Moduli problems in algebraic geometry

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Hacking's research focuses on moduli spaces of algebraic surfaces.These spaces are known to have complicated singularities ingeneral. However, Hacking proposes to describe fundamentalexamples which are well-behaved and understand the many newphenomena which do not occur for moduli of curves. Jointly withKeel and Tevelev, Hacking will describe natural compactificationsof the moduli spaces of del Pezzo surfaces. These spaces can bethought of as analogues of the moduli spaces of pointed stablecurves 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 suggeststhat moduli spaces of surfaces should have deformations ingeneral, and Hacking aims to construct an explicit example. Thetotal space of the deformation should be a moduli space ofgeneralised surfaces in some sense, e.g., noncommutative surfaces.Hacking also proposes to relate degenerations of del Pezzosurfaces and derived categories and to study noncommutativeanalogues of Kleinian singularities.Algebraic geometry is the study of algebraic varieties, i.e.,spaces defined by polynomial equations. Many of the spaces arisingin nature are algebraic varieties, so algebraic geometry isimportant in theoretical physics. A moduli space is a spaceparametrising all varieties of a given topological type. Modulispaces are again algebraic varieties and often have manyremarkable properties one cannot hope to observe on an arbitraryvariety. The moduli spaces of curves have been intensively studiedand are of fundamental importance in many contexts. The proposedresearch concerns moduli spaces of surfaces, which are only poorlyunderstood at present.