Topology in Norway

The term "topology" indicates, in a broad sense, a notion of geometric structure, and the basic approach in algebraic topology is to link geometric structures with algebraic structures. It often happens that the algebraic structures can be analyzed more effectively than their geometric counterparts and this is useful, for instance, when classifying various types of geometric phenomena. An incarnation of this principle is given by the algebraic K-theory introduced by D. Quillen: Questions in algebra, algebraic geometry, number theory, and geometric topology are encoded in topological objects known as spectra and are hence made accessible for analysis by the rich machinery of algebraic topology. The main objectives stated in the project description can be summarized as a quest to analyse and calculate the various forms of algebraic K-theory and to link the results with other areas of algebraic and geometric interest. The study of algebraic K-theory has been an extremely active area of research in algebra and topology during the last fifty years. One of the most successful approaches to the analysis of algebraic K-theory are the trace invariants introduced by Boksted-Hsiang-Madsen. These trace invariants take values in topological Hochschild homology (THH) and an equivariant refinement known as topological cyclic homology (TC). In favourable cases, algebraic K-theory is equivalent to TC, and the later has the advantage of being accessible to calculations by standard methods in algebraic topology. However, while THH and TC are often easier to calculate than algebraic K-theory, these theories do not have all the pleasant properties known from algebraic K-theory. Notably, the localization sequence for the algebraic K-theory of a Dedekind domain does not have an immediate analogue for THH. Motivated by earlier work by Hesselholt-Madsen and Rognes, it was singled out as a central goal in the project description to construct a logarithmic version of THH that does fit in a localization sequence for the kind of ring spectra relevant for the homotopical understanding of algebraic K-theory. Such a logarithmic theory has recently been developed by project participants J. Rognes and C. Schlichtkrull in collaboration with S. Sagave. In another but related direction, project participant B. Dundas has, in joint work with A. Lindenstrauss, B. Richter, and C. Ausoni, determined the higher and iterated THH in a number of cases including number rings and finite fields. These iterated theories are important for the understanding of the homotopical properties of algebraic K-theory. In order to understand the equivariant structure of iterated THH, and hence the passage to iterated TC, it is important to have a thorough understanding of the underlying building blocks, that is, the structure of equivariant smash powers. Building on the doctoral thesis by M. Stolz, project participants M. Brun and B. Dundas have in collaboration with M. Stolz produced a comprehensive account of this theory. Besides the trace invariants discussed above, the main approach to the analysis of algebraic K-theory is via the motivic homotopy theory developed by Morel and Voevodsky. This theory has made it possible to study algebraic varieties by topological methods and two of the main goals stated in this part of the proposal has been achieved by project participant P. A. Østvær and collaborators. The first of these goals was formulated as the questions on how to compute hermitian K-groups in terms of motivic stable homotopy groups and this question was settled in joint work with A. J. Berrick, M. Karoubi, and M. Schlichting. The second goal was to give a new proof of Milnor's conjecture for quadratic forms by an explicit calculation of the slice spectral sequence for the higher Witt-theory spectrum and this has been achieved in joint work with O. Rondigs. Apart from the work on the main scientific objectives listed above, there have been numerous scientific contributions in related areas by the project participants. These include the work by M. Szymik on non-linear Hochschild cohomology and derived centers and the work by M. Thaule and coworkers on n-angulated categories. There has also been an increasing interest among the project participants in the various applications of topological methods in data and numerical analysis. In this regard, A. Schmeding, who held one of the postdoctoral positions associated with the project, has played a bridging role in that he has collaborated both with the topology group and the numerical analysis group at NTNU. The existence of a well-developed topology network linking the institutions associated with the project (NTNU, UiB, UiO) is to a large extend due to support from the Research Council of Norway to this and previous projects. This has been extraordinary beneficial both for researchers and students and is reflected in the high number of joint projects and publications in the grant period.

The concept of symmetry is one of the most fundamental notions in mathematics and in the applications to neighboring scientific fields. One of the main scientific objectives of the project is to analyze the transfer of one fundamental form of symmetry, na mely the symmetry underlying the definition of a commutative algebra over the sphere spectrum, to another fundamental form of symmetry, the periodicity patterns arising from the chromatic view on the stable homotopy category. The concept of symmetry als o underlies the approaches to motivic homotopy theory and higher order structures in topology suggested by the project.