Mathematical Sciences: Gauge Theory in Geometry of Manifolds and Vector Bundles

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9626689 Leung The proposal lies in the area of complex differential geometry. The investigator proposes to compute Seiberg-Witten invariants for complex non-Kahler surfaces; to generalize Seiberg-Witten theory to include higher dimensional manifolds. Additionally, he is to investigate the relationship between the existence of Einstein-Hermitian metrics and stability of vector bundles a la Gieseker. In 1995 Seiberg and Witten introduced a system of differential equations on four dimensional spaces and defined certain invariant quantities (they are solutions to these equations); this work significantly impacted the classification theory of smooth four dimensional curved spaces. The proposed research attempts to continue this line of investigation with an emphasis on those four manifolds that are not algebraically defined.