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.

Abstractaward：DMS-0705428原理研究者：Craig C. Westerlandthisthis研究议程有三个部分。 首先，原理研究者提议在模量空间的几何形状中研究代数结构，尤其是从经营的角度来研究，尤其是从差分地理测定法，代数的几何形状和物理学的几个模量空间的家族中研究。 该提案的第二部分涉及该模量水平对象在同型理论中的应用，包括在歧管上的自由循环空间，弦拓扑操作的自由循环空间的同源性，循环同源性的拓扑拓扑操作以及分类空间的弦乐拓扑。 第三个主要项目将从稳定同义理论到Hurwitz空间的研究，与数字理论家合作。Moduli空间是描述其他几何对象的可变性的几何对象。 例如，所有在欧几里德二维空间中以原点为中心的球体的元素完全由圆的半径确定，因此，所有这些seaspheres的集合（每个二维对象）的集合被描述为实际数字线的正面一半（一较高的object）。 在这项和其他日益数学研究中特别感兴趣的模量空间描述了A表面的可变几何形状，例如在其上标记或标记的两个孔甜甜圈的表面，这是一种构造，可以访问量子场理论，代数和几何学的重要问题。