Exceptional vector bundles and degenerations of surfaces

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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.