Algebraic Cycles on Homogeneous Varieties

DMS-0355166 Alexander Merkkurjev该提案涵盖了代数的广泛方面，如代数几何，代数群和齐次簇。 研究者提出研究齐次簇的动机上同调群和动机分解，如射影齐次簇和代数群。调查人员将寻求产生与代数几何中出现的对象相关的结构，这些结构密切反映了代数循环的微妙方面。特别是，调查人员提出计算motivic上同调群的单纯对象相关联的投影齐性品种。该提案的第二个主题涉及引入基本维数和不可压缩簇的概念，这些概念提供了由代数圈给出的代数簇之间的新的相互关系。该项目的领域位于代数几何，数学的分支，致力于从图形多项式方程和称为代数簇的几何对象，和代数拓扑学，涉及连续变化的结构族，称为拓扑空间。将拓扑空间的拓扑方法转化为代数簇的拓扑方法，为代数几何问题的求解提供了新的工具。大部分的工作建议是使用拓扑技术和思想，以更好地理解一些问题的代数几何。

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.