Moduli problems in algebraic geometry

哈金的研究重点是代数曲面的模空间。这些空间通常具有复杂的奇点。然而，Hacking建议描述一些基本的例子，这些例子表现良好，并理解曲线模不会出现的许多新现象。与keel和Tevelev一起，Hacking将描述del Pezzo曲面模空间的自然紧化。这些空间可以被认为是与特殊根系相关的零属尖稳定曲线的模空间的类似物。Hacking已经证明曲线的模空间是刚性的，即不能变形。然而，由于Kapranov的启发式表明，曲面的模空间应该具有一般的变形，Hacking的目的是构造一个明确的例子。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。Hacking还建议将del pezzo曲面的退化与派生范畴联系起来，并研究Kleinian奇点的非交换类似物。代数几何是研究代数变量，即多项式方程所定义的空间。自然界中产生的许多空间都是代数的变体，因此代数几何在理论物理中很重要。模空间是对给定拓扑类型的所有变体进行空间参数化的空间。模空间又是代数的变种，而且常常具有许多人们不能指望在任意变种上观察到的显著性质。曲线的模空间已经得到了广泛的研究，并且在许多情况下具有重要的基础意义。所提出的研究涉及曲面的模空间，目前对其了解甚少。

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.