Algebraic Structures in Topology and Geometry

这个项目的目标是根据代数数据理解几何空间，并发展代数理论，从而开启对几何对象的定量和定性属性的有效分析。主要的工具来自代数拓扑学，这是一个通用的框架，允许人们根据关于代数结构(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。拓扑空间将按照具有特定运算的代数结构进行编码。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的相互作用的理论。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意义。该项目旨在从数学上解释拓扑学、几何学和代数学领域的等价性，以及它们不同的意义。通过研究这些场之间的相互作用而产生的计算工具和不变量在量子场论、弦理论和物理中的镜像对称性的数学公式中是有用的。该奖项为研究生参与这项研究的部分内容提供资金。PI将建立一个包容性和多样性的研究小组，并将促进针对目前在数学研究中未被充分代表的群体的倡议。在项目的第一部分，PI将通过从Koszul对偶理论的视角出发的E-无穷余代数的代数概念来刻画同伦类型。这一观点源于对PI和M.Zeinian的一个新的观察：空间上奇异链的E-无穷余代数结构决定了完全一般的基本群，并且这些数据被保存在映射下，这些映射在应用Cobar函子后变成准同构。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级别上描述Poincaré对偶的额外运算来增强，以便刻画同伦类型中的拓扑流形结构。该项目的第二部分涉及有关流形的弦拓扑的基础和计算问题。在弦拓扑中出现的一些代数结构，特别是与Goresky-Hingston余积有关的运算，能够在非单连通上下文中检测到超越同伦类型的精细几何信息。弦拓扑运算将使用Hochschild上同调和Tate上同调的框架来分析，如Pi和Z.Wang在之前的工作中发展的那样。PI旨在了解弦拓扑的完整代数结构，它对背景几何空间的依赖，以及出现的流形的新不变量。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.