Symplectic, Contact and Low-Dimensional Topology

9802533 Gompf这个项目的重点是构造辛流形、接触流形和四维流形。首席研究员正试图在任何具有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。他已经解决了四维的问题；所有维度的解，连同Donaldson最近的工作，将给出那些承认辛结构的流形的完整拓扑特征：它们正是那些承认Lefschetz铅笔的流形。（问题的关键在于如何恰当地用这种一般性来定义Lefschetz铅笔。）一个相关的主题涉及Stein曲面（复维数2的仿射解析变体）。这里有许多基本的存在性问题，例如，什么样的3-流形会作为Stein曲面的边界出现，有多少Stein曲面以给定的3-流形作为边界，在给定的3-流形上有多少接触结构是由这样的Stein曲面引起的？这些问题适用于首席研究员目前的技术，即在S1 x S2拷贝的连通总和中，将斯坦曲面描述为勒让德连杆图上的柄体。目前，首席研究员正在与a . Stipsicz一起撰写关于4-流形和Kirby微积分的研究生论文，并指导两名博士生和一名NSF博士后。n流形是一个在小尺度上看起来像n维欧几里得空间的空间。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。正因为如此，流形在数学、物理，以及在某种程度上，其他各种科学中扮演着核心角色。辛结构和接触结构首先出现在经典物理学（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。确定哪些流形可以有五环或接触结构是一个基本的未解决的问题。这个项目的目的是一个完整的答案，在辛的情况下，并进一步了解接触情况。该项目还应阐明关于3-和4-流形可能形状的基本未解决问题。＊＊＊

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. ***