Special Geometries

Just like geometry in the plane differs from geometry on the sphere, there are in higher dimensions many different geometries. Some of these are used to describe and explain the basic laws of physics. In this project we use special geometries to distinguish and find new algebraic and differentiable manifolds, and to develop motivic and stable homotopy theory as an analytic tool for the topology of such manifolds. During the project we have found new 4- and 6-dimensional HyperKähler manifolds, nef curve classes that are not pseudo effective and we have shown that the Koras-Russel threefolds are A1 contractible. On the theory side we will in particular mention two contributions in algebraic K-theory. The first one concerns a development of the trace map invented by Connes and Bökstedt, Hsiang and Madsen, extending the range of applications to the classification of a much larger class of manifolds then the original version. The second is a proof of some versions of the Beilinson -Lichtenbaum conjecture in motivic cohomology, also improving tools in the classification of manifolds.

Prosjektet oppnådde en revitalisering av de to forskningsgruppene i algebra/algebraisk geometri og geometri/topologi, både i form nyrekruttering av faste medarbeidere og en utvidelse av forskningsaktiviteteten til differensialgeometri. Prosjektet har lagt grunnlaget for nye satsinger med bidrag fra NFR og BFS i form av tre eksternt finansierte prosjekter som inkluderer ansettelse både av nye faste medarbeidere og en rekke PhD-er og postdocer. Samlet ser de to gruppene som danner en seksjon ved Matematisk Institutt en klar økning og fornying av sin forskningsaktivitet. Antall resultatindikatorene i prosjekter ligger godt over de forventede tallene ved starten av prosjektet.

The algebraic geometry and topology groups at UiO want to intensify research on special geometries, their applications to the classification of varieties and manifolds and to stable and motivic homotopy theory. The scientific program is divided in three work packages: -Projective models of hyperkähler and Calabi Yau varieties -Algebraic K-theory and stable homotopy theory -Motivic invariants for group schemes and stacks With new recruitments we aim at extending this list into the area of symplectic, complex or Kähler geometry.