Motivic homotopy theory of group scheme actions

We define an unstable equivariant motivic homotopy category for an algebraic group over a Noetherian base scheme. We show that equivariant algebraic $K$-theory is representable in the resulting homotopy category. Additionally, we establish homotopical purity and blow-up theorems for finite abelian groups.Comment: Final version, to appear in Journal of Topology. arXiv admin note: text overlap with arXiv:1403.191

Topological comparison theorems for Bredon motivic cohomology

We prove equivariant versions of the Beilinson-Lichtenbaum conjecture for Bredon motivic cohomology of smooth complex and real varieties with an action of the group of order two. This identifies equivariant motivic and topological invariants in a large range of degrees.Comment: Corrected indices in main theorem and a few minor changes. To appear, Transactions AM

Existence and uniqueness of E∞-structures on motivic K-theory spectra

The algebraic K-theory spectrum KGL, the motivic Adams summand ML and their connective covers have unique E1-structures refining their naive multiplicative structures in the motivic stable homotopy category. These results are deduced from T-homology computations in motivic obstruction theory

Homotopy Theory of C*-Algebras

Homotopy theory and C* algebras are central topics in contemporary mathematics. This book introduces a modern homotopy theory for C*-algebras. One basic idea of the setup is to merge C*-algebras and spaces studied in algebraic topology into one category comprising C*-spaces. These objects are suitable fodder for standard homotopy theoretic moves, leading to unstable and stable model structures. With the foundations in place one is led to natural definitions of invariants for C*-spaces such as homology and cohomology theories, K-theory and zeta-functions. The text is largely self-contained. I