Hacking's research focuses on moduli spaces of algebraic surfaces. These spaces are known to have complicated singularities in general. However, Hacking proposes to describe fundamental examples which are well-behaved and understand the many new phenomena which do not occur for moduli of curves. Jointly with Keel and Tevelev, Hacking will describe natural compactifications of the moduli spaces of del Pezzo surfaces. These spaces can be thought of as analogues of the moduli spaces of pointed stable curves of genus zero associated to the exceptional root systems. Hacking has shown that the moduli space of curves is rigid, i.e., cannot be deformed. However, a heuristic due to Kapranov suggests that moduli spaces of surfaces should have deformations in general, and Hacking aims to construct an explicit example. The total space of the deformation should be a moduli space of generalised surfaces in some sense, e.g., noncommutative surfaces. Hacking also proposes to relate degenerations of del Pezzo surfaces and derived categories and to study noncommutative analogues of Kleinian singularities.
Algebraic geometry is the study of algebraic varieties, i.e., spaces defined by polynomial equations. Many of the spaces arising in nature are algebraic varieties, so algebraic geometry is important in theoretical physics. A moduli space is a space parametrising all varieties of a given topological type. Moduli spaces are again algebraic varieties and often have many remarkable properties one cannot hope to observe on an arbitrary variety. The moduli spaces of curves have been intensively studied and are of fundamental importance in many contexts. The proposed research concerns moduli spaces of surfaces, which are only poorly understood at present.
