Paul Arne Østvær

Motivic slices and coloured operads

Colored operads were introduced in the 1970s for the purpose of studying homotopy invariant algebraic structures on topological spaces. In this paper we introduce colored operads in motivic stable homotopy theory. Our main motivation is to uncover hitherto unknown highly structured properties of the slice filtration. The latter decomposes every motivic spectrum into its slices, which are motives, and one may ask to what extend the slice filtration preserves highly structured objects such as algebras and modules. We use colored operads to give a precise solution to this problem. Our approach makes use of axiomatic setups which specialize to classical and motivic stable homotopy theory. Accessible t-structures are central to the development of the general theory. Concise introductions to colored operads and Bousfield (co)localizations are given in separate appendices.

Slices of hermitian K–theory and Milnor’s conjecture on quadratic forms

We advance the understanding of K-theory of quadratic forms by computing the slices of the motivic spectra representing hermitian K-groups and Witt-groups. By an explicit computation of the slice spectral sequence for higher Witt-theory, we prove Milnor’s conjecture relating Galois cohomology to quadratic forms via the filtration of the Witt ring by its fundamental ideal. In a related computation we express hermitian K-groups in terms of motivic cohomology.

Homotopy theory of C*-algebras

In this work we construct from ground up a homotopy theory of C ∗ -algebras. This is achieved in parallel withthe developmentofclassical homotopytheoryby firstintroducinganunstablemodelstructureandsecondastable modelstructure. The theory makes use of a full fledged import of homotopy theoretic techniques into the subject of C ∗ -algebras. The spaces in C ∗ -homotopy theory are certain hybrids of functors represented by C ∗ -algebras and spaces studied in classical homotopy theory. In particular, we employ both the topological circle and the C ∗ -algebra circle of complex-valued continuous functions on the real numbers which vanish at infinity. By using the inner workings of the theory, we may stabilize the spaces by forming spectra and bispectra with respect to either one of these circles or their tensor product. These stabilized spaces or spectra are the objects of study in stable C ∗ -homotopy theory. The stable homotopy category of C ∗ -algebras gives rise to invariants such as stable homotopy groups and bigraded cohomology and homology theories. We work out examples related to the emerging subject of noncommutative motives and zeta functions of C ∗ -algebras. In addition, we employ homotopy theory to define a new type of K-theory of C ∗ -algebras.

Motivic Brown–Peterson invariants of the rationals

Let BPhni, 0 n1 , denote the family of motivic truncated Brown‐Peterson spectra over Q. We employ a “local-to-global” philosophy in order to compute the bigraded homotopy groups of BPhni. Along the way, we produce a computation of the homotopy groups of BPhni over Q2 , prove a motivic Hasse principle for the spectra BPhni, and reprove several classical and recent theorems about the K ‐theory of particular fields in a streamlined fashion. We also compute the bigraded homotopy groups of the 2‐complete algebraic cobordism spectrum MGL over Q. 55T15; 19D50, 19E15

The Homotopy Type of Two-regular K -theory

We identify the 2-adic homotopy type of the algebraic K-theory space for rings of integers in two-regular exceptional number fields. The answer is given in terms of well-known spaces considered in topological K-theory.

Two-primary algebraic K-theory of two-regular number fields

Abstract. We explicitly calculate all the 2-primary higher algebraic K-groups of the rings of integers of all 2-regular quadratic number fields, cyclotomic number fields, or maximal real subfields of such. Here 2-regular means that (2) does not split in the number field, and its narrow Picard group is of odd order.

Two-complete stable motivic stems over finite fields

Let

A Norm Principle in Higher Algebraic K-Theory

\ell

Rigidity for equivariant pseudo pretheories

be a prime and

Unstable C*-homotopy theory

q = p^{\nu}