Algebraic Structures in Topology and Geometry

该项目的目标是通过代数数据来理解几何空间，并发展代数理论，从而有效地分析几何对象的定量和定性特性。主要的工具来自代数拓扑，这是一个通用的框架，它允许人们用代数结构（如向量空间）的等价问题来重新表述关于拓扑和几何的问题。拓扑空间将被编码为具有特定运算的代数结构。PI还将使用类似的代数框架来研究弦拓扑，这是一种关于几何空间中弦和环相互作用的理论。利用代数结构研究拓扑空间和几何空间在数学和数学物理中都具有重要的基础意义。该项目旨在从数学上解释拓扑学、几何和代数领域在哪些意义上是等价的，在哪些意义上是不同的。从研究这些领域之间的相互作用中产生的计算工具和不变量在量子场论、弦理论和物理中的镜像对称的数学公式中是有用的。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样化的研究小组，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。在项目的第一部分，PI将通过从Koszul对偶理论的透镜来看的e -∞协代数的代数概念来表征同伦类型。这一观点源于PI和M. Zeinalian的一个新观察：空间上奇异链的e -∞协代数结构决定了完全一般的基群，并且这些数据保存在应用cobar函子后变成拟同构的映射下。一旦通过这个框架理解了同伦类型，所得到的代数结构将通过在链水平上描述poincar<s:1>对偶性的额外操作得到增强，以便表征同伦类型中的拓扑流形结构。项目的第二部分是关于流形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，特别是与Goresky-Hingston余积相关的运算，能够在非单连通环境中检测到超越同伦类型的精细几何信息。弦拓扑操作将使用Hochschild同调和Tate上同调的框架进行分析，这是PI和Z. Wang在之前的工作中开发的。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，以及流形出现的新不变量。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.