The PI will work on several fundamental problems in algebraic geometry with a focus on compact moduli spaces of stable curves and stable surfaces (introduced by Kollar, Shepherd-Barron, and Alexeev). One of the goals is to introduce and describe various effective divisors on these moduli spaces and study how they affect their birational geometry. New methods of constructing stable surfaces of general type are proposed, with the moduli space of simply connected fake del Pezzo surfaces studied in detail.
Algebraic geometry studies algebraic varieties: shapes defined by systems of polynomial equations. Varieties are classified by discrete (topological) and continuous (geometric) parameters, called moduli. The study of the moduli space allows us to understand all possible geometric structures on a given shape. The proposed research program has several points of contact with other disciplines, for example introduction of Landau-Ginzburg models made moduli spaces of varieties of general type attractive to physicists.