Mathematical Sciences: Symplectic and Contact Structures and Low Dimensional Topology

数学科学：辛和接触结构以及低维拓扑

• 批准号: 9625654
• 负责人: Robert Gompf
• 金额: $ 3.92万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 1999-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9625654&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Symplectic; Contact; Structures

9625654 Gompf This project deals with the interplay between symplectic and contact geometry and low-dimensional topology. A main focus of the investigation is on the topology of Stein surfaces, or complex affine varieties that are also smooth 4-manifolds. These are automatically Kaehler, hence symplectic, and when their topology is finite, they have natural boundaries that are contact 3-manifolds. The principal investigator is studying Stein surfaces by means of their handle decompositions, addressing questions such as the following: Which 3-manifolds arise as boundaries of Stein surfaces, i.e., which 3-manifolds admit holomorphically fillable contact structures? Which homotopy classes of 2-plane fields on a given 3-manifold can be realized in this manner? How many Stein surfaces can have a given 3-manifold as boundary? Through the known connections between symplectic and 4-manifold topology (such as Seiberg-Witten theory), this should relate to the classification problem for smooth 4-manifolds as well as to other problems, such as the slice genus of classical knots. One of the cornerstones of modern mathematics and physics is the notion of a "manifold," that is, a space that looks locally like ordinary Euclidean space. Curves are 1-dimensional manifolds, and surfaces are 2-dimensional. The space in which we live is 3-dimensional, and the universe, space-time, is an example of a 4-manifold. A basic question of topology is what shapes such spaces can have. Ironically, the familiar dimensions 3 and 4 are by far the least understood -- even less understood than dimensions 5 and higher, where there is more "room" in which to work. Almost nothing was known about the shapes of 4-manifolds until 15 years ago, when the revolutionary breakthroughs of Freedman and Donaldson led to an explosive growth in knowledge and techniques. In the last few years, it has become apparent that topology in low dimensions (i.e., 3 and 4) is intimately linked to sympl ectic and contact geometry. These latter fields emerged from classsical physics (Hamiltonian mechanics), and in recent years they have become rapidly developing independent subjects with connections to various other areas of mathematics (for example, algebraic geometry). The principal investigator has been studying 4-manifolds since the initial breakthroughs in the field, and he is now examining the connections between low-dimensional topology and symplectic and contact geometry. Using topological techniques, he recently expanded the scope of symplectic geometry by showing that symplectic structures are much more common than originally believed. He is currently producing new examples of contact structures, and new tools for analyzing them. He anticipates that continued investigation of the interplay between these three areas should lead to further insight into all three. ***

小行星9625654 这个项目涉及辛和 接触几何和低维拓扑。 调查的一个主要重点是斯坦表面的拓扑结构，或复杂的仿射品种，也是光滑的4-流形。 它们自动地是Kaehler，因此是辛的，并且当它们的拓扑是有限的时，它们具有接触3-流形的自然边界。 主要研究人员正在研究斯坦曲面的处理分解的手段，解决问题，如以下：3流形出现的边界斯坦曲面，即，哪些三维流形允许全纯可填充的接触结构？ 在给定的3-流形上，2-平面场的哪些同伦类可以用这种方式实现？ 有多少Stein曲面可以以一个给定的三维流形为边界？ 通过辛和4-流形拓扑之间的已知联系（如Seiberg-Witten理论），这应该与光滑4-流形的分类问题以及其他问题有关，如经典纽结的切片亏格。 现代数学和物理学的基石之一是“流形”的概念，即局部看起来像普通欧几里得空间的空间。 曲线是一维流形，曲面是二维流形。 我们生活的空间是三维的，宇宙，时空，是四维流形的一个例子。 拓扑学的一个基本问题是这样的空间可以有什么形状。 具有讽刺意味的是，我们所熟悉的3维和4维是迄今为止最不被理解的--甚至比5维和更高的维度更不被理解，在5维和更高的维度中有更多的工作空间。 直到15年前，弗里德曼和唐纳森的革命性突破导致了知识和技术的爆炸性增长，人们对四维流形的形状几乎一无所知。 在过去的几年中，很明显，低维拓扑（即，3和4）是密切相关的辛几何和接触几何。 这些领域出现了经典物理学（哈密顿力学），近年来，他们已经成为迅速发展的独立学科与其他领域的数学（例如，代数几何）的联系。 主要研究者一直在研究4-流形，因为在该领域的初步突破，他现在正在研究低维拓扑和辛几何和接触几何之间的联系。 利用拓扑技术，他最近扩大了辛几何的范围，表明辛结构比原来认为的要普遍得多。 他目前正在制作接触结构的新例子，以及用于分析它们的新工具。 他预计，继续调查这三个领域之间的相互作用，应能进一步深入了解这三个领域。 ***