Algebraic Cycles on Homogeneous Varieties

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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.