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

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The purpose of the proposal is to study the geometry of algebraic varietiesand their invariants from different points of view. In particular theinvestigator and his colleagues studythe structure of fundamental groups and universal coverings of algebraicvarieties, the problemof placement of singularities, restrictions on the concentration of singularpoints in special fibers of the fibrations of algebraic varieties. In thelatter case the goal is to develop a sufficiently simple method which can begeneralized to the case of arithmetic family of curves. The study of theGalois groups of functional fields is going to provide with an alternativeapproach to the problems of birational geometry.The investigator suggests to continue a broad scope of research in the areaof algebraic geometry with potential applications to number theory andtheoretical physics. Of special interest is the study of a formulathat provides nontrivial estimates for a number of solutions in equationssimilar in form to the famous Fermat equation..