Algebraic Structures in Topology

DMS-0104004 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建议或激发的拓扑学中的新的代数结构。更具体地说，该项目的目的是在n维球空间的同调上发现一种新的代数结构，我们所说的n球空间是指从n维球到给定流形的连续映射的空间。项目的这一部分，与Dennis Sullivan合作，推广了Chas和Sullivan在n=1的情况下的工作，即通常的自由环空间的工作。另一个目的是在Chas-Sullivan的工作和Gromov-Witten不变量之间建立联系，我们认为Gromov-Witten不变量是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，查斯-沙利文的工作《弦拓扑学》可以被视为物理结构的拓扑版。该项目的这一部分建议通过发展无限维流形中半无限圈的融合交集理论来完成。最后，将上述结果与Kontsevich猜想联系起来，推广了Deligne的猜想，揭示了抽象n-代数的形变理论与(n+1)维欧氏空间中点的位形空间拓扑之间的深层联系.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联系起来来识别几何对象的拓扑性质.有时几何学太复杂，不能立即理解和处理对象，而代数信息本质上通常更简单。这个项目为球空间提出了一些新的代数结构，球空间是从n维球到流形的映射的空间。这类空间是相当复杂的，而且Chen、西格尔、Jones、Getzler、Burghelea、Fedorowicz、Goodwillie等人的经典工作不仅产生了环空间的同调的计算，这是n=1的球空间的特例，而且还揭示了与代数(HochschildComplex)的惊人联系。此外，弦理论的最新进展强调了与从2-球面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-球面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。

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.