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

New York University, New York NY

Investigators

Abstract

The purpose of the proposal is to study the geometry of algebraic varieties and their invariants from different points of view. In particular the investigator and his colleagues study the structure of fundamental groups and universal coverings of algebraic varieties, the problem of placement of singularities, restrictions on the concentration of singular points in special fibers of the fibrations of algebraic varieties. In the latter case the goal is to develop a sufficiently simple method which can be generalized to the case of arithmetic family of curves. The study of the Galois groups of functional fields is going to provide with an alternative approach to the problems of birational geometry. The investigator suggests to continue a broad scope of research in the area of algebraic geometry with potential applications to number theory and theoretical physics. Of special interest is the study of a formula that provides nontrivial estimates for a number of solutions in equations similar in form to the famous Fermat equation..