Algebraic Structures in Topology and Geometry

拓扑和几何中的代数结构

• 批准号: 2105544
• 负责人: Manuel Rivera
• 金额: $ 24.39万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2105544&HistoricalAwards=false
• 关键词: Algebraic; Structures; Topology; Geometry; Algebraic; Structures; Topology; Geometry

The goal of this project is to understand geometric space in terms of algebraic data and to develop algebraic theories that unlock effective analyses of quantitative and qualitative properties of geometric objects. The main tools come from algebraic topology, a general framework that allows one to reformulate questions about topology and geometry in terms of equivalent questions about algebraic structures such as vector spaces. Topological spaces will be encoded in terms of algebraic structures with particular operations. The PI will also use a similar algebraic framework to study string topology, a theory concerned with interactions of strings and loops in a geometric space. The study of topological and geometric spaces by means of algebraic structures is of fundamental importance in mathematics as well as in mathematical physics. The project aims to explain mathematically the sense in which the fields of topology, geometry, and algebra are equivalent, and the sense in which they are different. The computational tools and invariants that arise from studying the interplay between these fields are useful in the mathematical formulation of quantum field theory, string theory, and mirror symmetry in physics. The award provides funds for graduate students to be involved in parts of this research. The PI will build an inclusive and diverse research group and will promote initiatives directed towards groups that are currently underrepresented in mathematics research.In the first part of the project, the PI will characterize homotopy types through the algebraic concept of an E-infinity coalgebra viewed from the lens of Koszul duality theory. This viewpoint is motivated by a new observation of the PI and M. Zeinalian: the E-infinity coalgebra structure of the singular chains on a space determines the fundamental group in complete generality and this data is preserved under maps which become quasi-isomorphisms after applying the cobar functor. Once homotopy types are understood through this framework, the resulting algebraic structure will be enhanced with extra operations describing Poincaré duality at the chain level in order to characterize topological manifold structures in a homotopy type. The second part of the project is concerned with both foundational and computational questions regarding the string topology of manifolds. Some of the algebraic structures that arise in string topology, in particular the operations related to the Goresky-Hingston coproduct, are able to detect fine geometric information that go beyond the homotopy type in the non-simply connected context. String topology operations will be analyzed using the framework of Hochschild homology and Tate cohomology, as developed in previous work of the PI and Z. Wang. The PI aims to understand the full algebraic structure of string topology, its dependence on the background geometric space, and the new invariants for manifolds that arise.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.

该项目的目的是根据代数数据了解几何空间，并开发代数理论，以解锁几何对象的定量和定性特性的有效分析。主要工具来自代数拓扑，这是一个通用框架，可以从有关代数结构（例如向量空间）的同等问题来重新重新对有关拓扑和几何形状的问题进行重新制定问题。拓扑空间将根据具有特定操作的代数结构进行编码。 PI还将使用类似的代数框架来研究字符串拓扑，该理论与几何空间中的字符串和循环相互作用有关。通过代数结构对拓扑和几何空间的研究在数学和数学物理学中至关重要。该项目的目的是在数学上解释拓扑，几何和代数的领域等效的意义，以及它们与众不同的意义。研究这些字段之间的相互作用而产生的计算工具和不变性在物理学中的量子场理论，弦理论和镜像对称性的数学公式中很有用。该奖项为研究生提供了资金，以参与本研究的一部分。 PI将建立一个包容性和潜水员的研究小组，并将促进针对目前数学研究中代表不足的小组的举措。在该项目的第一部分中，PI将通过代数概念通过从Koszul Duality理论观察到的E-Infinity Coalbra的代数概念来表征同型类型。这种观点是由对PI和M. Zeinalian的新观察的动机：空间上奇异链的e-Infinity山结构结构决定了完全常规的基本组，并且在应用Cobar函数后将其保存在地图下，这些数据被保存在地图下。一旦通过此框架理解了同质类型，将通过描述链条级别的庞加莱二元性的额外操作来增强所得代数结构，以表征同型同型类型中的拓扑歧管结构。该项目的第二部分涉及有关歧管的弦拓扑的基础和计算问题。弦拓扑中出现的一些代数结构，尤其是与戈尔斯基 - 惠海共同体相关的操作，能够检测出在非相互连接的上下文中超越同型类型的精细几何信息。弦拓扑操作将使用Hochschild同源性和TATE共同体的框架进行分析，这是PI和Z. Wang的先前工作中开发的。 PI旨在了解弦拓扑的完整代数结构，其对背景几何空间的依赖以及出现的新型流形的新不变性。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响审查标准来通过评估来诚实地支持。