Mathematical Sciences: Symplectic and Contact Structures and Low Dimensional Topology

9625654 Gompf这个项目处理辛几何和接触几何与低维拓扑之间的相互作用。研究的一个主要焦点是Stein曲面的拓扑结构，或复杂的仿射变体，也是光滑的4流形。它们是自动的Kaehler，因此是辛的，当它们的拓扑是有限的，它们的自然边界是接触3流形。首席研究员正在研究Stein曲面的柄分解，解决以下问题：哪些3-流形作为Stein曲面的边界出现，即哪些3-流形允许全纯可填充的接触结构？在给定的3流形上，哪些2平面域的同伦类可以用这种方式实现？一个给定的3流形可以有多少个斯坦因曲面作为边界？通过已知的辛流形拓扑和4流形拓扑之间的联系（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（如经典结的切片属）有关。现代数学和物理学的基石之一是“流形”的概念，也就是说，一个局部看起来像普通欧几里得空间的空间。曲线是一维流形，而曲面是二维的。我们生活的空间是三维的，而宇宙，时空，是一个四流形的例子。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，我们熟悉的第3和第4维度是迄今为止最不为人所知的——甚至比第5和更高的维度更不为人所知，因为在第5和更高的维度上有更多的工作空间。直到15年前，人们对4流形的形状几乎一无所知，当时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式增长。在过去的几年里，很明显，低维（即3维和4维）的拓扑结构与对称几何和接触几何密切相关。这些后一个领域是从经典物理学（哈密顿力学）中产生的，近年来它们迅速发展成为独立的学科，与其他数学领域（例如代数几何）有联系。自该领域取得初步突破以来，首席研究员一直在研究4-流形，他现在正在研究低维拓扑与辛几何和接触几何之间的联系。利用拓扑技术，他最近扩展了辛几何的范围，表明辛结构比原先认为的要普遍得多。他目前正在制作接触结构的新例子，以及分析它们的新工具。他预计，对这三个领域之间相互作用的持续研究将导致对这三个领域的进一步了解。* * *

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. ***