Alexandr Bufetov

Limit theorems for parabolic flows (figures1, figures2, figures3, figures4)

The main results of the talk are limit theorems for two classes of measure-preserving flows of geometric origin.

The first class is given by translation flows on flat surfaces. The second class, for which the result re joint with Giovanni Forni, consists of horocycle flows on compact surfaces of constant negative curvature.

One of the main objects introduced in the talk is a special family of horocycle-invariant finitely-additive Hoelder measures on rectifiable arcs. An asymptotic formula for ergodic integrals of our flows is obtained in terms of the finitely-additive measures, and limit theorems follow as a corollary of the asymptotic formula.

Pierre Derbez

Chern Simons Theory and the volume of three-manifolds (slides)

We
give some
applications of the Chern-Simons gauge theory to the study of the set
vol(N,G) of volumes of all representations ρ
: π_{1 }N
→ G, where N is a closed
oriented 3-manifold and G is either or
Iso_{+}H^{3}. We
will focus on the following
questions:

(1) How to find non-zero values in
vol(N, G)? or
weakly how to find non-zero elements in vol(Ñ,
G) for some finite cover Ñ
of N?

(2) What kind of topological
information is enclosed in
the elements of vol(Ñ,
G)?

(3) Do these volumes satisfy the
covering property in the
sense of Thurston?

This is joint work
with Shicheng Wang.

François Guéritaud

Lorentzian manifolds with constant curvature (slides)

PSL

Kazuo Habiro

Unified quantum invariants of integral homology 3-spheres associated to simple Lie algebras (slides)

The
Witten-Reshetikhin-Turaev quantum 3-manifold invariants τ_{M}^{g}(ζ)
are defined for finite-dimensional simple Lie algebras g
and roots of unity ζ.

For an integral homology sphere M, we
define a "unified invariant" J_{m}^{g},
taking values in a completion of the polynomial ring **Z**[q],
which is called the cyclotomic completion of **Z**[q] or the ring
of analytic functions on the set of roots of unity.

The quantum
invariants τ_{M}^{g}(ζ)
at various roots of unity recover from this unified invariant by
setting q=ζ . This is a
joint work with Thang Le, and generalizes my previous result for the
sl_{2} case.

Kentaro Ito

On the topology of the space of Kleinian once-punctured torus groups (slides)

It is known that the topology of the space of Kleinian once-punctured torus groups is fairly complicated; it self-bumps and, moreover, it is not locally connected (due to Bromberg).

In this talk we first characterize sequences of punctured torus groups which induce self-bumping of the space.

We next explain Bromberg's theory from the viewpoints of the trace coordinate and the complex Fenchel-Nielsen coordinate.

Especially, the shape of linear slices which are close to the Maskit slice will be discussed.

Nariya Kawazumi

The completed Goldman-Turaev Lie bialgebra and mapping class groups (slides)

We introduce a completion of the Goldman-Turaev Lie bialgebra of a compact connected oriented surface with non-empty boundary. The smallest Torelli group of the surface in the sense of Putman is embedded into the kernel of the Turaev cobracket in the completed bialgebra. So a tensorial description of the completed bialgebra is desired. If the boundary of the surface is connected, a symplectic expansion introduced by Massuyeau relates the bialgebra to Kontsevich's "associative" Lie algebra. An approximative description of the Turaev cobracket in this case gives a geometric interpretation of the Morita traces on the cokernel of the Johnson homomorphism. On the other hand, if the genus of the surface is zero, a special expansion relates the bialgebra to the Lie algebra of special derivations on a free Lie algebra.

This talk is based on a joint work with Yusuke Kuno (Tsuda College).

Takahiro Kitayama

On an analogue of Culler-Shalen theory for higher dimensional representations (slides)

Culler and Shalen established a way to construct incompressible surfaces in a 3-manifold from ideal points of the SL

Cyril Lecuire

Extending mapping classes from the boundary of a handelbody (slides)

Consider a diffeomorphism f of a closed surface S and assume that S is the boundary of a handelbody. We want to know to which extent f is the restriction of a diffeomorphism of H. This leads us to define the biggest subcompression body of H to which f extends. We explain how one can find out from the invariants sets of f whether this compression body is trivial or not. This is a joint work with Ian Biringer.

Gwénaël Massuyeau

Splitting formulas for the LMO invariant (slides)

For rational homology 3-spheres, there are two universal finite-type invariants: the Le–Murakami–Ohtsuki invariant and the Kontsevich–Kuperberg–Thurston invariant. These invariants take values in the same space of “Jacobi diagrams”, but it is not known whether they are equal. In 2004, Lescop found some relations satisfied by the variations of the KKT invariant when one replaces embedded rational homology handlebodies by others in a “Lagrangian-preserving” way. In this talk, we shall review the LMO invariant and show that it satisfies exactly the same relations. Our proof is an application of the LMO functor, which is a kind of TQFT extending the LMO invariant.

Greg McShane

Dynamics of the mapping class group (slides)

Let G be a Lie group. The mapping class group of a surface S has a natural action on the representation space of the fundamental group into G.

We discuss the nature of the action (ergodic, proper discontinous) for different G (compact, SL(n,R)) and for surface with boundary and cone points.

Ken'ichi Ohshika

Subgroups of mapping class groups generated by Dehn twists around meridians on splitting surfaces (slides)

We consider a splitting surface S of a Heegaard decomposition or a bridge decomposition. We are interested in a subgroup of the mapping class group of S generated by Dehn twists around "meridians" on S. We shall show that this group has a natural free product decomposition, and that there is an open subset in the projective lamination space of S on which this group acts locally properly discontinuously and freely. This is joint work with Makoto Sakuma and partly with Brian Bowditch.

Joan Porti

Varieties of representations in SL(n,C) (slides)

Given a three manifold, we consider its variety or representations in SL(n,C) and its variety of characters.

We discuss its local structure for representations of geometric origin, for instance the representation

induced by the holonomy of a hyperbolic manifold, or representations of Seifert manifolds.

We will also discuss explicit examples.

Leonid Potyagailo

Quasiconvexity in relatively hyperbolic groups and its applications (slides)

This is a joint work with V. Gerasimov (Belo Horisonte). An action of a group G on a compactum X is called relatively hyperbolic (RH-action) if it is discontinuous on the space of distinct triples Θ

We provide several criteria for a subgroup H of G to be quasiconvex in a strong (absolute) and weak (relative) sense. As an application we obtain a stronger form of a recent theorem of Y. Matsuda, S. Oguni and S. Yamagata providing the existence of an equivariant homeomorphism between two RH-actions of G on different compacta.

Masatoshi Sato

On stable commutator length in the mapping class groups of punctured spheres (slides)

We give new upper and lower bounds on stable commutator length of some elements including half Dehn twists in the mapping class groups of punctured spheres. In particular, the lower bounds are obtained by calculating omega-signatures in Gambaudo-Ghys' paper. This talk is based on a joint work with Danny Calegari and Naoyuki Monden.

Yasushi Yamashita

The diagonal slice of SL(2,C)-character variety of free group of rank two (slides)

Let F

ρ → (trρ(A), trρ(B), trρ(AB)), where ρ is a representation of F