Mini-cours / Short courses


Kenneth Millett
Three aspects of the role of knotting in the study of polymers

Starting with the Sumners-Whittington, Pippenger solution of the Frisch-Wasserman Delbruck conjecture that the probability of knotting goes to one as the length of random walk or polygon goes to infinity, three facets of the role of knotting and linking in the study of macromolecules will be discussed. The first lecture will concern the presence and scale of knots, ephemeral knots and, slipknots in random walks and random polygons. The second lecture will discuss cumulative measures of the shape of polymer models and the effect of knotting on them. I will also describe current work with Rawdon and Stasiak on symmetry-breaking isodensity measures. The third lecture will explore work with Panagioutou on measures of entanglement applied to models of polymer melts.



Joan Porti
Varieties of representation and three-manifolds

The goal of these lectures is to give an introduction to varieties ofrepresentations and varieties of characters for three-manifolds and Culler-Shalen theory. I plan to compute explicit examples of varieties of characters and some applications to topology.



Dale Rolfsen
Three dimensional manifolds and their groups

The theme of these lectures is to explore the various ways in which the topology of a 3-manifold is reflected in its fundamental group. I will begin with a quick review of the history of this connection. For example if the group splits as a free product, then the manifold is a connected sum, existence of certain subgroups implies the existence of special surfaces, and so on. We will also discuss some of the consequences of Perelman's solution to the Poincare conjecture and Thurston's geometrization conjecture.
In the second two lectures I will concentrate on another property of groups that is connected with the topology of the manifold -- orderability. This is related to the existence of certain nice foliations, and also has relevance to manifolds which fibre over the circle. I will also focus on fibred knot groups, surgery and L-spaces (in the sense of Ozsvath and Szabo) and the relevance of ordering to these issues.


Peter Scott
The JSJ decomposition of 3-manifolds

 The decomposition of the title was proved to exist in the 1970's in work of Jaco and Shalen and independent work of Johannson. Almost simultaneously, Thurston announced that many of the pieces of the decomposition had hyperbolic structures. About 1980, he refined this statement to his Geometrization Conjecture which has recently been proved by Perelman.
In my talks, I will not assume familiarity with any of the above. In the first talk I will discuss the Geometrization Conjecture ( but not its proof). In the remaining talks I will discuss an approach to the proof of the existence and uniqueness of the JSJ decomposition using the theory of least area surfaces.


 
Exposés / Talks

 Daniele Alessandrini
Sur les espaces de Teichmüller des surfaces de type infini

Les espaces de Teichmüller pour surfaces de type infini peuvent être étudié avec des techniques géométriques comme le découpage en pantalons et les coordonnée de Fenchel-Nielsen. Dans ce cas il y a une infinité dénombrable de pantalons et de coordonnées. Avec ces techniques on peut décrire, sous certaines hypothèses la topologie donnée par la distance de Teichmüller et la topologie donnée par la distance "length spectrum".



Charalampos Charitos
Generalized Teichmüller space of non-compact 3-manifolds and Mostow rigidity

Consider a 3-dimensional manifold N by gluing a finite number of ideal hyperbolic tetrahedra, via isometries along their faces. By varying the isometry type of each tetrahedron but keeping fixed the gluing pattern of them we define a space T of complete hyperbolic metrics on N. We prove that T is homeomorphic to an euclidean space and we compute its dimension. By means of examples, we examine if the elements of T are uniquely determined by the angles around the edges of N.



François Gautero
Rapid Decay and 3-manifolds groups

A group G satisfies the Rapid Decay (RD) property if there exist k and s such that for any complex-valued function f on G with finite support (f in CG) the operator norm of f (f acts on  CG by left-convolution) is bounded above by k times the l2-norm of f weighted with the parameter s. This property, introduced by Haagerup, holds for free groups (Haagerup) and hyperbolic groups (de la Harpe). It has been useful in the context of Novikov and Baum-Connes conjectures. After recalling basic definitions and known results, we will try to characterize the 3-manifolds whose fundamental group has property (RD)



Shanti Gibert
Seifert rational homology spheres and taut foliations.

We focus on the existence of taut foliations on compact 3-manifolds. David Gabai solved the problem for 3-manifolds with non-trivial homology, saying that they always admit a taut foliation. It remains the case of rational homology spheres. Here, we will study the case of Seifert rational homology spheres.
The main result is that a Seifert integral homology sphere which is neither S3 nor Poincaré homology sphere, always admits a taut foliation. Nevertheless, the result is completely different when we only suppose that the rational homology of the manifold is trivial: whatever may be the number of exceptional fibers (greater than or equal to 3), there exist infinitely many manifolds without taut foliations, and there exist infinitely many manifolds which admit one.
Moreover, we will discuss the relationship between the geometry and the existence of taut foliation.



Martin Lustig
Are large distance Heegaard splittings generic?

A precise definition of "generic" in the given contex will be given, and it will be shown that with this definition, the answer to the question raised is positive. However, there are a number of delicacies involved when passing over to the precise statement proven, which will be specified in the talk. If time permits, we will also compare our result with parallel approaches to the question by other methods. (with Y. Moriah)



Gregor Masbaum
How to approximate quantum representations of the mapping class group by finite groups

The Witten-Reshetikhin-Turaev TQFT-invariants of 3-manifolds give rise to finite-dimensional representations of mapping class groups of surfaces. I will show how to approximate these representations by representations into finite groups, using the theory of Integral TQFT developed in joint work with P. Gilmer.


Yoav Moriah
Heegaard splitting with large subsurface distance

Subsurfaces of a Heegaard surface for which the relative Hempel distance of the splitting is sufficiently high have to appear in any Heegaard surface of genus bounded by half that distance.



Delphine Moussard
Caractérisation des modules d'Alexander des noeuds homologiquement triviaux dans les sphères d'homologie rationnelle

Les modules apparaissant comme modules d'Alexander de noeuds dans les sphères d'homologie entière ont été caractérisés par Levine. On généralisera ce résultat au cas des noeuds homologiquement triviaux dans les sphères d'homologie rationnelle.


Saul Schleimer
The graph of handlebodies

We introduce the graph of handlebodies and prove that it is quasi-isometric to an electrification of the curve complex. We show that this graph is Gromov-hyperbolic and of infinite diameter. As an application of our results we give a new, sharper proof of a theorem of Bonahon: there is a pseudo-Anosov map f so that no non-trivial power of f extends over any handlebody. (Joint work with Joseph Maher.)



Vladimir Verchinine
Tresses virtuelles de trois brins

Nous étudions diverses propriétés du groupe de tresses virtuelles pure PV3. De sa présentation, nous obtenons une décomposition de PV3 comme un produit libre. En conséquence, nous montrons que PV3 est résiduellement nilpotent sans torsion, ce qui implique que l'ensemble des invariants du type fini dans le sens de Goussarov-Polyak-Viro est complet pour tresses virtuelles pures de trois brins. De plus nous prouvons que la présentation de PV3 est sphérique. On détermine aussi l'anneau de cohomologie et l'algèbre de Lie associée graduée de PV3. Enfin, nous décrivons quelques variétés de résonance de cohomologies de ce groupe. (L'exposé est basé sur l'article avec V. Bardakov, R. Mikhailov et Jie Wu, arXiv:0906.1743.)