Moduli Spaces and Algebraic Structures in Homotopy Theory

同伦理论中的模空间和代数结构

• 批准号: 0705428
• 负责人: Jordan Ellenberg
• 金额: $ 10.62万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-15 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0705428&HistoricalAwards=false
• 关键词: Moduli; Spaces; Algebraic; Structures; Homotopy

AbstractAward: DMS-0705428Principal Investigator: Craig C. WesterlandThis research agenda has three parts. In the first, theprincipal investigator proposes to study algebraic structuresinherent in the geometry of moduli spaces, particularly thosethat have not been studied from an operadic point of view andincluding several families of moduli spaces from differentialgeometry, algebraic geometry, and physics. The second part ofthe proposal concerns applications of this study of moduli spacesto objects in homotopy theory, including equivariant homology forthe free loop space on a manifold, string topology operations fora topological version of cyclic homology, and string topology ofclassifying spaces of Lie groups. The third major project willapply techniques from stable homotopy theory to the study ofHurwitz spaces, in a collaboration with number theorists.Moduli spaces are geometric objects that describe the variabilityof other geometric objects. For example, any element of thecollection of all spheres centered at the origin in Euclideanthree-dimensional space is completely determined by the radius ofthe circle, a positive number, so the collection of all thesespheres (each of which is a two-dimensional object) is describedby the positive half of the real number line (a one-dimensionalobject). A moduli space of particular interest in this and otherongoing mathematical research describes the variable geometry ofa surface such as the surface of a two-holed doughnut withseveral points labeled or marked on it, a construction whichprovides access to important questions of quantum field theory,algebra, and geometry.

摘要奖项：DMS-0705428 首席研究员：Craig C. Westerland 本研究议程分为三个部分。 首先，主要研究者建议研究模空间几何中固有的代数结构，特别是那些尚未从操作角度研究过的代数结构，包括来自微分几何、代数几何和物理学的几个模空间族。 该提案的第二部分涉及模空间研究在同伦理论中对象的应用，包括流形上自由循环空间的等变同调、循环同调拓扑版本的弦拓扑运算以及李群分类空间的弦拓扑。 第三个主要项目将与数论学家合作，将稳定同伦理论的技术应用于赫维茨空间的研究。模空间是描述其他几何对象的可变性的几何对象。 例如，欧几里得三维空间中以原点为中心的所有球体集合中的任何元素完全由圆的半径（正数）决定，因此所有这些球体（每个球体都是一个二维物体）的集合为由实数轴的正半部分（一维物体）描述。 在这个和其他正在进行的数学研究中特别感兴趣的模空间描述了表面的可变几何形状，例如带有标记或标记的两个孔甜甜圈的表面，这种结构提供了量子场论、代数、和几何。