Geometry and Topology of Symplectic Four Manifolds

建议：DMS-9975469PI：Lian-jun Lian-jun摘要：本项目的重点是应用微分拓扑学、几何分析和代数几何的方法研究辛四流形。对于辛四流形，Seiberg-Witten不变量与辛Gromov-Taube不变量的等价性在辛拓扑中得到了许多令人惊叹的结果。在与刘的合作下，Li建立了参数化的Seiberg-Witten理论，并推导出了一个跨越墙的公式。李建议进一步发展参数化的Seiberg-Witten理论和参数化的Gromov-Taubes理论，并证明这两个理论对于辛族是等价的。参数化理论对于研究微分同构和辛同构都是有用的。这两个理论之间的等价性有望在辛四流形的分类中发挥重要作用，特别是那些具有扭转正则类的流形。最近，光滑Lefschetz纤维被证明是四个辛流形的拓扑、曲线模空间的几何和映射类群的代数之间的纽带。研究人员计划探索美丽而丰富的相互作用，以促进对所有这些对象的理解。流形是一个局部看起来像n维欧几里得空间的空间。例如，我们生活的时空宇宙是一个四维流形。辛结构是一种非常基本的结构，它几乎涵盖了经典物理和量子物理中的所有方程。辛四流形是具有辛结构的四流形。因此，辛四流形在数学和物理中起着核心作用。基本问题是对所有的辛四流形进行分类。作者的目的是对辛四流形的一般形式有一些了解。

Proposal: DMS-9975469PI: Tian-Jun LiAbstract: The focus of this project is to apply methods from differential topology,geometric analysis and algebraic geometry to study symplectic four manifolds.For symplectic four manifolds, the equivalence between Seiberg-Witten invariants and the symplectic Gromov-Taubes invariants has led to many striking results in symplectic topology. In collaboration with Liu, Li has set up the parametrized Seiberg-Witten theory and derived a wall-crossing formula.The parametrized Seiberg-Witten theory is particularly interesting forfamilies of symplectic manifolds. Li proposes to develop further the parametrized Seiberg-Witten and the parametrized Gromov-Taubes theories and show that these two theories are equivalent for symplectic families.The parametrized theories should be useful for studying the isotopies of diffeomorphismsas well as symplectomorphisms. And the equivalence between the two theories is expected to play an important role in the classification of symplectic four manifolds, especially those with torsion canonical classes. Recently, it has been shown that smooth Lefschetz fibrationsprovide a link between the topology of symplectic four manifolds,the geometry of the moduli space of curves and the algebra ofthe mapping class groups. The investigator plans to explore thebeautiful and rich interplay to advance understandingof all these objects.An n manifold is a space that locally looks like Euclidean spaceof dimension n. For example, the space-time universe we live inis a four manifold. A symplectic structure is a very basic structure that underliesalmost all the equations of classical and quantum physics. A symplectic four manifold is a four manifold witha symplectic structure. Thus symplectic four manifoldsplay a central role in mathematics and physics.The fundamental problem is to classify all symplecticfour manifolds. The investigator aims to gain some understanding of the general shapeof symplectic four manifolds.