Moduli Spaces and Algebraic Structures in Homotopy Theory

项目编号：dms -0705428首席研究员：Craig C. westerland本研究议程分为三部分。首先，主要研究者建议研究模空间几何中固有的代数结构，特别是那些没有从运算的角度研究的结构，包括微分几何、代数几何和物理学中的几个模空间族。第二部分讨论了模空间在同伦理论中的应用，包括流形上自由环空间的等变同调，循环同调的拓扑形式的弦拓扑运算，李群分类空间的弦拓扑。第三个主要项目将应用稳定同伦理论的技术来研究赫维茨空间，与数论专家合作。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中的任何元素都完全由圆的半径（一个正数）决定，因此所有这些球体的集合（每个球体都是一个二维物体）由实数线的正一半（一个一维物体）来描述。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，例如带有几个标记或标记的双孔甜甜圈的表面，这种结构为量子场论，代数和几何的重要问题提供了途径。

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.