What does a geometric 3-manifold look like? How do geometric decompositions mimic combinatorial ones?

Francis Bonahon, USC

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Quantum hyperbolic geometry
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Abstract:
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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*

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Abstract
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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.

(Joint work with Robert Meyerhoff and Peter Milley) We discuss an approach towards addressing the Thurston, Weeks, Matveev - Fomenko conjecture that complete low volume hyperbolic 3-manifolds are of low topological complexity.

Dick Canary, University of Michigan

*Introductory Bumponomics: Deformation spaces of hyperbolic
3-manifolds.*

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Abstract
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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

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Some problems in Diophantine geometry over groups.
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Peter Shalen, University of Illinois at Chicago

*Hyperbolic volume and classical topology.*

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Abstract
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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.

To every finite subgroup of SU(2) there is associated a finite-dimensional algebra, the cross-product of the group algebra with the exterior algebra on two generators. The braid group naturally acts on the homotopy category of complexes of modules over that algebra. We show how this leads to a categorification of the Burau representation of the braid group and explain why the categorified action is faithful.

Jeff Lagarias, University of Michigan

*Computational Topology of 3-Manifolds.*

** Abstract: **

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.*

An overview of how one uses Ricci flow on the space of metrics on a closed three-dimensional manifold to establish the existence of the decomposition of the manifold into pieces admitting homogeneous geometries. We will review the foundational work of Hamilton on Ricci flow and the revolutionary advances of Perelman.

Bruce Kleiner, University of Michigan

Hyam Rubinstein, University of Melbourne

*Polyhedral and geometric structures on 3-manifolds.*

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Abstract
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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.*

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Abstract
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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).