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

我们提议召开一次会议，主题是评估三维几何和拓扑学的未来发展方向。会议的目的是回顾三维流形理论的重要最新发展，并讨论它们将如何影响未来的研究进程。会议将在密歇根大学安娜堡分校举行。会议计划于2005年5月6日至5月9日举行。我们预计将有50到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做。几个可能影响三维几何和拓扑学未来的重要最新发展1。Grisha Pereleman关于Ricci流下三维流形上度量演化的工作仍在被数学界吸收和审查。然而，很明显，它有可能从根本上改变三维拓扑学和相关领域的研究方向和技术。这次会议的目的不是关注佩雷尔曼方法的细节，而是它们的影响。利玛窦的流动技术还有哪些值得研究的问题？传统的3-流形理论中的哪些定理和结果被归入几何化？如果佩雷尔曼的几何化结果成立，什么问题就变得重要起来。这对几何群论和三维流形中曲面的微分几何等相关领域有何启示？随着这个重大问题的解决，整个三维流形领域会变得不那么重要，还是会变得更加重要？2.几何群论和群论中的拓扑学方法未来可能的发展是什么，这两个领域主要基于三维流形理论的思想。首先是Stallings，最近Rips和Sela已经证明了3-流形在许多方面都是一般有限表示群的原型。Ozsvath和Szabo发展了与闭3-流形的Heegaard分解相关的全纯曲线理论，得到了他们所说的“Heegaard Floer同调”。这将对理解三维流形和其他领域产生什么影响？与此相关的是Khovanov和Rasmussen定义和应用Jones多项式同调的工作。规范理论等四维技术在三维流形问题中有一定的应用。因此，Kronheimer和Mrowkagve证明了性质P(也隐含在几何化中)。这些技巧可能会对三维流形理论产生什么进一步的影响？5.在过去的二十年里，已经发展了许多关于纽结和三维流形的不变量，包括纽结多项式，有限类型不变量等。这些对理解三维流形有多大帮助？它们与流形的几何性质(如双曲体积)有什么联系？6.将三维流形理论的方法应用于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存在重大的新兴机会？3-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?