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

点击翻译按钮获取中文摘要

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?