Shapes of Symplectic and Legendrian Submanifolds

摘要奖：DMS-9971374主要研究员：丽莎·特雷诺贯穿辛几何所有分支的一个中心主题是辛流形的特殊辛子流形和接触流形的特殊树形子流形的重要性。这个提案概述了一些项目，这些项目探索了这些特殊的子流形的形状的刚性和灵活性之间有趣的细微差别。项目包括研究欧几里德空间的子集或环面的余切丛的辛包的可能效率，以及研究辛球或传奇环和缠结在辛变换或接触变换下的可能演化。技术包括通过母函数和伪全纯曲线定义的组合学和同调理论。此外，基于观察到的传奇曲线和DNA的重合行为，提出了辛几何和生物学之间的联系。辛几何和接触几何起源于物理学：它们是理解旋转的陀螺、水下机器人的机械和行星轨迹的背景。一个基本的问题是了解系统如何在接触或辛结构施加的自然运动下演化。理解动力学的一种方法是从标准物体如球或环状的弦开始，并研究这种基本形状可能达到的构型。结果可能会相当令人惊讶。例如，尽管球可能发生许多变形，但一个经典的结果是，它永远不能在特定方向上被挤压。这是物理学测不准原理的几何版本。这个项目的目的是进一步洞察这些规范转换的灵活性和刚性。

AbstractAward: DMS-9971374Principal Investigator: Lisa TraynorA central theme throughout all branches of symplectic geometry isthe importance of the special symplectic or lagrangiansubmanifolds of a symplectic manifold and of the speciallegendrian submanifolds of a contact manifold. This proposaloutlines a number of projects that explore the interesting fineline between the rigidity and flexibility of the shapes of thesespecial submanifolds. Projects include studying the possibleefficiency of symplectically packing subsets of euclidean spaceor cotangent bundles of tori and studying the possible evolutionof symplectic balls or legendrian links and tangles undersymplectic or contact transformations. Techniques includecombinatorics and homology theories defined via generatingfunctions and pseudo-holomorphic curves. In addition,connections between symplectic geometry and biology are proposedbased on observed coincidental behaviors of legendrian curves andDNA.Symplectic and contact geometry have their origins in physics:they are the setting for understanding spinning tops, mechanicsof underwater vehicles, and planetary trajectories. A basicproblem is to understand how a system can evolve under thenatural motions imposed by a contact or symplectic structure.One approach to understanding the dynamics is to start with astandard object such as a ball or a looped piece of string and tostudy the possible configurations that this basic shape canattain. The results can be quite surprising. For example,although many deformations of a ball are possible, a classicalresult is that it can never be squeezed in particular directions.This is a geometric version of the Uncertainty Principle fromphysics. This project aims is to give further insight into theflexibility and rigidity of these canonical transformations.