String Topology and the Algebraic Topology of Moduli Spaces

这个建议包括几个项目，使用代数拓扑技术来研究几何问题所产生的弦拓扑，和拓扑的模空间。 弦拓扑理论由查斯和沙利文在1999年首次提出，现在涉及到流形的路径和回路空间上的大量丰富结构，以及从曲面到流形的映射。 在这份提案中，科恩将强调 应用 弦拓扑的研究，以及模空间同伦型研究的最新突破。这些应用将是两个几何（了解格罗莫夫-威滕理论的余切丛，和cobordism型模空间的全纯曲线），并在代数拓扑（K -理论的循环空间的分类空间，扭曲等变K -理论，和瓦尔德豪森的代数K -理论的空间）。 科恩与I.马德森，也将研究同伦型的模空间映射的表面，延长马德森和韦斯最近的工作广义芒福德猜想。 他们提出了一个长期的项目，利用这一知识，以及适应弦拓扑方法，了解bordism类所代表的紧凑模空间的稳定全纯曲线在辛流形。 这个建议包括几个项目，调查新的研究领域，称为“弦拓扑”，以及相关的问题。 弦拓扑学是Chas和Sullivan在1999年首次提出的一个理论，研究路径、环和曲面空间上的结构。 这种结构是由物理学中的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）如何随时间演变。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曲面来测量的，这些曲面跨越这些环。 在这个项目中，科恩倾向于研究这些空间的循环和表面的相交性质，特别是拓扑空间的表面映射到一个背景空间。 他的目标是 将这些构造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.