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

三维拓扑未来方向会议；

• 批准号: 0455864
• 负责人: Joel Hass
• 金额: $ 2万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-01-01 至 2006-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0455864&HistoricalAwards=false
• 关键词: Conference; Future; Directions; Dimensional; Topology

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?

我们提议召开一次会议，会议的主题是评估三维几何和拓扑学发展的未来方向。会议的目的是回顾三维流形理论最近的重大发展，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阿伯举行。会议计划于2005年5月6日至5月9日举行。我们预计有50至80名与会者，其中约三分之一将是研究生。一些讲座将专门为研究生量身定制。 Grisha Pereleman关于Ricci流下三维流形上度规的演化的工作仍然被数学界所吸收和审查。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研究中使用的方向和技术。这次会议的目的不是关注佩雷尔曼方法的细节，而是关注它们的含义。 还有哪些问题可以用里奇流动技术来研究？ 3-流形理论中的哪些传统定理和结果被几何化了？如果佩雷尔曼的几何化结果成立，什么问题变得重要。对相关领域，如几何群论和三维流形中曲面的微分几何有什么影响？随着这个主要问题的解决，三维流形的整个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方法的未来可能发展是什么，这些领域主要基于3-流形理论的思想。 Stallings首先，最近Rips和Sela证明了3-流形在许多方面是一般群的原型。Ozsvath和Szabo发展了一个与封闭三维流形的Heegaard分解相关的全纯曲线理论，导致了他们所谓的“Heegaard Floer同调”。这对理解三元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源性的工作。四维技术，如规范理论和相关的主题已经有一些应用到三维流形的问题。因此Kronheimer和Mrowkagive证明了性质P (also由几何化暗示）。这些技术可能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，已经发展了许多纽结和三维流形的不变量，包括纽结多项式，有限型不变量等。这些在理解三维流形方面有多大用处？它们与流形的几何性质（如双曲体积）有什么联系？6.是否有重要的新兴机会，应用3流形理论的方法，如计算几何和宇宙学以及领域，如计算机辅助设计和计算复杂性。三维流形理论和这些学科之间有什么联系？