Manifolds, Moduli Spaces and Homotopy Theory

流形、模空间和同伦理论

• 批准号: 1105058
• 负责人: Soren Galatius
• 金额: $ 37.42万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105058&HistoricalAwards=false
• 关键词: Manifolds; Moduli; Spaces; Homotopy; Theory; Manifolds; Moduli; Spaces; Homotopy; Theory

The proposed research 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. Their result concerns the group of symmetries of closed two-dimensional manifolds or, equivalently, it concerns the moduli space of such manifolds. A major part of Galatius' proposed activity will study the extension of this theory to manifolds of higher dimension. 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.Galatius' research will concern the study of manifolds and their symmetries. A manifold is a generalized versions 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 real parameters (for example, the surface of the earth is locally parametrized by two parameters, longitude and latitude). A classic 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. A related classical problem is the classification of symmetries of manifolds (for example, rotation about an axis is a symmetry of the sphere). Galatius' research will apply new methods to study these classic questions.

拟议的研究将继续开发一个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。他们的结果涉及封闭的二维流形的对称性组，或者等效地涉及此类歧管的模量空间。 加拉图斯提出的活动的主要部分将研究该理论的扩展到更高维度的流形。该主题在于代数拓扑结构与数学的其他分支之间的重叠，在代数几何形状，符号几何形状以及可能的理论物理学中具有预期的应用。 歧管是一个广义的空间版本，遍布数学和科学。 歧管的定义属性是，它通过有限数量的实际参数在局部参数化（例如，地球表面通过两个参数，经度和纬度在局部参数化）。 Galatius的研究领域是代数拓扑的一个经典主题，是流形的分类：一个人如何决定两个流形是否是同构，以及如何编写所有可能的歧管列表。 一个相关的经典问题是歧管对称性的分类（例如，轴旋转是球体的对称性）。 Galatius的研究将采用新方法来研究这些经典问题。