Interactions of homotopy theory and algebra

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

The project has three components, all concerning new interactions between algebra and topology. In the first component, the investigator will explore the role of orthogonal Grassmannians in motivic homotopy theory. The focus will be on their relationship with Hermitian K-theory and on their motivic cohomology, the latter being related to motivic characteristic classes for quadratic bundles. One goal will be to geometrically construct such classes and to investigate their basic properties. In the second part of the project the investigator will explore (with D. Biss and D. Isaksen) spaces of zero-divisors in Cayley-Dickson algebras, in an attempt to completely classify these. This may eventually have applications to the Kervaire invariant one problem, and this will be explored. Finally, the researcher will continue joint work with B. Shipley which studies the relationship between differential graded algebras and ring spectra. The last ten years have seen the discovery of many new, surprising ways in which topology and algebra interact. Historically, algebra has always been used to give information about topology; but what is amazing about recent developments is that they often use topology to give information about algebra. This project focuses on three instances of this. The main component of the research involves motivic cohomology, which is a very exciting area that is growing quickly. At its core, it involves the application of topological techniques to the study of systems of polynomial equations. The objective of the project is to study the motivic cohomology of certain basic objects which are well-understood from a topological viewpoint, and to investigate how the algebra and topology interact in these important examples. The results should be very important for the further development of the field.