Symplectic, Contact and Low-dimensional Topology

辛、接触和低维拓扑

• 批准号: 0102922
• 负责人: Robert Gompf
• 金额: $ 31.93万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-15 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0102922&HistoricalAwards=false
• 关键词: Symplectic; Contact; Low; dimensional; Topology

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.

摘要奖：DMS-0102922主要研究者：Robert E.这个项目的重点是构造辛流形、切触流形和四维流形。 主要研究者定义了一个拓扑结构称为“超铅笔”的偶数维封闭流形，推广了一个线性系统的曲线上的复杂代数流形。 他已经表明，任何超铅笔规范确定辛结构上的流形。 在小于8维的情况下，这种对应关系将超铅笔集映射到流形上所有辛形式的稠密子集上（直到合痕和标度）。 类似的陈述在高维中似乎是可能的，并且将提供那些流形的完整拓扑特征，这些流形具有辛结构。 如果这种对应的纤维可以拓扑地指定，那么结果将是所有闭流形上的全辛结构的稠密子集的纯拓扑描述。 除此之外，Stein曲面及其接触3流形边界将通过Legendrian Kirby演算等方法进行研究。 奇异R^4流形和其他光滑4-流形的拓扑也将被研究。n-流形是一个在小区域看起来像维欧氏空间的空间。 点、曲线和曲面分别是0维、1维和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，尽管近年来已经取得了重大进展，但3-和4-流形比它们的高维对应物要少得多。 流形上的辛结构和接触结构是通过经典物理学（分别是哈密顿力学和光学）发现的，但它们现在被认为在量子物理、复分析、微分几何和拓扑学等不同领域都很重要。 例如，一个在三维空间中摆动的刚性摆决定了一个辛流形。 这个球在一个球面上移动，所以它的位置由2个变量（这个球面上的纬度和经度）指定。 为了完整地说明系统的状态，还必须包括摆锤的动量。 在任何位置，它的动量都与球体相切，因此由另外两个变量指定。 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4-流形有一个特殊的辛结构，其作用是将每个位置变量与相应的动量变量联系起来。 这种联系最终导致在汉密尔顿方程描述的运动下，一个指定的力场的影响下的钟摆。 还有许多其他的情况下，流形和他们的辛和contactstructure出现。 虽然近年来我们对这种结构的理解有了很大的进步，但一些最基本的问题仍然有待回答。 哪些流形具有辛或切触结构，给定流形上有多少个这样的结构？ 有多少个四维流形（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。