Algebraic Structures in Topology

拓扑中的代数结构

• 批准号: 0227974
• 负责人: Alexander Voronov
• 金额: $ 3.13万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-03-28 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0227974&HistoricalAwards=false
• 关键词: Algebraic; Structures; Topology

DMS-0104004Alexander A. VoronovThe goal of the project is to discover and study new algebraicstructures in topology suggested or motivated by mathematical physics, in particular, quantum field theory and string theory. More specifically, the project aims at discovering a new algebraicstructure on the homology of an n-sphere space, by which we mean the space of continuous maps from the n-dimensional sphere to a given manifold. This part of the project, joint with Dennis Sullivan, generalizes the work pioneered by Chas and Sullivan in the case n=1, i.e., that of a usual free loop space. Another goal is to establish connection between Chas-Sullivan's work and Gromov-Witten invariants, which we believe to be a holomorphic version of Chas-Sullivan's algebraic structure. Gromov-Witten invariants come from sigma model of quantum field theory, and Chas-Sullivan's work "String Topology" may be regarded as a topological version of the physical construction. This part of the project is suggested to be completed by developing a fusion intersection theory of semi-infinite cycles in infinite dimensional manifolds. Finally, part of the project isdedicated to relating the above to Kontsevich's Conjecture, which generalizes Deligne's Conjecture and unravels a deep relation between deformation theory of abstract n-algebras and the topology of configuration spaces of points in an (n+1)-dimensional Euclidean space.The main idea of Algebraic Topology is to be able to recognizetopological properties of a geometric object by associating algebraic data or structure to the geometric object. Sometimes the geometry is too complicated to allow immediate understanding and work with the object, while the algebraic information is usually simpler by its nature. This project suggests some new algebraic structure for a sphere space, the space of maps from an n-dimensional sphere to a manifold. Such spaces are quite complicated and the classical work of Chen, Segal, Jones, Getzler, Burghelea, Fedorowicz, Goodwillie, and others, produced not only the computation of the homology of loop spaces, which are the particular case of sphere spaces for n=1, but also revealed amazing connections with algebra (Hochschildcomplex). Also, recent progress in string theory emphasized the importance of invariants associated to holomorphic maps from the 2-sphere to a manifold (Gromov-Witten invariants). In this project we undertake an analogous study of continuous maps from the n-sphere to a manifold, which for n=1 has already enabled significant progress in topology.

alexander A. voronov该项目的目标是发现和研究由数学物理，特别是量子场论和弦理论提出或激发的拓扑中的新代数结构。更具体地说，该项目旨在发现n球空间的同调上的一个新的代数结构，我们指的是从n维球到给定流形的连续映射空间。这部分项目与Dennis Sullivan共同推广了Chas和Sullivan在n=1的情况下（即通常的自由循环空间）所做的工作。另一个目标是建立查斯-沙利文的工作和Gromov-Witten不变量之间的联系，我们认为Gromov-Witten不变量是查斯-沙利文代数结构的全纯版本。Gromov-Witten不变量来源于量子场论的sigma模型，而charles - sullivan的“弦拓扑”可以看作是物理结构的拓扑版本。这部分项目建议通过发展无限维流形中半无限循环的融合交集理论来完成。最后，该项目的一部分致力于将上述内容与Kontsevich猜想联系起来，该猜想推广了Deligne猜想，并揭示了抽象n-代数的变形理论与（n+1）维欧几里德空间中点的构型空间拓扑之间的深刻关系。代数拓扑学的主要思想是通过将代数数据或结构与几何对象相关联来识别几何对象的拓扑性质。有时几何图形太复杂，无法立即理解和处理对象，而代数信息通常本质上更简单。这个项目提出了一个球空间的一些新的代数结构，即从n维球到流形的映射空间。这样的空间是相当复杂的，Chen、Segal、Jones、Getzler、Burghelea、Fedorowicz、Goodwillie等人的经典工作不仅计算了环空间的同构，这是n=1时球空间的特殊情况，而且揭示了与代数（Hochschildcomplex）的惊人联系。此外，弦理论的最新进展强调了与从2球到流形的全纯映射相关的不变量（Gromov-Witten不变量）的重要性。在这个项目中，我们进行了从n球到流形的连续映射的类似研究，对于n=1，这已经使拓扑学取得了重大进展。