Applications of Morava K-theory to algebraic groups and quadrics.

For a prime number p and a non-negative integer n we consider a Morava K-theory K(n) with the coefficient ring ?p. This is a universal oriented cohomology theory in the sense of Levine-Morel with a pn-typical formal group law which has height n modulo p. It turns out that K(n) is strongly related to cohomological invariants of algebraic groups in the sense of Serre. This is our starting point to compute the Chow groups of quadrics from the powers Im+2 of the fundamental ideal of the Witt ring up to codimension 2m. Moreover, the Morava K-theory gives a conceptional explanation of the nature of the obtained answer. This is a joint work with Pavel Sechin.