3-Manifolds After Perelman; March 2006; Edinburgh, UK

项目编号：dms -0601251项目负责人：Cameron McA。这个项目是一个关于三维拓扑的研讨会，将于2006年3月在英国爱丁堡的国际数学科学中心举行。瑟斯顿的几何化猜想（Thurston’s Geometrization Conjecture）指导了三维拓扑研究近三十年，现在已经成立，因此有必要为该主题的未来研究制定一个计划。这就是研讨会的目标。该项目将围绕Jeff Brock、Dave Gabai、Marc Lackenby、PeterOzsvath和Peter Shalen的五个系列讲座展开，每个讲座时长3小时。此外，还将邀请其他主要研究人员进行9场1小时的演讲。为了阐明几何化之外的三维流形理论，本文将讨论三维拓扑学中最近的重要发展，包括：单调和终结层叠猜想的证明、heeggaard Floer同调理论、虚哈肯猜想的各种方法，以及三维拓扑学与双曲群理论之间的联系。三维拓扑学是纯数学一般领域中一个庞大而活跃的课题。它的目标是研究所有可能的三维“空间”，以及可以放置在其中的各种物体，比如结。在过去30年左右的时间里，这门学科有了许多令人兴奋的发展，揭示了它与许多其他数学分支以及量子物理学的联系，甚至与DNA的行为（在结的情况下）的联系。在此期间，这一学科的指路明灯是威廉·瑟斯顿（william Thurston）在20世纪70年代中期提出的几何化猜想（geometric ization Conjecture），该猜想断言，由罗巴切夫斯基（Lobachevsky）在19世纪发现的非欧几里得几何——双曲几何在描述这些三维空间中起着关键作用。这个猜想最近被Grisha Perelman证明了。因此，似乎是时候对这个问题进行评估，找出仍然存在的主要问题，并指出未来工作的潜在技术和方向。该计划的核心是由该领域的五位国际知名专家主持的五个系列讲座，每次3小时。预计这将使研讨会对年轻研究人员特别有用。会议网址：https://www.icms.org.uk/meetings/2006/3-manifolds/index.html。

AbstractAward: DMS-0601251Principal Investigator: Cameron McA. Gordon, Alan W. ReidThe project is a workshop on 3-dimensional topology that will be held atthe International Centre for Mathematical Sciences in Edinburgh, UK, inMarch, 2006. Thurston's Geometrization Conjecture, which has guidedresearch in 3-dimensional topology for almost three decades, has nowbeen established, and consequently there is a need to set out a programfor future research in the subject. This is the objective of theworkshop. The program will be centered around five series of three1-hour lectures, given by Jeff Brock, Dave Gabai, Marc Lackenby, PeterOzsvath, and Peter Shalen. There will also be nine additional invited1-hour talks by other leading researchers. Among the recent importantdevelopments in 3-dimensional topology that will be discussed, with aview to illuminating the theory of 3-manifolds beyond Geometrization,are: the proofs of the Tameness and Ending Lamination Conjectures,Heegaard Floer homology theory, various approaches to the virtual HakenConjecture, and the connection between 3-dimensional topology and thetheory of word hyperbolic groups.3-dimensional topology is a large and active subject within the generalfield of pure mathematics. Its goal is the study of all possible3-dimensional "spaces", and of the various objects that can be situatedin them, such as knots. Over the last 30 years or so this subject hasseen many exciting developments, revealing connections to many otherbranches of mathematics as well as quantum physics and (in the case ofknots) even to the behavior of DNA. The guiding light of the subjectduring this period has been the Geometrization Conjecture, put forwardWilliam Thurston in the mid 1970's, which asserts that hyperbolicgeometry, the non-euclidean geometry discovered by Lobachevsky in the19th century, plays a key role in the description of these 3-dimensionalspaces. This conjecture was recently proved by Grisha Perelman. It seemstimely, then, to take stock of the subject, identifying the majorproblems that remain and indicating potential techniques and directionsfor future work. The core of the program will be five series of three1-hour lectures, given by five internationally renowned experts in thesubject. It is expected that this will make the workshop especiallyuseful to young researchers.The conference web site ishttp://www.icms.org.uk/meetings/2006/3-manifolds/index.html.