IAN AGOL

*TAMENESS OF HYPERBOLIC 3-MANIFOLDS*

- We show that a hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact manifold with boundary. This answers a question of Marden (conjectured by Canary). Many cases were already known, including geometrically finite manifolds (Marden), indecomposable fundamental group (Bonahon), and limits of tame hyperbolic manifolds (Brock-Souto, extending work of Canary-Minsky, Evans, Ohshika). Applications will be discussed as well to conjectures and problems of Ahlfors, Bers, Canary, Hass, Simon, and Thurston.

IAN AGOL

*WORKSHOP ON 3-MANIFOLD THEORY*:
Geometrization conjecture and covers of 3-manifolds

- I'll give an outline of the geometrization conjecture for 3-dimensional closed manifolds. Then I'll explain how this gives a classification of the universal covers of compact manifolds, implying the sphere theorem and the Poincare conjecture.

JEFF BROCK

*ENDING LAMINATIONS AND THE WEIL-PETERSSON VISUAL SPHERE*

- Recent advances in the combinatorial topology of surfaces have led to new classification results for infinite volume hyperbolic 3-manifolds via their 'ending laminations'. These developments have ramifications toward another goal, namely, to develop an analogous classification for geodesic rays in the Weil-Petersson metric on Teichmüller space. I will discuss recent results in the CAT(0) geometry of the Weil-Petersson completion and analogies that arise between its geodesics and the geometry of hyperbolic 3-manifolds. In particular I will address cases in which a kind of 'ending lamination conjecture' for Weil-Petersson rays can be shown. Portions of this talk represent joint work with Yair Minsky and Howard Masur.

NATHAN DUNFIELD

*DOES A RANDOM 3-MANIFOLD FIBER
OVER THE CIRCLE*

- I'll discuss the question of when a tunnel number one 3-manifold fibers over the circle. In particular, I will discuss a criterion of Brown which answers this question from a presentation of the fundamental group. I will describe how techniques of Agol, Hass and W. Thurston can be adapted to calculate this very efficiently by using that the relator comes from an embedded curve on the boundary of a genus 2 handlebody. I will then describe some experiments which strongly suggest the answer to the question: Does a random tunnel-number one 3-manifold fiber over the circle? (Joint work with Dylan Thurston.)

JOHN ETNYRE

*INVARIANTS OF EMBEDDINGS VIA CONTACT GEOMETRY*

- I will describe a method to define, hopefully new, invariants of any embedded submanifold of Euclidean space. To define this invariant we will need to take an excursion into the realm of contact geometry and a recent generalization of Floer homology called contact homology. More specifically, after recalling various notions from contact geometry, I will show how to associate a Lagrangian submanifold to any embedded submanifold of Euclidean space. The invariant of the embedding will be the contact homology of this Lagrangian. Though the definition of this invariant is somewhat complicated it is possible to compute it for knots in Euclidean 3-space. Lenny Ng has combinatorially studied this invariant for such knots and has shown that it does not seem to be determined by previously known invariants but non the less has some connections with the classical Alexander polynomial of a knot. I will concentrate on the more geometric aspects of the invariant and ongoing work of Tobias Ekholm, Michael Sullivan and myself aimed at a better understanding of the invariant (in particular, showing that it is well defined in some generality).

ROSTISLAV GRIGORCHUK

*GROUPS OF BRANCH TYPE AND
FINITELY PRESENTED GROUPS*

- We dicsuss finite presentability of branch groups and (nice) embeddings of branch groups into finitely presented groups. Some interesting notions will be discussed, such as amenability, growth, and residual finiteness, along with their relation to finitely presented groups. We will pay particular attention to some problems of geometric nature regarding branch groups.

PETER KRONHEIMER

*PROPERTY P FOR KNOTS*

- According to Bing, a knot K has "Property P" if no surgery on K can yield a counterexample to the Poincare conjecture. Bing asked whether every knot has Property P. We give an affirmative answer to this question, drawing on techniques and recent results from gauge theory and symplectic topology.

YULI RUDYAK

*CATEGORY WEIGHT AND THE ARNOLD
CONJECTURE ON FIXED POINTS OF
SYMPLECTOMORPHISMS*

- The well-known Arnold conjecture claims that, for every closed
symplectic manifold M, the number of fixed points of any Hamiltonian
symplectomorphism of M is at least the minimal number of critical
points of a smooth function on M. We prove that this conjecture
holds provided π
_{2}(M)=0 (more precisely, if the first Chern class and the symplectic form vanish on π_{2}(M)). This improves the results of Floer and Hofer.

RICH SCHWARTZ

*SPHERICAL CR GEOMETRY AND DEHN SURGERY?*

- A complete spherical CR 3-manifold is the manifold at infinity for a discrete group acting isometrically on the complex hyperbolic plane. I will describe how Dehn filling arises in the context of these manifolds: Under fairly general conditions, when such a discrete group is perturbed, it remains discrete and the manifold at infinity undergoes a Dehn filling. This phenomenon is, of course, parallel to Thurston's well-known hyperbolic Dehn surgery theorem. As an application of the spherical CR Dehn surgery theorem I will show that a positive density subset of the Dehn fillings of the Whitehead link complement result in closed 3-manifolds which bound complex hyperbolic 4-orbifolds.

DENNIS SULLIVAN

*STRING BACKGROUND IN ALGEBRAIC TOPOLOGY*

- Using transversality and classical algebraic topology in manifolds, one can construct a string background or string algebra. Then more sophisticated theories like Gromov-Witten invariants of symplectic manifolds and the symplectic field theory for contact manifolds can be formulated in terms of solutions to equations written in the string background. Hopefully there are similar backgrounds for other appearances of quantum invariants in topology. A background is to a given theory as say, the deRham algebra of forms on a manifold is to the cohomology algebra. Namely, higher order products or invariants can be constructed from the background.

ZOLTAN SZABO

*HEEGAARD DIAGRAMS AND HOLOMORPHIC DISKS*

- The aim of this talk is to give a quick introduction to Heegaard Floer homology and discuss some applications to knots. (This is a joint work with Peter Ozsvath.)

BILL THURSTON

*WHAT NEXT?*

- Abstract not yet available.

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Last modified: Thu May 06 17:56:35 Eastern Daylight Time 2004