Manifolds and Moduli Spaces

流形和模空间

• 批准号: 1405001
• 负责人: Soren Galatius
• 金额: $ 30.32万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405001&HistoricalAwards=false
• 关键词: Manifolds; Moduli; Spaces

Parts of the research conducted under this grant will concern the study of manifolds and their moduli. Manifolds are a generalized concepts of space, appearing all over mathematics and science. The defining property of a manifold is that it is locally parametrized by a finite number of parameters (for example, the surface of the earth can locally be parametrized by two parameters, namely longitude and latitude). A classical topic in algebraic topology, Galatius' field of research, is the classification of manifolds: How does one decide whether two manifolds are isomorphic, and how does one write a list of all possible manifolds? The research conducted here concerns the related question of classifying how manifolds may vary in families: what are their symmetries and what are their deformations. The research conducted under this grant will apply new methods to study these classical and important questions. Another part of the research conducted under this grant concerns deformations of objects arising from number theory, studied using methods from homotopy theory.The proposed activity will continue the development of a relatively new area of mathematics in which homotopy theory is used to study various moduli spaces. An important early result this area is the solution, by Madsen and Weiss, of a conjecture of Mumford. Significant new developments have happened since then, many of which, such as the solution to a 1995 conjecture of Hatcher, are due to the principal investigator. This subject lies in the overlap between algebraic topology and other branches of mathematics, with expected applications in algebraic geometry, symplectic geometry, and possibly theoretical physics. Another part of the project concerns applications of homotopy theoretic ideas in number theory. Working in a derived setting is a very familiar idea in homotopy theory, and has often been successfully imported into algebra and other fields. Recently, the use of a derived setting for the study of deformations has received much attention. The project will develop and apply these techniques to Galois representations, objects of fundamental importance in number theory.

根据该赠款进行的部分研究将涉及对流形及其模量的研究。 歧管是一个广义的空间概念，遍布数学和科学。 歧管的定义属性是，它是由有限数量的参数在局部参数化的（例如，地球表面可以通过两个参数（即经度和纬度）在局部参数化）。 代数拓扑中的一个经典主题，即加拉图斯的研究领域，是流形的分类：一个人如何决定两个流形是否同构，以及如何编写所有可能的流形列表？ 这里进行的研究涉及分类多种流形可能在家庭中如何变化的相关问题：它们的对称性是什么，它们的变形是什么。 根据该赠款进行的研究将采用新方法来研究这些经典和重要的问题。 根据该赠款进行的研究的另一部分是使用数字理论引起的对象的变形，该方法是使用同型理论中的方法进行了研究的。拟议活动将继续发展相对较新的数学领域，其中使用同型理论来研究各种模量空间。 一个重要的早期结果是Madsen和Weiss的解决方案，是Mumford的猜想。 从那时起，发生了重大的新发展，其中许多（例如1995年的Hatcher的解决方案）归因于首席研究员。 该主题在于代数拓扑结构与数学的其他分支之间的重叠，并在代数几何形状，符号几何形状以及可能的理论物理学中进行了预期的应用。 该项目的另一部分涉及同义理论思想在数理论中的应用。 在派生的环境中工作是同质理论中非常熟悉的想法，并且经常被成功地导入代数和其他领域。 最近，将派生设置用于变形研究引起了很多关注。 该项目将开发并将这些技术应用于Galois表示形式，这是数量理论中基本重要性的对象。