Algebraic Structures in Topology

DMS-0104004Alexander A.该项目的目标是发现和研究数学物理学，特别是量子场论和弦理论提出或激发的拓扑学中的新代数结构。更具体地说，该项目的目的是发现一个新的algebraicstructure上的同调的n-sphere空间，我们的意思是空间的连续映射从n维球面到一个给定的流形。项目的这一部分，与Dennis Sullivan联合，概括了Chas和Sullivan在n=1的情况下开创的工作，即，一个普通的自由循环空间。另一个目标是建立Chas-Sullivan的工作和Gromov-Witten不变量之间的联系，我们认为这是Chas-Sullivan代数结构的全纯版本。Gromov-Witten不变量来自量子场论的sigma模型，Chas-Sullivan的工作“弦拓扑”可以被看作是物理结构的拓扑版本。这部分的项目建议通过发展一个融合相交理论的半无限循环在无限维流形。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，推广了Deligne猜想，揭示了抽象n-代数的形变理论与（n+1）-代数拓扑的主要思想是能够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属性。几何物体。有时，几何形状太复杂，无法立即理解和处理对象，而代数信息通常本质上更简单。这个项目提出了一些新的代数结构的球面空间，空间的地图从一个n维球面的流形。这样的空间是相当复杂的和经典的工作陈，西格尔，琼斯，Getzler，Burghelea，Fedorowicz，Goodwillie，和其他人，产生不仅计算的同源性循环空间，这是特殊情况下的领域空间n=1，但也揭示了惊人的连接与代数（Hochschildcomplex）。此外，弦理论的最新进展强调了从2-球面到流形的全纯映射的不变量（Gromov-Witten不变量）的重要性。在这个项目中，我们对从n-球面到流形的连续映射进行了类似的研究，对于n=1，这已经使拓扑学取得了重大进展。

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.