This supports the work of Professor Fedor Bogomolov to work on the geometry of algebraic varieties and related problems in birational geometry. He proposes to work on constructing minimal dominant classes of algebraic varieties, asymptotic formulas for cohomology of lines bundles and a counter-example to Lang's conjecture in higher dimensions. He also wants to work on the Sylow subgroups of Galois Groups of certain function fields. This is research in the field of algebraic geometry, yet it directly connects to one of the greatest advances made in the solution of equations--Galois theory. Algebraic geometry itself 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.
