Morava motives of projective quadrics

The present Ph. D. thesis is devoted to Morava motives of projective quadrics, meaning that we replace the Chow theory by another oriented cohomology theory. We consider arbitrary oriented cohomology theories as we wish to obtain invariants that are simpler than Chow motives. In fact, there exists a series of theories, more precisely, Morava K-theories K(n)*, which starts from K^0 and tends to CH*. The most important and interesting results are the following ones: Theorem (Theorem 1.3.9) Let Q be a generic quadric of dimension D > 0, and n > 1; we denote N = 2^n for D = 2d even, or N = 2^n-1 for D = 2d+1 odd. Then K(n)-motive of Q has an indecomposable summand of rank min(N, 2d + 2), and max(0, 2d + 2 - N) summands isomorphic to Tate motives. Theorem (Theorem 2.0.1) For a group G_m = Spin_m or G_m = SO_m with m > 2^(n+1), n > 1, the canonical map K(n)*(G_m; F_2) → K(n)*(G_(m+2); F_2) is an isomorphism. We also describe several algorithms useful for computer computations of K(n)-motives of small-dimensional varieties.