Symplectic Topology via Lagrangian Fibrations

DMS-0204368玛格丽特·F·赛明顿首席研究员建议使用奇异拉格朗日纤维来洞察辛流形的拓扑结构。最终的目标是开发一种有效的语言来构造和分析辛四流形，类似于光滑四流形的Kirby演算。这项工作位于环形几何、可积系统和光滑四流形拓扑的交点上。这个项目的进展将允许将辛四流形表示为众所周知的流形的广义和。例如，这种方法允许首席研究者将光滑运算专门用于辛范畴，从而确定具有奇异光滑结构的四流形无限族上的辛结构的存在性。进一步的工作将包括扩展到六维，其中这种纤颤出现在开创性的镜像对称理论中。流形是曲面到其他维度的推广。要使流形成为辛流形，它的内部结构必须类似于摆等机械系统的位置和速度空间。辛流形在拓扑学、几何学和物理学中是普遍存在的。近年来，人们对辛流形的认识取得了很大的进展，尤其是在四维辛流形上，但许多基本问题仍然没有得到回答。例如，给定一个流形，没有一般的方法来确定它是否允许它成为辛流形所需的内部结构。解决这个问题的一种方法是在流形上要求一些额外的结构。首席调查员建议通过求助于测量高程的高维地形图来加深我们对这些流形的理解。额外的结构可以被认为是等高线的模拟。本质上，首席调查员计划开发二维地图，以产生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.