Symplectic Topology via Lagrangian Fibrations

DMS-0204368玛格丽特F.主要研究者提出使用奇异拉格朗日纤维化来深入了解辛流形的拓扑结构。 最终目标是发展一种有效的语言来构造和分析辛四维流形，类似于光滑四维流形的Kirby演算，这项工作是复曲面几何、可积系统和光滑四维流形拓扑学的交叉点。 在这个项目上的进展将允许辛四流形的介绍作为广义和良好理解的流形。 例如，这种方法使主要研究者能够专门对辛范畴进行光滑手术，从而确定具有奇异光滑结构的四流形的无限族上辛结构的存在。 进一步的工作将包括扩展到六维，在那里这种纤维化出现在开创性的镜像理论中。流形是表面到其他维度的推广。 对于一个流形是辛的，它必须有一个类似于一个机械系统（如钟摆）的位置和速度空间的内部结构。辛流形在拓扑学、几何学和物理学中是普遍存在的，近年来人们对辛流形，特别是四维辛流形的认识取得了很大的进展，但仍有许多基本问题没有得到解决，例如，给定一个流形，它是否允许其内部结构是辛的，目前还没有一个通用的方法来确定. 解决这个问题的一个方法是在流形上增加一些额外的结构。首席研究员建议通过使用更高维的地形图来测量高程，以加深我们对这些流形的理解。 附加的结构可以被认为是恒定高度的线的模拟。 从本质上讲，首席研究员计划开发二维地图，提供足够的信息来完全确定地形（四维流形）和使这些地图可解释的图例。

DMS-0204368Margaret F. SymingtonThe Principal Investigator proposes to use singular Lagrangian fibrations to gain insight into the topology of symplectic manifolds. The ultimate goal is to develop an effective language for constructing and analyzing symplectic four-manifolds, analogous to Kirby calculus for smooth four-manifolds.This work lies at the intersection of toric geometry, integrable systemsand smooth four-manifold topology. Progress on this project will allow the presentation of symplectic four-manifolds as generalized sums of well-understood manifolds. For instance, this approach has allowed the Principal Investigator to specialize a smooth surgery to the symplectic category and thereby determine the existence of symplectic structures on an infinite family of four-manifolds with exotic smooth structures. Further work will include extensions to dimension six where such fibrations arise in the ground-breaking theory of mirror symmetry.A manifold is the generalization of a surface to other dimensions. For a manifold to be symplectic it must have an internal structure akin to the space of positions and velocities of a mechanical system such as a pendulum. Symplectic manifolds are ubiquitous in topology, geometry and physics.Recently great progress has been made in understanding symplectic manifolds,especially of dimension four, but many basic questions remain unanswered.For instance, given a manifold there is no general way to determine if it permits the internal structure required for it to be symplectic. One way to attack such a question is to require some additional structure on the manifold.The Principal Investigator proposes to deepen our understanding of these manifolds by appealing to a higher-dimensional analog of topographic maps for measuring elevation. The additional structure could be thought of as the analog of lines of constant elevation. In essence, the Principal Investigator plans to develop two-dimensional maps that yield enough information to completely determine the terrain (the four-dimensional manifold) and a legend that make these maps interpretable.