Geometry and Topology of Symplectic Four Manifolds

摘要：本课题的重点是应用微分拓扑、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-Taubes不变量的等价性在辛拓扑中得到了许多显著的结果。与Liu合作，Li建立了参数化Seiberg-Witten理论，并推导了一个过墙公式。参数化Seiberg-Witten理论对于辛流形族来说是特别有趣的。Li提出进一步发展参数化Seiberg-Witten理论和参数化Gromov-Taubes理论，并证明这两种理论对于辛族是等价的。参数化理论对差同型同位素和复同型同位素的研究都是有用的。这两种理论之间的等价性在辛四流形的分类，特别是具有扭转正则类的辛四流形的分类中具有重要的意义。最近，光滑的Lefschetz纤维提供了辛四流形的拓扑、曲线模空间的几何和映射类群的代数之间的联系。研究人员计划探索美丽而丰富的相互作用，以促进对所有这些物体的理解。一个n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流形。辛结构是一种非常基本的结构，它构成了几乎所有经典物理学和量子物理学方程的基础。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学中起着中心作用。基本问题是对所有辛四流形进行分类。研究者的目的是对辛四流形的一般形状有一些了解。

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.