Abstract:

The category of Chow motives defined by Grothendieck has plenty of various applications
to quadratic forms, and, more generally, to projective homogeneous varieties. However,
there are many open questions about the behaviour of Chow motives. In contrast, if we
change the Chow group by Grothendieck's K-theory in the definition of motives, the resulting
category behaves much more simply. One can define the category of motives corresponding to
any oriented cohomology theory and obtain invariants that are simpler than Chow motives but
keep more information than K-theory motives. In the talk, I will consider categories of motives
constructed with respect to Morava K-theories and describe several recent results in this area.