The Topology of Manifolds of Dimensions 3 and 4

3 维和 4 维流形的拓扑

• 批准号: 0229035
• 负责人: Cameron Gordon
• 金额: $ 2.35万
• 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: 2003
• 资助国家: 美国
• 起止时间: 2003-04-01 至 2005-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0229035&HistoricalAwards=false
• 关键词: Topology; Manifolds; Dimensions

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.

摘要奖：DMS 0229035首席研究员：卡梅隆·戈登该项目是一场三维和四维拓扑学的会议。20世纪60年代末，S对维度大于或等于5的流形做出了令人满意的描述。从那时起，人们的注意力自然集中在3维和4维上。尽管取得了巨大的进展(瑟斯顿、弗里德曼、唐纳森、琼斯、威腾……)，但这些低维的许多主要问题仍然没有解决，例如经典的3维庞加莱猜想及其光滑的4维类比。弗里德曼在20世纪80年代初实现了1连通4维流形的拓扑分类，但在光滑的情况下，甚至还有一幅猜想图。相比之下，瑟斯顿的几何化猜想提供了对所有3-流形的美丽而连贯的描述，但还远远没有建立起来。在1980年代中期，S，琼斯发现了他的多项式链接不变量，与Witten的工作一起，导致了量子物理方法的三维拓扑的引入。低维拓扑学的现状是有许多不同的方向和方法，但对它们之间的关系却知之甚少。因此，在3维空间中，我们有双曲几何、叶和薄片、正规曲面、组合几何方法、量子不变量、有限类型不变量、Floer同调等，而在4维空间中，我们有Donaldson和Seiberg-Witten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之间的互动和思想交流提供一个论坛。几何拓扑学旨在了解n维流形，这些流形局部看起来像普通的n维欧氏空间(其点由n坐标描述)，但其全局结构可能相当复杂。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，一个引人注目的事实是，正是这些维度在数学上是反常的。在过去的二十五年里，三维和四维拓扑学领域已经有了许多发展和相当大的进步，但在这两个维度上，仍然缺乏完整的图景。会议的目的之一是鼓励专家之间在几个不同方面和目前在低维拓扑学中追求的技术之间的互动和合作，特别是那些在维度3和维度4工作的人之间的互动和合作，在这两个领域的方法往往是完全不同的，但他们之间有一些联系的迹象。该学科目前的不同状态使得开始研究低维拓扑学的人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，我们打算会议也应该为年轻的研究人员提供一个机会，让他们对当前的知识状况、开始追求的方向和主要的未决问题有一个广泛的视角。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。