Exceptional vector bundles and degenerations of surfaces

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。当一个光滑的复曲面退化为一个具有普通双点的奇异曲面时，存在一个所谓的消失循环：一个二维球体坍缩为一个点。然而，在曲面模理论中，自然会出现更复杂的退化，其中不存在消失循环。在某些情况下，Hacking描述了光滑纤维上的刚性全纯矢量束，类似于消失循环。哈金将利用这种构造来研究曲面模空间的边界和曲面上稳定向量束的分类。在自然界中，空间的形状可以随时间连续变化；一个很好的例子是一个大肥皂泡的表面。空间的几何也可以经历突变或退化。例如，一个气泡可以分成两个较小的气泡。Hacking将根据给定空间的几何性质研究其可能的退化。之前对这个问题的研究使用了a ？消失的周期?-给定空间的一个子集，它在退化中收缩到一个点。黑客将使用给定空间上的线性空间束，其结构在退化下变得更简单。此方法适用于旧技术无法提供任何信息的重要示例。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).When a smooth complex surface degenerates to a singular surface with an ordinary double point there is a so called vanishing cycle: a two dimensional sphere which collapses to a point. However, in the theory of moduli of surfaces, more complicated degenerations naturally arise for which there are no vanishing cycles. In some of these cases Hacking has described a rigid holomorphic vector bundle on the smooth fibre which is analogous to a vanishing cycle. Hacking will use this construction to study the boundary of the moduli space of surfaces and the classification of stable vector bundles on surfaces.In nature, the shape of a space can vary continuously with time; a good example is the surface of a large soap bubble. The geometry of a space can also undergo an abrupt change or ?degeneration?, for example, a bubble can divide into two smaller bubbles. Hacking will study the possible degenerations of a given space in terms of its geometric properties. Previous work on this problem used the notion of a ?vanishing cycle? - a subset of the given space which shrinks to a point in the degeneration. Hacking will use bundles of linear spaces over the given space whose structure becomes simpler under the degeneration. This method applies to important examples where the old techniques do not provide any information.