Symplectically oriented cohomology theories of algebraic varieties

A^1-同伦理论的多功能性及其上同调技巧的范围使其成为数学的一个重要分支。最近出现了一些根本性的发展，这些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(Fields Medal 2002)、M.Rost、A.Suslin、I.Panin、M.Levine、F.Morel.的名字有关。这项建议的主要目的是研究Nenashev-Panin-Walter最近提出的关于代数簇的面向辛上同调理论。在一个给定的环上同调理论上，我们计划分别建立辛取向、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一个面向辛上同调理论上给出辛迹结构的一个显式描述。这一理论由沃沃茨基的辛代数余边法、埃尔米特代数K理论和其他例子来说明。研究将在斯旺西大学数学系进行。

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.