Mathematical Sciences: Homotopy Theory

9504530麦克卢尔是首席研究员，他正在研究同伦理论中的各种问题。他继续研究拓扑Andre-Quillen同调及其与不稳定同伦理论的关系。他还在研究A-无穷环谱上类似于代数中环中心的结构的性质；这似乎可能适用于Mahowald、Ravenel和Shick在他们关于望远镜猜想的工作中提出的一个问题。他打算用割圈同调(Bokstedt，Goodwillie，Hsiang和Madsen定义的一种结构，它在代数K-理论中有重要的应用)做一些计算。他正在研究面向复数的E-无穷大环的偶数悬挂的楔形是否又是E-无穷大环的问题，以及这一性质将会产生的一些结果。他打算研究几个将同伦理论与C*-代数联系起来的问题。他正在研究Adams-Wilkerson和Atiyah提出的一些老问题，涉及Atiyah-Hirzebruch谱序列中Adams运算和Steenrod运算之间的关系。他正在寻求更好地理解扩展权力的莫拉瓦K理论。拓扑学是研究形状可能具有的某些属性，这些属性不依赖于详细的测量(众所周知的例子是，对于拓扑学家来说，甜甜圈和咖啡杯是相同的，因为一个可以拉伸到看起来完全相同)。代数拓扑学是一种通过某种计算来描述形状的方法。例如，甜甜圈和咖啡杯的表面可以被分成不重叠的三角形，然后可以计算三角形的数量加上顶点的数量减去边的数量；结果在这两种情况下是相同的，这一事实表明(通过一个非平凡的定理)一个形状可以拉伸成另一个形状。为了从这些计算中获得更多的信息，人们通过将它们与群或环联系起来，将它们结合到抽象代数的框架中。在过去的十年里，这门学科最有趣的趋势之一是通过将形状本身与群或环联系起来，在这一过程的早期阶段引入抽象代数；这就是短语“A-无穷环谱”所指的。这个项目的主要目的是继续发展这一想法。***

9504530 McClure The principal investigator is working on a variety of questions in homotopy theory. He is continuing his work on topological Andre-Quillen homology and its relation to unstable homotopy theory. He is also investigating the properties of a construction on A-infinity ring spectra that is analogous to the center of a ring in algebra; this seems likely to have applications to a question posed by Mahowald, Ravenel and Shick in their work on the telescope conjecture. He intends to do some calculations with cyclotomic homology (a construction that was defined by Bokstedt, Goodwillie, Hsiang and Madsen, which has important applications in algebraic K-theory). He is investigating the question of whether the wedge of the even suspensions of a complex-oriented E-infinity ring is again an E-infinity ring, and some consequences that this property would have. He intends to work on several problems which relate homotopy theory to C*-algebras. He is investigating some old questions posed by Adams-Wilkerson and Atiyah involving the relationship between Adams operations and Steenrod operations in the Atiyah-Hirzebruch spectral sequence. He is seeking a better understanding of the Morava K-theory of extended powers. Topology is the study of certain properties that shapes can have that do not depend on detailed measurement (the well-known example is that a doughnut and a coffee cup are the same to a topologist, because one can be stretched to look exactly like the other). Algebraic topology is a method for describing shapes by means of certain kinds of calculation. For example, the surfaces of the doughnut and the coffee cup could be divided into non-overlapping triangles, and one could then calculate the number of triangles plus the number of vertices minus the number of edges; the result would be the same in both cases, and this fact shows (by means of a nontrivial theorem) that one shape can be stretched into the other. In order to get more info rmation from these calculations, one incorporates them into the framework of abstract algebra by relating them to groups or rings. One of the most interesting trends in the subject in the last decade has been to bring in abstract algebra at an earlier stage of the process by relating the shapes themselves to groups or rings; that is what the phrase "A-infinity ring spectrum" refers to. The main purpose of this project is to continue the development of this idea. ***