Mathematical Sciences: Homotopy Theory

9504530麦克卢尔 首席研究员正在研究同伦理论中的各种问题。 他正在继续他的工作拓扑安德烈-奎伦同源性及其关系不稳定同伦理论。 他还调查的性质建设的A-无限环光谱，类似于中心的一环代数;这似乎有可能有应用程序提出的问题Mahowald，Ravenel和希克在他们的工作望远镜猜想。 他打算用分圆同调（一种由Bokstedt、Goodwillie、Hsiang和Madsen定义的结构，在代数K理论中有重要应用）做一些计算。 他正在调查的问题，是否楔甚至悬浮液的复杂定向E-无限环再次是一个E-无限环，以及一些后果，这一性质将有。 他打算工作的几个问题有关同伦理论的C*-代数。 他正在调查一些老问题所提出的亚当斯，威尔克森和阿蒂亚涉及之间的关系亚当斯业务和Steenrod业务的阿蒂亚，Hirzebruch频谱序列。 他正在寻求更好地理解摩拉瓦K理论的扩展权力。 拓扑学是研究形状可以具有的某些性质，这些性质不依赖于详细的测量（众所周知的例子是，甜甜圈和咖啡杯对拓扑学家来说是一样的，因为一个可以被拉伸到看起来完全像另一个）。 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;结果在两种情况下是相同的，这一事实表明（通过非平凡定理）一个形状可以延伸到另一个形状。 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，将它们纳入抽象代数的框架中。 在过去的十年里，这个学科最有趣的趋势之一就是在抽象代数的早期阶段，通过将形状本身与群或环联系起来，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个想法。 ***

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. ***