Algebraic Cycles on Homogeneous Varieties

DMS-0355166 Alexander Merkurjev代数几何、代数群和齐次簇等代数领域的广泛内容。作者建议研究齐次簇的基元上同调群和基元分解，如射影齐次簇和代数群。研究人员将寻求产生与代数几何中出现的对象相关联的结构，这些结构紧密地反映了代数循环的微妙方面。特别地，研究者建议计算与射影齐次簇相关的单纯对象的动机上同调群。这项计划的第二个主题是引入本质维度和不可压缩簇的概念，它们提供了由代数循环给出的代数簇之间的新的相互关系。这个项目的范围介于代数几何学和代数拓扑学之间，代数几何学是一门数学分支，专门研究由多项式方程作图而来的几何对象，称为代数簇，而代数拓扑学涉及连续变化的称为拓扑空间族的结构。将拓扑学的方法从拓扑空间转化为代数变体，为解决代数几何中的问题提供了新的工具。许多建议的工作是使用拓扑技术和思想来更好地理解代数几何的一些问题。

DMS-0355166Alexander MerkkurjevThe proposal covers a wide range of aspects in algebra such as algebraic geometry, algebraic groups, and homogeneous varieties. The investigator proposes to study motivic cohomology groups and motivic decomposition of homogeneous varieties such as projective homogeneous varieties and algebraic groups. The investigator will seek to produce constructions associated to objects arising in algebraic geometry, which closely reflect subtle aspects of algebraic cycles. In particular, the investigator proposes to compute motivic cohomology groups of simplicial objects associated to projective homogeneous varieties. The second topic of the proposal involves introduction of the notion of essential dimension and incompressible varieties that provide new interrelations between algebraic varieties given by algebraic cycles.The area of this project lies between algebraic geometry, the branch of mathematics devoted to geometric objects coming from graphing polynomial equations and called algebraic varieties, and algebraic topology that concerns continuously varying families of structures called topological spaces. Translating the methods of topology from topological spaces to algebraic varieties gives new tools to solve problems in algebraic geometry. Much of the proposed work is to use topological techniques and ideas to get better understanding of some problems of algebraic geometry.