String Topology and the Algebraic Topology of Moduli Spaces

本提案由几个使用代数拓扑技术来研究弦拓扑和模空间拓扑中出现的几何问题的项目组成。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路空间上的大量丰富结构，以及从曲面到流形的映射。在这个提议中，Cohen将强调弦拓扑的应用，以及模空间同伦类型研究的最新突破。这些应用将在几何（理解Gromov-Witten的共切束理论，以及全纯曲线模空间的共模类型）和代数拓扑（分类空间的环空间的K理论，扭曲等变K理论和Waldhausen的代数空间K理论）中得到应用。Cohen与i.m adsen合作，还将研究曲面映射的模空间的同伦类型，扩展Madsen和Weiss最近关于广义芒福德猜想的工作。他们提出了一个更长期的项目，利用这些知识，以及对弦拓扑方法的适应，来理解由辛流形中稳定全纯曲线的紧模空间所表示的边界类。该提案由几个项目组成，研究被称为“弦拓扑”的新研究领域，以及相关问题。弦拓扑是由Chas和Sullivan于1999年首次提出的理论，研究路径、环路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。这个想法是为了理解背景空间中的循环（或路径）是如何随时间演变的。循环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表面来测量的，这些表面跨越了这些循环。在这个项目中，Cohen倾向于研究这些环与曲面空间的相交特性，特别是曲面空间映射到背景空间的拓扑结构。他的目标是将这些结构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.