Shapes of Symplectic and Legendrian Submanifolds

摘要：关于辛几何的特殊辛流形或拉格朗日子流形以及接触流形的特殊拉格朗日子流形的重要性，是贯穿辛几何所有分支的中心主题。该提案概述了一些探索这些特殊子流形形状的刚性和灵活性之间有趣的细线的项目。项目包括研究欧几里得空间或环面的协切束的辛填充子集的可能效率，以及研究辛球或legendrian连杆和缠结在辛变换或接触变换下的可能演化。技术包括通过生成函数和伪全纯曲线定义的组合学和同调理论。此外，根据观察到的legendrian曲线和dna的巧合行为，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：它们是理解旋转陀螺、水下航行器力学和行星轨迹的基础。一个基本问题是理解一个系统如何在接触或辛结构施加的自然运动下进化。理解动力学的一种方法是从标准物体开始，比如一个球或一根绕圈的绳子，研究这个基本形状可能达到的构型。结果可能相当令人惊讶。例如，尽管一个球的许多变形都是可能的，一个经典的结果是它永远不会在特定的方向上被挤压。这是物理学中测不准原理的几何版本。这个项目的目的是进一步深入了解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.