Mathematical Sciences: Homotopy Theory

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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. ***