Problems in Homotopy Theory and Group Cohomology

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

Professor Kuhn has a long record of developing homotopy and representation theoretic methods to solve problems in both topology and algebra. Recently, he has discovered some remarkable connections between two of the main strands of homotopy theory as studied over the past 25 years: homotopy as organized into its `periodic' layers (as in the work of M.Hopkins), and homotopy as organized using sophisticated homotopical algebraic decomposition techniques (as in the work of T.Goodwillie). The largest part of this project concerns the continued exploration of these connections, and the continued development of tools for exploiting this. A second part will focus on some problems in finite group cohomology exploiting primitives associated to central extensions. Specific older work of his to be brought to bear on these questions include his work on the Whitehead Conjecture, Bousfield--Kuhn telescopic functors, Hopkins--Kuhn--Ravenel generalized characters, and unstable modules over the Steenrod algebra. Homotopy theory and group cohomology are mathematical subjects in which one is trying to discover, and ultimately classify, fundamental "building blocks" of structure: homotopy dealing with deformations of geometric objects such as higher dimensional surfaces, and group cohomology being concerned with analyzing symmetries of both geometric and algebraic objects. A sample homotopy problem is to understand the "shape" of all continuous functions from a surface to itself. A sample group cohomology problem is to understand the way in which very basic subgroups of symmetries "control" the behavior of more complicated symmetry groups. Professor Kuhn is studying these subjects by developing a variety of state-of-the-art algebraic and homotopy theoretic tools. By their very nature, many of these tools involve "universal" constructions of interest to researchers in other scientific disciplines ranging from computer science to robotics.