Motives and Algebraic Groups

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The PI of this project will continue to work on problems in algebraic group theory using three algebraic cohomology theories having topological origin: motivic cohomology, algebraic K-theory and algebraic cobordism. The PI proposes to work on three relatively independent topics. The first topic is devoted to the rationality problem of algebraic groups and deals with motivic cohomology. The PI expects to find a Postnikov tower for the motive of a simply connected group involving motives of projective homogeneous varieties. The second topic is related to algebraic cobordism. The PI sees the opportunity to use this theory in order to approach the general Rost's degree formula. The latter has many applications in the theory of homogeneous spaces of algebraic groups. The third topic, the essential dimension, although seemingly different from the others, nevertheless, involves the phenomenon of compression of algebraic varieties and hence is closely related to the second topic. The PI proposes to use degree formulas for the computation of essential dimensions of algebraic groups.The area of this project lies between algebraic geometry, the branch of mathematics devoted to geometric objects called algebraic varieties and described by polynomial equations, and algebraic topology where one studies 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 this project is about using techniques that are of a topological nature to obtain a better understanding of certain problems in algebraic geometry.