The PI will study surfaces and their variations. From the computational point of view, the best surfaces are those that can be described by a polynomial equation. Not every surface is such but Nash proved that every surface can be approximated by such polynomially defined surfaces. The main aim of the proposal is to understand how surfaces vary if we change the coefficients of the defining polynomial equations. The PI would especially aim to understand situations when a family of surfaces degenerates to a very complicated, highly singular surface. The main aim is to develop methods that can be used to simplify such singularities. As part of this project, the PI also aims to study families of curves on surfaces, especially near the singular points of the surface.
The PI aims to study families of algebraic surfaces of general type. There is a universal space for all families, called the moduli space. This moduli space is not compact, the PI intends to prove that a good compactification is given by considering surfaces that have semi-log canonical singularities and ample canonical class. The PI aims to develop a similar theory in higher dimensions, here the canonical models of varieties of general type provide the basic objects. As part of this project, the PI aims to understand the structure of semi-log canonical singularities, especially the combinatorial structure of their resolution. The PI aims to understand the structure of the dual complex of the resolution. A closely related but in principle independent project is to understand the structure of the Nash space of arcs through a singularity. The original conjectures of Nash were disproved in dimensions 3 and up, but there is a modification that takes these example into account. The PI aims to study both some very concrete examples and some general phenomena.