Interactions of homotopy theory and algebra

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The project has three components, all concerning new interactionsbetween algebra and topology. In the first component, theinvestigator will explore the role of orthogonal Grassmannians inmotivic homotopy theory. The focus will be on their relationship withHermitian K-theory and on their motivic cohomology, the latter beingrelated to motivic characteristic classes for quadratic bundles. Onegoal will be to geometrically construct such classes and toinvestigate their basic properties. In the second part of the projectthe investigator will explore (with D. Biss and D. Isaksen) spaces ofzero-divisors in Cayley-Dickson algebras, in an attempt to completelyclassify these. This may eventually have applications to the Kervaireinvariant one problem, and this will be explored. Finally, theresearcher will continue joint work with B. Shipley which studies therelationship between differential graded algebras and ring spectra.The last ten years have seen the discovery of many new, surprisingways in which topology and algebra interact. Historically, algebrahas always been used to give information about topology; but what isamazing about recent developments is that they often use topology togive information about algebra. This project focuses on threeinstances of this. The main component of the research involvesmotivic cohomology, which is a very exciting area that is growingquickly. At its core, it involves the application of topologicaltechniques to the study of systems of polynomial equations. Theobjective of the project is to study the motivic cohomology of certainbasic objects which are well-understood from a topological viewpoint,and to investigate how the algebra and topology interact in theseimportant examples. The results should be very important for thefurther development of the field.