Conference on Future Directions in 3-Dimensional Topology; May 6-9, 2005; Ann Arbor, MI

我们提议召开一次以评估三维几何和拓扑的未来发展方向为主题的会议。会议的目的是回顾最近3流形理论的重要发展，并讨论它们将如何影响未来的研究过程。会议将在密歇根大学安娜堡分校举行。会议的计划日期是2005年5月6日至5月9日。我们预计有50至80人参加，其中约三分之一是研究生。一些讲座将专门为研究生量身定制。可能影响三维几何和拓扑学未来的几个重要的最新发展。Grisha Pereleman关于Ricci流下三维流形度规演化的工作仍在被数学界吸收和回顾。然而，很明显，它有可能从根本上改变三维拓扑和相关领域研究中使用的方向和技术。这次会议的目的不是关注Pereleman方法的细节，而是关注它们的含义。还有哪些问题可以用里奇流技术来研究？三流形理论的哪些传统定理和结果被纳入几何化？如果Pereleman的几何化结果成立，哪些问题会变得重要？对相关领域如几何群论和3流形曲面的微分几何有什么影响？随着这个主要问题的解决，整个3-流形领域会变得不那么重要，还是会变得更加重要？几何群论和群论中的拓扑方法在很大程度上是基于3流形理论的思想的领域中可能的未来发展是什么？首先是Stallings，最近Rips和Sela已经证明了3流形在很多方面都是一般有限表示群的原型。Ozsvath和Szabo发展了一个与封闭3流形Heegaard分解相关的全纯曲线理论，导致了他们所谓的“Heegaard花同调”。这对理解3-流形和其他领域有什么影响？与此相关的是Khovanov和Rasmussen定义和应用琼斯多项式同调的工作。四维技术如规范理论和相关主题已经在三维流形问题中得到了一些应用。因此，Kronheimer和mrowk给出了性质P（也由几何化隐含）的证明。这些技术可能对三流形理论有什么进一步的影响？在过去的二十年里，结和3流形的不变量得到了发展，包括结多项式、有限型不变量等。这些对理解3流形有多大帮助？它们与流形的几何性质有什么联系，比如双曲体积？将3流形理论的方法应用到计算几何和宇宙学等领域以及计算机辅助设计和计算复杂性等领域是否有重大的新机会？三流形理论和这些学科之间正在发展什么联系？

We are proposing a conference centered on the theme of evaluating futuredirections in the development of 3-dimensional geometry and topology.The purpose of the conference is to review significant recent developmentsin the theory of 3-manifolds and to discuss how they will affect the courseof future research. The conference will be held at the University ofMichigan at Ann Arbor. The planned dates for the conference are May 6 to May 9, 2005.We expect 50 to 80 participants, about one third of whom will be graduate students.Some talks will be tailored specifically for graduate students.Several important recent developments that are likely to influence thefuture of 3-dimensional geometry and topology.1. The work of Grisha Pereleman on the evolution of a metric on a 3-dimensional manifold under the Ricci flow is still being absorbed and reviewed by the mathematical community. It is clear however that it has the potential to radically change the directions and techniques used in the study of 3-dimensional topology and related fields. The aim for this conference is not to focus on the details of Pereleman's methods but rather on their implications. What other problems are amenable to study by Ricci flow techniques? Which of the traditional theorems and results of 3-manifold theory become subsumed in geometrization? What questions become important if Pereleman's geometrization results hold up. What are the implications to related areas such as geometric group theory and the differential geometry of surfaces in 3-manifolds? Will the entire field of 3-manifolds become less important with this major problem solved, or will go on to have even greater importance?2. What are the likely future developments in Geometric Group Theory, and inTopological Methods in Group Theory, areas largely based on ideas from 3-manifoldtheory. Stallings first, and more recently Rips and Sela have shown that 3-manifoldsare prototypes in many ways for general finitely presented groups.3. Ozsvath and Szabo have developed a theory of holomorphic curves associatedto Heegaard decompositions of closed 3-manifolds, leading to what they call"Heegaard Floer homology". What impact will this have in understanding 3-manifoldsand to other areas? Related to this is the work of Khovanov and Rasmussen definingand applying a Jones polynomial homology.4. Four dimensional techniques such as gauge theory and related topics have hadsome applications to problems in 3-manifolds. Thus Kronheimer and Mrowkagive a proof of Property P (also implied by geometrization). What furtherimpact on 3-manifold theory are such techniques likely to have?5. Many invariants of knots and 3-manifolds have been developed over the last two decades, including knot polynomials, finite type invariants etc. How useful are these in understanding 3-manifolds? What ties do they have to geometric properties of manifolds such as hyperbolic volume?6. Are there significant emerging opportunities to apply the methods of 3-manifoldtheory to areas such as Computational Geometry and Cosmology as well as fieldssuch as Computer Aided Design and Computational Complexity. What ties are developing between the theory of 3-manifolds and such disciplines?