In 2003-2004, the seminar was organised by Choi Young Eun.

Kasra Rafi (Univ of California, Santa Barbara, USA)

*Geometry of Teichmuller space and the complex of curves*

**Abstract**:
Using techniques developed by Masur and Minsky, we study geodesics in
Teichmuller space. We give a characterization of short curves along a
geodesic and give an estimate for the shortest length of a curve. As a
consequence we provide a combinatorial formula for the distance between
two points in Teichmuller space.

Javier Aramayona (Southampton)

*The Weil-Petersson geometry of the five-times punctured sphere*

Reza Chamanara (Indiana)

*Simultaneous bending of projectively convex piecewise circular Jordan arcs (CONTINUED)*

Reza Chamanara (Indiana University, USA)

*Simultaneous bending of projectively convex piecewise circular Jordan arcs*

Mary Rees (Liverpool)

*The geometric model and coarse Lipschitz equivalence direct from Teichmuller geodesics*

**Abstract**:
We shall explain a proof of the Ending Lamination Conjecture which uses
Teichmuller geodesics directly, restricted, for simplicity, to the case
when the ending laminations data is a pair of minimal laminations.

Caroline Series (Warwick)

*Thurston's bending measure conjecture for once punctured torus groups*

**Abstract**:
We prove Thurston's bending measure conjecture for quasifuchsian once
punctured torus groups. The conjecture states that the bending measures
of the two components of the convex hull boundary uniquely determine
the group.

Cliff Earle (Cornell University)

*A dynamical approach to the conformal barycenter*

**Abstract**:
The conformal barycenter of a probability measure on the unit circle is
a central ingredient in Douady and Earle's theory of the barycentric
extension of circle homeomorphisms. Recently, Abikoff developed an
iteration scheme, called the MAY iterator, for calculating the
conformal barycenter. New work by Abikoff, Mitra, and me uses the MAY
iterator to provide a new definition of the conformal barycenter. This
definition applies to a wider class of probability measures and allows
the extension of continuous degree one monotone maps of the circle that
are not necessarilty homeomorphisms.

Al Marden (University of Minnesota)

*On PSL(2,C)-representation varieties for Kleinian groups*

**Abstract**:
I will give a survey of some results centered on the represention
variety of a geometrically finite kleinian group. The most detailed
information available is for the special case of a fuchsian surface
group. For this case the principal tool to be discussed is the covering
of the representation variety by the bundle of complex projective
structures, in particular, grafting. The method of complex scaling
offers another approach. I will give McMullen's proof of a prototypical
case of ``bumping'' of components of the discreteness locus of the
representation variety. Apart from this proof the talk will be entirely
expository.

Cyril Lecuire (Warwick)

*Bending laminations and convergence of metrics*

**Abstract**:
In this talk I will conclude the proof of the main theorem of the first
talk which gave necessary conditions for a geodesic measured lamination
to be the bending lamination of some convex cocompact hyperbolic metric.

Cyril Lecuire (Warwick)

*Bending laminations and algebraic convergence*

**Abstract**:
In this third talk, I will show that if we have a sequence of faithful
and discrete representations with converging bending measures (with
some restrictions on the limit) then there is a subsequence that
converges algebraically.

Cyril Lecuire (Warwick)

*Convex cores of hyperbolic 3-manifolds and bending measured geodesic laminations (continued)*

Cyril Lecuire (Warwick)

*Convex cores of hyperbolic 3-manifolds and bending measured geodesic laminations*

**Abstract**:
This is the first talk in a series of four. I will give some basic
definitions, especially that of a convex core, bending measured
laminations and characteristic submanifolds of hyperbolic 3-manifolds.
I will then explain some properties of the bending geodesic measured
laminations.

Emmanuel Dufraine (Warwick)

*Actions of Surface Groups on Real Trees -- Skora's Theorem (Otal's book, chapter 8)*

Kentaro Ito (Nagoya University, Japan)

*Bumping of components of quasi-fuchsian projective structures*

**Abstract**:
Let S be a closed hyperbolic surface. Let P(S) be the space of marked
projective structures on S and let Q(S) be the subset of P(S)
consisting of projective structures with quasi-fuchsian holonomy. It is
known that Q(S) has infinitely many components: there is only one
component called "standard" and all other components are called
"exotic".

In this talk, we explain that each exotic component bumps into the standard component. Moreover we show that each exotic component self-bumps. As a consequence, we can also show that any two components bump. One of the main tools used here is the grafting operation on a projective surface.

Caroline Series (Warwick)

*Chapter 6 of Otal's book "The Hyperbolisation Theorem for Fibred 3-Manifolds"*

Caroline Series (Warwick)

*Chapter 6 of Otal's book "The Hyperbolisation Theorem for Fibred 3-Manifolds"*

Akira Ushijima (Warwick, Kanazawa University)

*Volumes of Hyperbolic Tetrahedra*

*Reading seminar of "The hyperbolization theorem for fibered 3-manifolds" by Jean-Pierre Otal*

Co-organized with Emmanuel Dufraine.