Motives of twisted flag varieties is an important tool in the study of splitting properties of torsors, in the geometric theory of quadratic forms and central simple algebras with involutions. For instance, motives of Pfister quadrics played a key role in the proofs of the Milnor conjecture on quadratic forms and the Bloch-Kato conjecture. Another applications include cohomological invariants, the theory of canonical and essential dimensions of linear algebraic groups.

The course will survey some of this research. It will be started with an introduction to the theory motives of twisted flag varieties and conclude with a discussion of open problems. We will introduce and study the discrete motivic invariant of a torsor (the J-invariant), explain relations to canonical dimensions, K-theory, Chow groups, algebraic cobordism of twisted flag variaties and linear algebraic groups.