The Geometry and Dynamics of Symplectic Manifolds

摘要奖：DMS-0905191首席研究员：Dusa McDuff由于辛结构允许人们测量二维曲面的面积，因此辛流形的二维子流形自然是其全局结构的关键元素。辅助几乎复结构J的选择决定了一类特别有趣的此类曲面，即J-全纯曲面。利用它们可以建立许多有趣的同调理论，例如量子上同调或辛场理论。最近，关于空间上辛同态的动力学性质(特别是空间具有多大的对称性)与其量子同调环的结构之间的关系的有趣的联系被揭示出来。McDuff最近发现，如果空间具有圆对称性，那么它是无规则的，这意味着对于每个J的选择，空间中的每一个点都有AJ-全纯球面。她提议的一个项目将对这种联系进行更深入的研究。另一类是研究具有极大交换对称群的辛流形的结构。她还提出了一个与Schlenk的联合项目，该项目将阐明一个非常基本的辛刚性现象。这试图准确地理解一个四维辛椭球何时可以被挤压到球内。这引出了一些非常有趣的数论问题，也指出了射影平面爆破中J-全纯曲线的组合与嵌入接触同调中作为指数出现的数之间的联系。空间可以有几种基本几何结构之一，例如测量距离和角度的方法(如欧几里得几何)或测量二维对象的面积的方法(如辛几何)。辛几何中研究的结构很重要，因为它们不仅是经典能量守恒系统(如行星系统)方程的基础，而且是许多现代物理学理论(如弦理论)的重要组成部分。这个项目旨在加深我们对辛空间的基本理解。一种是关于大空间结构(如上同调)对空间动力学性质的影响的问题，例如，研究在空间的任意运动下固定的点的数量和性质。另一种是研究人们可能认为的辛空间小块的结晶性质；在压力下，这些小块是如何折叠的，以占据无上空间？这第二条线引出了初等数论和组合学中一些非常有趣的问题，这是辛几何中第一次出现这些领域之间的关系。这些问题可以向高中生解释，因此将提供一个极好的方式，向年轻人解释当今研究数学家所做的一些事情，并激发他们对这一领域的兴趣。

AbstractAward: DMS-0905191Principal Investigator: Dusa McDuffSince a symplectic structure allows one to measure the areas oftwo dimensional surfaces, it is natural that the two dimensionalsubmanifolds of a symplectic manifold are key elements of theirglobal structure. The choice of an auxiliary almost complexstructure J determines a specially interesting class of suchsurfaces, namely those that are J-holomorphic. Using them it ispossible to build many interesting homology theories, such asquantum cohomology or symplectic field theory. Recently veryintriguing connections have come to light concerning the relationbetween the dynamical properties of the symplectomorphisms on aspace (in particular, how much symmetry the space has) and thestructure of its quantum homology ring. McDuff recentlydiscovered that if the space has a circle symmetry then it isuniruled, which implies that for every choice of J there is aJ-holomorphic sphere though every point in the space. One of herproposed projects will investigate such connections in moredepth. Another will investigate the structure of toricmanifolds, which are symplectic manifolds with maximal abeliansymmetric group. She also proposes a joint project with Schlenkthat will illuminate a very basic symplectic rigidity phenomenon.This attempts to understand exactly when a four dimensionalsymplectic ellipsoid can be squeezed inside a ball. This givesrise to some very interesting number-theoretic questions, andalso indicates a connection between the combinatorics ofJ-holomorphic curves in the blow up of the projective plane andthe numbers that appear as indices in embedded contact homology.A space can have one of several fundamental geometric structures,for example a way of measuring distance and angle (as inEuclidean geometry) or a way of measuring the area of twodimensional objects (as in symplectic geometry.) The structuresstudied in symplectic geometry are important because they notonly underlie the equations of classical energy-conservingsystems such as the planetary system, but also appear as a vitalcomponent of many of the modern theories in physics such asstring theory. This project aims to further our basicunderstanding of symplectic spaces. One line of inquiryconcentrates on questions about the influence of structures inthe large (such as cohomology) on the dynamical properties of thespace, investigating for example the number and nature of thepoints that are fixed under an arbitrary movement of the space.Another line of inquiry investigates what one might think of asthe crystalline nature of small pieces of a symplectic space;under pressure how do such small pieces fold so as to take upless space? This second line of inquiry leads to some veryinteresting questions in elementary number theory andcombinatorics, the first appearance in symplectic geometry of arelation between these fields. These questions can be explainedto high school students, and so will provide an excellent way toexplain to young people something of what research mathematiciansdo today and to stimulate their interest in the field.