The Topology of Manifolds of Dimensions 3 and 4

[摘要]奖项：DMS 0229035首席研究员：Cameron gordon该项目是一个三维和四维拓扑的会议。对于大于或等于5维的流形的令人满意的计算在20世纪60年代末得到了实现。从那时起，注意力自然集中在第三和第四维度上。尽管取得了巨大的进步（Thurston, Freedman, Donaldson, Jones, Witten……），但在这些低维度上的许多主要问题仍然没有得到解决，例如经典的三维庞加莱猜想及其光滑的四维类比。1连通4流形的拓扑分类是由Freedman在20世纪80年代初实现的，但在光滑情况下甚至没有一个猜想图。相比之下，瑟斯顿的几何猜想为所有3流形提供了一个美丽而连贯的描述，但还远远没有建立起来。在20世纪80年代中期，琼斯发现了他的多项式连杆不变量，与威滕的工作一起，导致了量子物理学方法的三维拓扑的引入。目前低维拓扑研究的现状是研究方向和方法很多，但对它们之间的关系了解甚少。因此，在3维中，我们有双曲几何、叶状和层状、法线曲面、组合几何方法、量子不变量、有限型不变量、花同调等等。，而在第四维有唐纳森和塞伯格-威滕理论，辛结构，....会议的目的是讨论这些主题，以及它们之间的联系，并为不同领域的专家提供一个互动和交流思想的论坛。几何拓扑学旨在理解n维流形，这些流形是局部看起来像普通的n维欧几里得空间（其点由n个坐标描述）的对象，但其整体结构可能相当复杂。维度3和维度4的情况特别有趣，它们是我们的空间和时空宇宙的维度，一个惊人的事实是，正是这些维度在数学上是反常的。在过去的25年里，三维和四维拓扑领域有了许多发展和相当大的进步，但是，在这两个维度上，仍然缺乏一个完整的图景。会议的目的之一是鼓励目前在低维拓扑中所追求的几个不同方面和技术的专家之间的互动和合作，特别是那些在维度3和维度4中工作的人之间的互动和合作，这两个领域的方法往往是完全不同的，但它们之间有一些联系的暗示。目前这个学科的不同状态使得人们开始研究低维拓扑很难对该领域有一个很好的概述，因此，通过鼓励研究生和博士后研究人员的参与，我们希望这次会议也能为年轻的研究人员提供一个机会，让他们对目前的知识状态、开始追求的方向和主要的开放性问题有一个广阔的视角。受邀的演讲者包括一些世界领先的三维和四维拓扑学家，他们的专业知识涵盖了这两个领域的所有主要方面。

AbstractAward: DMS 0229035Principal Investigator: Cameron GordonThe project is a conference in 3- and 4-dimensional topology.Satisfactory accounts of manifolds of dimension greater than or equalto 5 were achieved by the late 1960's. Since then, attention hasnaturally focused on dimensions 3 and 4. Despite dramatic advances(Thurston, Freedman, Donaldson, Jones, Witten...), many major problemsin these low dimensions remain unsolved, for example the classical3-dimensional Poincare Conjecture and its smooth 4-dimensional analog.The topological classification of 1-connected 4-manifolds was achievedby Freedman in the early 1980's, but in the smooth case there is noteven a conjectural picture. By contrast, Thurston's GeometrizationConjecture provides a beautiful and coherent description of all3-manifolds, but is far from established. In the mid 1980's, Jones'discovery of his polynomial link invariant, together with work ofWitten, led to the introduction into 3-dimensional topology of methodsfrom quantum physics. The current situation in low-dimensionaltopology is that there are many different directions and methods, butlittle understanding of the relations between them. Thus in dimension3 we have hyperbolic geometry, foliations and laminations, normalsurfaces, combinatorial geometric methods, quantum invariants, finitetype invariants, Floer homology,..., while in dimension 4 there arethe Donaldson and Seiberg-Witten theories, symplectic structures,....The aim of the conference is to address these topics, and theconnections between them, and to provide a forum for interaction andexchange of ideas between experts in the different areas.Geometric topology aims to understand n-dimensional manifolds, whichare objects that locally look like ordinary n-dimensional Euclideanspace (whose points are described by n co-ordinates), but whose globalstructure might be quite complicated. The cases of dimensions 3 and 4are particularly interesting, being the dimensions of our spatial andspatial-temporal universes, and it is a striking fact that it isprecisely these dimensions that are mathematically anomalous. Therehave been many developments and considerable progress in the fields of3- and 4-dimensional topology over the last twenty-five years, but, inboth dimensions, a complete picture is still lacking. One of the aimsof the conference is to encourage interaction and collaborationbetween experts in the several different aspects and techniques thatare currently being pursued in low-dimensional topology, and inparticular between those people working in dimension 3 and those indimension 4, where the methods in the two areas tend to be quitedifferent, but where there are several hints of connections betweenthem. The current disparate state of the subject makes it difficultfor people beginning research in low-dimensional topology to get agood overview of the area, and so by encouraging the participation ofgraduate students and postdoctoral researchers, we intend that theconference should also provide an opportunity for young researchers toget a broad perspective of the present state of knowledge, directionsbegin pursued, and the main open problems. The invited speakersinclude some of the world's leading 3- and 4-dimensional topologists,whose expertise together covers all the major aspects of the twofields.