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

University Of California-Davis, Davis CA

Investigators

Abstract

We are proposing a conference centered on the theme of evaluating future directions in the development of 3-dimensional geometry and topology. The purpose of the conference is to review significant recent developments in the theory of 3-manifolds and to discuss how they will affect the course of future research. The conference will be held at the University of Michigan 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 the future 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 in Topological Methods in Group Theory, areas largely based on ideas from 3-manifold theory. Stallings first, and more recently Rips and Sela have shown that 3-manifolds are prototypes in many ways for general finitely presented groups. 3. Ozsvath and Szabo have developed a theory of holomorphic curves associated to Heegaard decompositions of closed 3-manifolds, leading to what they call "Heegaard Floer homology". What impact will this have in understanding 3-manifolds and to other areas? Related to this is the work of Khovanov and Rasmussen defining and applying a Jones polynomial homology. 4. Four dimensional techniques such as gauge theory and related topics have had some applications to problems in 3-manifolds. Thus Kronheimer and Mrowka give a proof of Property P (also implied by geometrization). What further impact 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-manifold theory to areas such as Computational Geometry and Cosmology as well as fields such as Computer Aided Design and Computational Complexity. What ties are developing between the theory of 3-manifolds and such disciplines?