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

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.

该项目的目标是从代数数据的角度理解几何空间，并发展代数理论，以有效分析几何对象的定量和定性特性。主要工具来自代数拓扑，这是一种允许人们重新表述问题的通用框架。关于拓扑和几何的相关代数结构（例如向量空间）的等效问题将根据具有特定运算的代数结构进行编码。拓扑，一种涉及几何空间中弦和环相互作用的理论，通过代数结构研究拓扑和几何空间在数学和数学物理中具有根本重要性。其中拓扑、几何和代数的领域是等价的，而它们在不同的意义上是不同的。通过研究这些领域之间的相互作用而产生的计算工具和不变量在量子场论、弦理论、和镜像对称该奖项为研究生参与部分研究提供资金，PI 将建立一个包容性和多元化的研究小组，并将促进针对目前在数学研究中代表性不足的群体的倡议。项目中，PI 将通过从 Koszul 对偶理论的角度来看的 E-无穷代数的代数概念来表征同伦类型，这一观点是由 PI 和 M. Zeinalian 的新观察所激发的：E-无穷大。空间上奇异链的余代数结构决定了完全一般性的基本群，并且该数据被保存在应用 cobar 函子后成为准同构的映射中，一旦通过该框架理解了同伦类型，所得到的代数结构将得到增强。在链级上描述庞加莱对偶性的额外操作，以便表征同伦类型的拓扑流形结构。该项目的第二部分涉及有关弦拓扑的基础和计算问题。弦拓扑中出现的一些代数结构，特别是与 Goresky-Hingston 余积相关的运算，能够检测超出非单连通弦拓扑运算中的同伦类型的精细几何信息。使用 PI 和 Z. Wang 之前的工作中开发的 Hochschild 同调和 Tate 上同调框架进行分析。 PI 旨在理解弦的完整代数结构。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

The simplicial coalgebra of chains determines homotopy types rationally and one prime at a time

链的单纯余代数有理地确定同伦类型并一次确定一个素数

• DOI: 10.1090/tran/8579
• 发表时间: 2022
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Rivera, Manuel;Wierstra, Felix;Zeinalian, Mahmoud
• 通讯作者: Zeinalian, Mahmoud

Categorical models for path spaces

• DOI: 10.1016/j.aim.2023.108898
• 发表时间: 2022-01
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Emilio Minichiello;M. Rivera;M. Zeinalian

Adams' cobar construction revisited

重新审视亚当斯的科巴结构

• DOI: 10.4171/dm/895
• 发表时间: 2022
• 期刊: Documenta Mathematica
• 影响因子: 0.9
• 作者: Rivera, Manuel