Investigating connections between homotopy theory and algebra

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The project focuses on three areas: motivic stable homotopy groups, new foundations for etale homotopy theory, and topological equivalences of differential graded algebras. In the first area the PI will continue his work with Isaksen on the motivic Adams spectral sequence, and will investigate specific relations amongst the motivic Hopf elements that arise from Cayley-Dickson algebras. The etale homotopy theory project involves using quasi-categories to create an Artin-Mazur style homotopy theory of homotopically cofiltered diagrams of spaces. Finally, in joint work with Shipley the PI will continue the investigation of situations in which differential graded algebras can give rise to weakly equivalence Eilenberg-MacLane spectra.Homotopy theory is the part of mathematics that studies geometric objects in higher-dimensional space via algebraic techniques. The flow of information can go in both directions: typically one starts with algebraic knowledge and uses homotopy theory to deduce geometric information, but especially in recent years it has become possible to start with geometric knowledge and deduce some sophisticated algebraic information. This research project will try to add to our knowledge about several topics in which algebra and geometry interact in mysterious ways. By coming to a deeper understanding of these connections, we will be in a better position to understand certain elaborate geometric patterns and possibly to attack some difficult conjectures.