Francis Bonahon, USC
Quantum hyperbolic geometry
Surprising connections between hyperbolic geometry and quantum algebraic invariants of knots and 3-manifolds appear to be emerging. A typical example is the Volume Conjecture, which connects the growth of the Jones polynomials of a link to the hyperbolic volume of its complement. I will describe a framework which mixes hyperbolic geometry with quantum algebra (namely the science of the relation XY = q YX, where X and Y are variables and q is a scalar) to construct topological invariants of 3-manifolds, surface diffeomorphisms, etc... There will be speculations about where this kind of ideas could be heading.
Rachel Roberts, Washington University
Foliations and group actions on R
This is joint work with Sergio Fenley. We show that there are infinitely many 3-manifolds Y such that $\pi_1(Y)\subgroup Homeo^+(\mathbb R)$ but which contain no $\mathbb R$-covered foliations. These examples can be found among closed hyperbolic manifolds obtained by surgery on once-punctured torus bundles.
Dick Canary, University of Michigan
Introductory Bumponomics: Deformation spaces of hyperbolic 3-manifolds.
Abstract We will survey recent results on the global topology of the space AH(M) of all (marked) hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M. The interior of this space is quite well-understood. Each component of the interior is a manifold parameterized by natural analytical data and the components are enumerated by the set of (marked) hyperbolizable 3-manifold homotopy equivalent to M. The discovery that the components of the interior can bump has led to an extensive study of the pathological behavior of the topology of AH(M). On the other hand some more recent results indicate that the topology is ``nice'' at many points. We will survey these results and, in the spirit of the conference, speculate on future avenues for research (i.e. intermediate bumponomics)
Zlil Sela, Hebrew University, Jerusalem
Some problems in Diophantine geometry over groups.
Peter Shalen, University of Illinois at Chicago
Hyperbolic volume and classical topology.
Perelman's work on the Ricci flow appears to confirm the intimate connection between topology and hyperbolic geometry in dimension three. I believe that as a result of this work we are beginning to see even richer interactions between the deep methods developed in these two areas than those that we have seen over the last 30 years. I will illustrate this by discussing some of my recent work with Marc Culler relating hyperbolic volume to such classical topological invariants as homology and Heegaard genus. In this work, topological methods, such as the tower arguments developed by Papakyriakopoulos and Shapiro-Whitehead, and estimates for homology of covering spaces proved in my paper with P. Wagreich, have interacted with diverse geometrical methods and results. These include the Marden conjecture (recently theorem by Agol and Calegari-Gabai), the paradoxical decomposition arguments initiated in my joint work with Culler and further developed in our collaboration with Anderson and Canary, and the recent results proved by Agol, Dunfield, Storm and Thurston, in which Perelman's work on the Ricci flow is applied to the study of hyperbolic volume. I will also discuss prospects for future work in this direction, including the potential interactions between hyperbolic geometry and deep topological questions about Heegaard splittings.
Jeff Lagarias, University of Michigan
Computational Topology of 3-Manifolds.
The interaction between topology and computation traces back to work of Dehn from 1907--1912 on combinatorial topology and knot invariants. Dehn formulated the word problem and conjugacy problem for finitely presented groups, which were some of the first problems later shown to be computationally undecidable. Three-dimensional topology problems seem to lie at the dividing line between computationally tractible and intractible problems. This talk surveys these connections and describes results obtained in the last few years on the computational difficulty of such problems as detecting unknottedness, computing the genus of a knot, and the complexity of computing various knot and 3-manifold invariants such as the Jones polynomial and Vassiliev invariants. Exciting future directions for research include connections with quantum computation.`
Peter Kropholler, Glasgow
Graham Niblo, Southampton
On the algebraic torus theorem.
Bruce Kleiner, University of Michigan
Hyam Rubinstein, University of Melbourne
Polyhedral and geometric structures on 3-manifolds.
Abstract An interesting problem is to find a good polyhedral version of a geometric structure. I will talk about several possible approaches, including Casson and Rivin's angle structures, good triangulations. and higher dimensional cubical complexes. Another piece of evidence is a direct version of Swarup's result that the fundamental group of an atoroidal Haken manifold is word hyperbolic, using very short hierarchies.
Vincent Guirardel, University Paul Sabatier
Intersection numbers and convex cores.
We present the construction of a kind of "convex core" for the action of a group in the product of two trees. This geometric construction allows us to generalise and unify Scott's intersection number of a pair of splittings, the intersection number of a pair of measured foliations on a surface, and the apparition of surfaces in Fujiwara-Papasoglu's construction of the JSJ-splitting. We will discuss potential applications to the description of more general JSJ splittings, and to the study of Out(F_n).