Homotopy Theory and Moduli Spaces

同伦理论和模空间

• 批准号: 0805843
• 负责人: Soren Galatius
• 金额: $ 26.56万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805843&HistoricalAwards=false
• 关键词: Homotopy; Theory; Moduli; Spaces

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.

提议者旨在研究各种模量空间的同型理论特性。 这些是具有几何结构的歧管的模量空间，例如复杂的结构。 复杂尺寸的情况下给出了Riemann表面的模量空间，到目前为止，该空间已经相对众所周知。 相比之下，关于更高维复合歧管的模量空间的理解更少。 这个项目将使用同义理论的方法研究有关此问题和相关空间的说法。 该项目的另一个目标是研究具有奇异性的歧管的模量空间。 在这里，最基本的情况是Riemann Moduli空间的Deligne-Mumford紧凑型。 对于许多应用，这个空间比未编译的模量空间更重要，并且同型理论理解应该是富有成果的。 特别是，提案者将研究对格罗莫夫 - 理论理论的应用。A歧管是空间的一般概念。 歧管是几何，拓扑，分析和其他数学领域的基础。 在物理学中，它们是爱因斯坦一般相对论中的基本几何结构，并且在力学，量子场理论以及可能的其他许多领域中扮演着基本角色。 从数学角度研究歧管可以通过两种方式进行：您可以一次研究每个歧管的几何特性，一个一次，也可以一次研究所有歧管收集的几何特性。 “所有歧管的收集”本身可以被认为是一种空间，每个单独的歧管都是这个空间中的一点。 这个空间是模量空间的一个示例，当前的项目旨在了解该模量空间及其变化的几何和拓扑特性。