Homotopy Theory and Moduli Spaces

本文旨在研究各种模空间的同伦理论性质。这些是具有几何结构的流形的模空间，例如复杂结构。复维1的情况给出了黎曼曲面的模空间，到目前为止，这是相对较好的理解。相比之下，人们对高维复流形的模空间知之甚少。本课题将利用同伦理论的方法来研究这个空间和相关空间。本课题的另一个目标是研究具有奇点的流形的模空间。这里，最基本的情况是黎曼模空间的Deligne-Mumford紧化。在许多应用中，这个空间比非紧化模空间更重要，同伦理论的理解应该是有成果的。特别地，提议者将研究Gromov-Witten理论的应用。流形是空间的一般概念。流形是几何学、拓扑学、分析学和其他数学领域的基础。在物理学中，它们是爱因斯坦广义相对论中潜在的几何结构，在力学、量子场论以及许多其他领域发挥着重要作用。从数学的角度研究流形有两种方式：你可以研究每个流形的几何性质，一次一个，或者你可以一次研究所有流形集合的几何性质。“所有流形的集合”本身可以被认为是一种空间，每个单独的流形都是这个空间中的一个点。这个空间是模空间的一个例子，目前的项目旨在了解这个模空间及其变化的几何和拓扑性质。

The proposer aims to study homotopy theoretical properties of various moduli spaces. These are moduli spaces of manifolds with a geometric structure, for example a complex structure. The case of complex dimension one gives the moduli space of Riemann surfaces, which by now is relatively well understood. In contrast, much less is understood about moduli spaces of higher dimensional complex manifolds. This is project will investigate what can be said about this and related spaces using methods of homotopy theory. Another goal of this project is to study moduli spaces of manifolds with singularities. Here, the most basic case is the Deligne-Mumford compactification of Riemann's moduli space. For many application this space is more important than the uncompactified moduli space, and a homotopy theoretic understanding should be fruitful. In particular, the proposer will study applications to Gromov-Witten theory.A manifold is a general notion of space. Manifolds are fundamental in geometry, topology, analysis, and other areas of mathematics. In physics, they are the underlying geometric structure in Einstein's general theory of relativity, and play fundamental roles in mechanics, quantum field theory, and probably many other areas. Studying manifolds from a mathematical perspective can be done in two ways: You can study the geometric properties each manifold, one at a time, or you can study geometric properties of the collection of all manifolds at once. The "collection of all manifolds" can itself be thought of as a kind of space, and each individual manifold is a point in this space. This space is an example of a moduli space, and the current project aims at understanding geometric and topological properties of this moduli space and its variations.