Moduli problems in algebraic geometry

哈金的研究主要集中在代数曲面的模空间上，这些空间一般都有复杂的奇点。然而，黑客建议描述行为良好的基本例子，并理解许多新现象，这些现象不会出现在曲线的模数上。他将与Keel和Tevelev一起描述del Pezzo曲面的模空间的自然紧致性。这些空间可以看作是与例外根系有关的亏格为零的点稳定曲线的模空间的类比。哈奇证明了曲线的模空间是刚性的，即不能变形。然而，Kapranov的一个启发式建议曲面的模空间应该具有一般性的形变，而黑客的目的是构造一个显式的例子。变形的总空间在某种意义上应该是广义曲面的模空间，例如，非交换曲面。哈克还提出将Del PezzosSurface的退化与派生范畴联系起来，并研究Kleian奇点的非交换类似。代数几何是研究代数簇，即由多项式方程定义的空间。自然界中出现的许多空间都是代数变体，因此代数几何在理论物理中很重要。模空间是表示给定拓扑型的所有簇的空间参数。模空间也是代数簇，通常具有许多人们不能期望在任意簇上观察到的显著性质。曲线的模空间已经得到了深入的研究，并且在许多情况下具有重要的意义。所提出的研究是关于曲面的模空间的，目前对它的了解还很少。

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.