Algebraic cycles on homogeneous varieties

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Cohomological methods in mathematics were introduced and applied in the middle of the last century by Solomon Lefschetz to study complex algebraic varieties. During the next decade they were essentially developed by Andre Weil and Alexander Grothendieck leading to the proof of the famous Weil's Conjectures by Pierre Deligne. Since then cohomological methods have become a fundamental technique in mathematics which in many cases led to the resolution of classical algebraic and geometric problems. Among the major recent achievements in this direction one should mention: the oriented cohomology theory, the theory of algebraic cobordism by Levine-Morel and the theory of motives by Grothendieck, Manin and Voevodsky. Another central object of the modern mathematics is the notion of a (linear algebraic) group G acting on a variety or manifold X. The basic example is the group of linear transformations acting on a vector space. The geometry of G and X has been a subject of intensive investigations for almost a century, with important contributions by Armand Borel, Claude Chevalley, Michel Demazure, Jean-Pierre Serre, Jaques Tits, and others.