9625658 Recently some progress has been achieved in the construction of the basic theory of (mixed) motives. The PI is in possession now of a number of techniques which make it possible to approach at least some of the "hard" motivic conjectures. These conjectures can be divided into two classes depending on whether they deal with rational or finite coefficients. The goal of the research described in this proposal is to understand the structure of the "Motivic world" for finite coefficients. It turns out that the usual "motivic t-structure" intuition does not work in the finite coefficients case. One of the possible replacements for it seems to be the "chromatic" intuition which was very successfully employed in algebraic topology in the last decade. This approach was used in the recent PI's preprints where it was shown that a number of standard conjectures about motives with finite coefficients could be solved if we had a good definition of algebro-geometrical analogs of the higher Morava K-theories. To construct these theories seems to be a nontrivial but relatively straight forward problem. The steps to be taken include the description of the cohomological operations in motivic cohomology, construction of algebraic cobordisms through an algebro-geometrical analog of Thom spectrum, computation of its "coefficients ring" using "motivic" Adams spectral sequence and finally the formal construction of algebraic Morava K-theories from the algebraic cobordisms. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics .
