Theory of motives and algebraic cycles

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1. Motives of varieties: Let D(k) be the category of mixed motives over a field k. We produced a functor from the category of quasi-projective varieties into D(k). The construction uses the method of cubical hyperresolution.2. Motivic decomposition theorem: It is of interest to formulate and prove the motivic analogue of the topological decomposition theorem (of Beilinson, Bernstein and Deligne). In the case of the universal family of abelian varieties over the Hilbert modular variety, we showed the existence of the expected motivic decomposition, and deduced from it the Grothendieck-Murre conjecture for the fiber variety.3. Homology correspondence at chain level: It is well-known to consider homological correspondences and their compositions. We considered this at the chain level. Namely we gave a complex of abelian groups which gives cohomology, and produced the composition map as a map of complexes. This construction is applied to produce the cohomology realization functor from mixed motives.4. Mixed motivic sheaves: We sketched the construction of the triangulated category of mixed motives over a quasi-projective variety.