Algebraic Varieties, Birational Geometry and the Structure of the Galois Groups

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Bogomolov 9801591 The principal investigator intends to continue his work on the geometry of algebraic varieties, the structure of Galois groups, and related problems in birational geometry. He plans to work on constructing minimal dominant classes of algebraic varieties, asymptotic formulas for cohomology of line bundles, the description of fundamental groups and universal coverings of algebraic and symplectic manifolds, and the structure of complex symplectic and hyperkahler manifolds. He also proposes to work on the structure of Sylow subgroups of the Galois groups and on the effective version of the stabilization procedure for group cohomology. 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.