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

9626689梁这项建议涉及复杂微分几何领域。研究者建议计算复杂非Kahler曲面的Seiberg-Witten不变量；推广Seiberg-Witten理论以包括高维流形。此外，他还将研究爱因斯坦-厄米特度规的存在与向量丛的稳定性之间的关系。1995年，Seiberg和Witten在四维空间上引入了一个微分方程组，并定义了某些不变量(它们是这些方程的解)，这项工作对光滑四维曲线空间的分类理论产生了重大的影响。建议的研究试图继续这条线，重点放在那四个非代数定义的流形上。

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.