Symplectic, Contact and Low-dimensional Topology

摘要奖：DMS-0102922主要研究员：罗伯特·E·贡普夫这个项目致力于构造辛流形、接触流形和4维流形。首席研究者在偶数维闭流形上定义了一种称为“超铅笔”的拓扑结构，推广了复代数流形上的线性曲线系。他已经证明，任何超铅笔都规范地决定流形上的辛结构。在维度小于8的情况下，这种对应关系将超铅笔的集合映射到流形上所有辛形的稠密子集上(直到保序和尺度)。类似的陈述似乎在更高的维度上也是可能的，并将提供那些流形上允许辛结构的完整的拓扑特征。如果这种对应的纤维可以在拓扑上指定，则结果将是对所有闭流形上的所有辛结构的稠密子集的纯拓扑描述。除了这项研究外，还将用Legendrian Kirby微积分等方法研究Stein曲面及其接触的三维流形边界。还将研究奇异的R^4‘S和其他光滑4-流形的拓扑。n-流形是指在小区域中看起来像是维的欧几里德空间的空间。点、曲线和曲面分别是0、1和2维的流形。我们生活的空间是一个三维流形，而宇宙(时空)是一个四维流形。令人惊讶的是，尽管最近几年取得了重大进展，但三维和四维流形比它们的高维流形要少得多。流形上的辛结构和接触结构是通过经典物理(分别是哈密顿力学和光学)发现的，但现在它们在量子物理、复分析、微分几何和拓扑学等不同领域中都被认为是重要的。例如，在三维空间中摆动的刚性摆决定了辛流形。浮子在球面上运动，因此它的位置由2个变量(球体上的纬度和经度)指定。为了完整地指定系统的状态，还必须包括bob的时刻。在任何位置，它的动量都与球体相切，因此由另外两个变量来确定。因此，系统的状态集是一个4-流形(由4个变量局部指定)。这个四维流形有一个特殊的辛结构，它的作用是将每个位置变量与相应的动量变量联系起来。这种联系最终导致了描述钟摆在特定力场影响下的运动的哈密顿方程。流形以及它们的辛结构和接触结构的出现还有很多其他的情况。虽然我们对这种结构的了解近年来有了很大的进步，但一些最基本的问题仍有待回答。哪些流形有辛或接触结构，给定流形有多少这样的结构？有多少个4-流形(如果有的话)满足给定的描述？诸如此类的问题构成了这个项目的基础。

AbstractAward: DMS-0102922Principal Investigator: Robert E. GompfThis project focuses on constructing symplectic, contact and4-dimensional manifolds. The Principal Investigator has defineda topological structure called a "hyperpencil" on aneven-dimensional closed manifold, generalizing a linear system ofcurves on a complex algebraic manifold. He has shown that anyhyperpencil canonically determines a symplectic structure on themanifold. In dimensions less than 8, this correspondence mapsthe set of hyperpencils onto a dense subset of all symplecticforms on the manifold (up to isotopy and scale). A similarstatement seems likely in higher dimensions, and would provide acomplete topological characterization of those manifoldsadmitting symplectic structures. If the fibers of thiscorrespondence can be topologically specified, the result will bea purely topological description of a dense subset of allsymplectic structures on all closed manifolds. In addition tothis investigation, Stein surfaces and their contact 3-manifoldboundaries will be studied by methods such as Legendrian Kirbycalculus. The topology of exotic R^4's and other smooth4-manifolds will also be investigated.An n-manifold is a space that in small regions looks just liken-dimensional Euclidean space. Points, curves and surfaces aremanifolds of dimensions 0,1 and 2, respectively. The space inwhich we live is a 3-manifold, and the universe (space-time) is a4-manifold. Surprisingly, 3- and 4-manifolds are much less wellunderstood than their higher-dimensional counterparts, althoughmajor progress has been made in recent years. Symplectic andcontact structures on manifolds were discovered through classicalphysics (Hamiltonian mechanics and optics, respectively), butthey are now seen to be important in such diverse areas asquantum physics, complex analysis, differential geometry andtopology. For example, a rigid pendulum swinging in3-dimensional space determines a symplectic manifold. The bobmoves on a spherical surface, so its position is specified by 2variables (latitude and longitude on this sphere). To completelyspecify the state of the system, one must also include themomentum of the bob. At any position, its momentum is tangent tothe sphere and thus specified by two more variables. Hence, theset of states of the system is a 4-manifold (locally specified by4 variables). This 4-manifold has a special symplectic structurewhose role is to link each position variable to the correspondingmomentum variable. This linkage ultimately results in Hamilton'sequations describing the motion of the pendulum under theinfluence of a specified force field. There are many othersituations in which manifolds and their symplectic and contactstructures arise. While our understanding of such structures hasadvanced enormously in recent years, some of the most basicquestions remain to be answered. Which manifolds have symplecticor contact structures, and how many such structures are there ona given manifold? How many 4-manifolds (if any) are theresatisfying a given description? Questions such as these form thebasis of this project.