Symplectically oriented cohomology theories of algebraic varieties

The versatility of A^1-homotopy theory and its associated range of cohomological techniques has made it an important branch of mathematics. Recently there have been several fundamental developments which have been used to solve a number of longstanding problems. The new strategically important developments are related to the names of V. Voevodsky (Fields Medal 2002), M. Rost, A. Suslin, I. Panin, M.Levine, F. Morel.The principal aim of this proposal is the study of symplectically oriented cohomology theories on algebraic varieties introduced recently by Nenashev-Panin-Walter. We plan to construct bijective correspondences between symplectic orientations, Pontriagin structures, Pontriagin-Thom structures and symplectic trace structures (symplectic integrations) respectively on a given ring cohomology theory. Another aim of the project is to give an explicitdescription for symplectic trace structures on a symplectically oriented cohomology theory. The theory is illustrated by the symplectic algebraic cobordism of Voevodsky, the hermitian algebraic K-theory and by other examples.The research will be undertaken in the Department of Mathematics, Swansea University.

The main beneficiaries will be the departments of pure mathematics of UK universities. The project will be of great interest for experts in algebraic geometry, algebraic topology, K-theory. There is a growing interest in various parts of Mathematics to applications of contemporary algebraic geometry and Voevodsky motivic homotopy theory. This project will add new values to applications of oriented cohomology theory to algebraic geometry and algebraic topology. Besides of its research merits, this project is aimed to enhance the collaboration between UK-universities and their counterpart in Russia. Discussion and dissemination of the involved mathematical ideas will be undoubtedly beneficial for mathematical communities in both countries.