Symplectic, Contact and Low-Dimensional Topology

辛、接触和低维拓扑

• 批准号: 9802533
• 负责人: Robert Gompf
• 金额: $ 7.26万
• 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: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2002-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802533&HistoricalAwards=false
• 关键词: Symplectic; Contact; Low; Dimensional; Topology

9802533 Gompf The focus of this project is on constructing symplectic, contact, and 4-dimensional manifolds. The principal investigator is attempting to construct a symplectic structure on any manifold endowed with a Lefschetz pencil (a kind of singular fibration originally discovered in algebraic geometry). He has already solved the problem in dimension 4; a solution in all dimensions, together with recent work of Donaldson, will give a complete topological characterization of those manifolds admitting symplectic structures: They are precisely those that admit Lefschetz pencils. (A key part of the problem is suitably to define Lefschetz pencils in this generality.) A related topic concerns Stein surfaces (affine analytic varieties of complex dimension 2). There are many fundamental existence questions here, for example, what 3-manifolds arise as boundaries of Stein surfaces, how many Stein surfaces have a given 3-manifold as boundary, and how many contact structures are induced on a given 3-manifold by such Stein surfaces? These questions are amenable to the principal investigator's current techniques of describing Stein surfaces as handlebodies on Legendrian link diagrams in connected sums of copies of S1 x S2. The principal investigator is also currently writing a graduate text on 4-manifolds and Kirby calculus with A. Stipsicz and is advising two PhD students and an NSF Postdoctoral Fellow. An n-manifold is a space that on small scales looks like Euclidean space of dimension n. For example, the space in which we live is a 3-manifold, and the universe (space-time) is a 4-manifold. As such, manifolds play a central role in mathematics, physics, and to some extent, various other sciences. Symplectic and contact structures first appeared in classical physics (Hamiltonian dynamics) but have since become important in pure mathematics (geometry, topology, analysis) as well. It is a basic unsolved problem to determine which manifolds can have s ymplectic or contact structures. This project aims for a complete answer in the symplectic case and for additional understanding of the contact case. The project should also shed light on basic unsolved problems regarding the possible shapes of 3- and 4-manifolds. ***

9802533 Gompf这个项目的重点是构造辛流形、接触流形和4维流形。主要研究人员试图在任何具有Lefschetz铅笔(一种最初在代数几何中发现的奇异纤维)的流形上构造辛结构。他已经解决了4维的问题；所有维的解，加上Donaldson最近的工作，将给出那些允许辛结构的流形的完整的拓扑特征：它们正是那些允许Lefschetz铅笔的流形。(问题的一个关键部分是在这种一般性的情况下恰当地定义Lefschetz铅笔。)一个相关的主题涉及Stein曲面(复数维2的仿射解析簇)。这里有许多基本的存在问题，例如，哪些三维流形是作为Stein曲面的边界产生的，有多少Stein曲面具有给定的三维流形作为边界，以及由这样的Stein曲面在给定的三维流形上诱导了多少接触结构？这些问题适用于首席研究人员目前的技术，即在S1 x S2的副本的连通和中将Stein曲面描述为Legendrian链接图上的手柄。这位首席研究员目前还在与A.Stipsicz一起撰写一篇关于4-流形和Kirby微积分的研究生论文，并为两名博士生和一名NSF博士后研究员提供建议。N-流形是一个在小尺度上看起来像n维欧几里德空间的空间。例如，我们生活的空间是一个3-流形，而宇宙(时空)是一个4-流形。因此，流形在数学、物理以及某种程度上的各种其他科学中扮演着核心角色。辛结构和接触结构最早出现在经典物理学(哈密顿动力学)中，但后来在纯数学(几何、拓扑、分析)中也变得重要起来。确定哪些流形具有S辛结构或接触结构是一个基本的悬而未决的问题。这个项目的目的是在辛情形下得到一个完整的答案，并对接触情形有更多的了解。该项目还应阐明有关三维和四维管汇可能形状的基本悬而未决的问题。***