Symplectically oriented cohomology theories of algebraic varieties

A^1-同伦理论的多功能性及其相关的一系列上同调技巧使其成为数学的一个重要分支。最近有几个基本的发展，已被用来解决一些长期存在的问题。新的具有重要战略意义的发展与V. Voevodsky（2002年菲尔兹奖），M。Rost，A.苏斯林岛Panin，M.Levine，F.这个建议的主要目的是研究最近由Nenashev-Panin-Walter介绍的代数簇的辛上同调理论。在给定的环上同调理论下，我们分别构造了辛定向、Pontriagin结构、Pontriagin-Thom结构和辛迹结构（辛积分）之间的双射对应.该项目的另一个目的是在辛上同调理论上给出辛迹结构的明确描述。该理论是由辛代数配边Voevodsky，埃尔米特代数K-理论和其他的例子。研究将在数学系，斯旺西大学进行。

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.