Mathematical Sciences: Smooth Structures of Algebraic Surfaces

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9400729 Qin Since the fundamental work of Simon Donaldson about a decade ago, gauge theory has been very sucessfully applied to study smooth 4-manifolds, especially complex surfaces. The main concern of this project is to study the interplay between the complex geometry and the differential topology of complex surfaces, by utilizing smooth invariants introduced by Simon Donaldson and Dieter Kotschick, by investigating stable vector bundles and their moduli spaces on complex surfaces, and by using recent work of Jun Li and Robert Friedman. In particular, the investigator wishes to study Donaldson invariants for simply connected algebraic surfaces with trivial geometric genus (including the complex projective plane). This project is in the field of gauge theory, which interacts heavily with other areas of mathematics such as topology, algebraic geometry, and even mathematical physics. Progress could take many different forms. Surprisingly, recent work in supersymmetric Yang-Mills theory by some physicists (notably Edward Witten) has led to many mathematical insights in the differential topology of complex surfaces. Quite possibly, a deeper understanding of some of the algebraic geometry and algebraic topology at issue in this project will in turn shed light on problems of concern to the community of quantum physicists. ***