Moduli Spaces and Algebraic Structures in Homotopy Theory

摘要奖：DMS-0705428首席研究员：克雷格·C·韦斯特兰本研究议程分为三个部分。在第一部分中，主要研究者建议研究模空间几何中固有的代数结构，特别是那些没有从运算的角度研究的代数结构，包括从微分几何、代数几何和物理的几类模空间。第二部分讨论了模空间在同伦理论中的应用，包括流形上自由循环空间的等变同调，循环同调拓扑型的弦拓扑运算，李群分类空间的弦拓扑。第三个主要项目将与数学家合作，将稳定同伦理论的技术应用到Hurwitz空间的研究中。模空间是描述其他几何对象的可变性的几何对象。例如，在欧几里得三维空间中，以原点为中心的所有球体的集合的任何元素完全由圆的半径(正数)确定，因此所有这些球体(每个球体都是二维对象)的集合由实数线(一维对象)的正半部分来描述。在这项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，例如在上面标记或标记了几个点的两孔甜甜圈的表面，这种结构提供了进入量子场论、代数和几何的重要问题的途径。

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.