Algebraic Cycles on Homogeneous Varieties

alexander merkkurjev该提案涵盖了代数的广泛方面，如代数几何、代数群和齐次变异。研究者建议研究齐次簇的动机上同群和动机分解，如射影齐次簇和代数群。研究者将寻求产生与代数几何中产生的物体相关的结构，这些结构紧密地反映了代数循环的微妙方面。特别是，研究者建议计算与射影齐次变量相关的简单对象的动机上同调群。该提案的第二个主题涉及引入基本维数和不可压缩变量的概念，这些概念提供了代数循环给出的代数变量之间的新相互关系。这个项目的领域介于代数几何和代数拓扑之间，代数几何是数学的一个分支，致力于研究多项式方程的几何对象，称为代数变异，代数拓扑涉及连续变化的结构族，称为拓扑空间。将拓扑方法从拓扑空间转化为代数变量，为求解代数几何问题提供了新的工具。许多被提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.