String Topology and the Algebraic Topology of Moduli Spaces

这项建议由几个项目组成，使用代数拓扑技术来研究弦拓扑中出现的几何问题，以及模空间的拓扑。弦拓扑理论最早由Chas和Sullivan于1999年提出，现在涉及到流形的路径和环空间上的大量丰富结构，以及从曲面到流形的映射。在这项建议中，Cohen将强调弦拓扑的应用，以及最近在模空间的同伦型研究方面的突破。这些应用既适用于几何(理解余切丛的Gromov-Witten理论和全纯曲线的模空间的余边型)，也适用于代数拓扑(分类空间的循环空间的K-理论、扭曲等变K-理论和Waldhausen的代数K-理论)。Cohen和I.Madsen还将研究曲面映射的模空间的同伦型，推广了Madsen和Weiss最近关于广义Mumford猜想的工作。他们提出了一个更长期的项目，利用这一知识以及弦拓扑方法的改编，来理解由辛流形中稳定全纯曲线的紧致模空间表示的边界类。这项提议包括几个项目，调查被称为“弦拓扑学”的新研究领域，以及相关问题。弦拓扑是由查斯和沙利文于1999年首次提出的一种理论，它研究路径、环和曲面空间上的结构。这种结构是由物理学中弦理论的形式主义所激发的。这个想法是为了了解背景空间中的环路(或路径)如何随时间演变。环路可以通过改变大小甚至分裂来进化。这些现象是通过研究映射到背景空间的曲面来测量的，这些曲面横跨这些循环。在这个项目中，Cohen倾向于研究这些环和曲面空间的相交性质，特别是映射到背景空间的曲面空间的拓扑。他的目标是将这些构造法应用于各种几何问题。

This proposal consists of several projects using algebraic topological techniques to study geometric questions arising in String Topology, and the topology of moduli spaces. The theory of String Topology, first introduced by Chas and Sullivan in 1999, now involves a vast array of rich structure on spaces of paths and loops of manifolds, as well as maps from surfaces to manifolds. In this proposal, Cohen will emphasize applications of string topology, and the recent breakthroughs in the study of the homotopy type of moduli spaces. These applications will be both to geometry (understanding the Gromov-Witten theory of cotangent bundles, and the cobordism type of moduli spaces of holomorphic curves), and within algebraic topology (the K -theory of loop spaces of classifying spaces, twisted equivariant K -theory, and Waldhausen's algebraic K -theory of spaces). Cohen, in collaboration with I. Madsen, will also study the homotopy type of moduli spaces of maps of surfaces, extending Madsen and Weiss's recent work on the generalized Mumford conjecture. They propose a longer term project to use this knowledge, as well as an adaptation of string topology methods, to understand bordism classes represented by the compact moduli spaces of stable holomorphic curves in a symplectic manifold. This proposal consists of several projects investigating the new area of research known as "String Topology", as well as related questions. String topology, a theory that was first introduced by Chas and Sullivan in 1999, studies structures on spaces of paths, loops, and surfaces. This structure was motivated by formalisms in string theory in physics. The idea is to understand how loops (or paths) in a background space can evolve in time. Loops can evolve by changing in size and even breaking apart. These phenomena are measured by studying surfaces mapping to the background space, that span these loops. In this project, Cohen tends to study the intersection properties of these spaces of loops and surfaces, and in particular the topology of the space of surfaces mapping to a background space. His goal is to apply these constructions to a variety of geometric questions.