朗兰兹对偶重新表述和证明为倒立轮的纯拓扑对偶。最后一个项目将建立分支和扭曲的非阿贝尔Hodge对应的函数性，这些问题的解决将巩固和揭开几何学、辛拓扑和场论中现有的几个量子化方案的神秘面纱。该项目为理解代数变种的基本结构奠定了基础，以适合在广泛的应用中实用的方式。除了对代数拓扑学的自然应用之外，所提出的工作将直接与范畴理论、几何表示理论、可积系统理论、弦理论、规范理论、量子重力和宇宙学中的深层问题相关。该项目概述了镜像对称性物理、重整化群流和规范理论量子化的具体跨学科应用。该项目还旨在组织一项集中努力，以增强和建立适用于代数循环理论、辛拓扑和高能物理的新几何库。这将通过在数学和物理方面培训一批年轻的研究人员和研究生，并通过开发一门关于派生辛几何的课程和一门关于非对易Hodge理论的课程来实现。讨论了研究生和博士后在几何和弦理论的界面上的具体研究机会。拟议的工作将通过在多学科会议上的演讲、研究研讨会和同行评议的科学期刊上的出版物来传播。

This is a research in the field of algebraic geometry. The project fuses techniques from geometry and quantum field theory to unravel the hidden complexity of algebraic singularities, to extract new enumerative invariants of varieties, and to construct novel dualities in representation theory. Five problems will be studied. The first one aims to construct new algebraic invariants of moduli spaces and of symplectic singularities. The building and computation of these invariants requires a theoretical development of the formalism of shifted symplectic structures in non-commutative geometry. In the second project a new method is proposed for constructing motivic enumerative invariants by shifted quantization of the moduli of objects in differential graded categories of Calabi-Yau type. The third project analyzes the way in which non-commutative Hodge theory controls the deformations of Landau-Ginzburg models and proposes a very general unobstructedness theorem. The fourth project gives a strategy for reformulating and proving the classical limit Langlands duality as a purely topological duality for perverse sheaves. The last project will establish the functoriality of the ramified and the twisted non-abelian Hodge correspondences.The resolution of these questions will consolidate and demystify several existing quantization schemes in geometry, symplectic topology, and field theory. The project sets the stage for understanding the basic structure of algebraic varieties in a way suitable for pragmatic use in a broad spectrum of applications. Aside from the natural applications to algebraic topology, the work proposed will be immediately relevant to deep questions in category theory, geometric representation theory, the theory of integrable systems, string theory, gauge theory, quantum gravity and cosmology. The project outlines concrete cross-discipline applications to the physics of mirror symmetry, renormalization group flow, and the quantization of gauge theories. This project also aims to organize a concentrated effort on enhancing and building a new geometric arsenal of techniques applicable to the theory of algebraic cycles, symplectic topology, and high energy physics. This will be achieved by training a group of young researchers, and graduate students in mathematics and physics, and by a curriculum development of a course on derived symplectic geometry, and a course on non-commutative Hodge theory. Specific research opportunities on the interface of geometry and string theory for graduate students and postdocs are discussed. The proposed work will be disseminated through talks at multidisciplinary conferences, research seminars and publications in peer reviewed scientific journals.