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A poster session will be organized if at least 4 participants are willing to present a poster. In case you are interested in participating and presenting a poster, please indicate it when registering.

**Jeff Brock****Leo Brunswic****Steve Carlip****François Guéritaud****Colin Guillarmou****Kirill Krasnov****Catherine Meusburger****Daniel Monclair****Carlos Scarinci****Andrea Seppi****Richard Wentworth**

In this talk I will describe recent developments that provide a satisfying analytic explanation for the connection between volume of quasi-Fuchsian convex cores and Weil-Petersson distance, as well as volumes of hyperbolic 3-manifolds that fiber over the circle and Weil-Petersson translation distance of pseudo-Anosov automorphisms of Teichmüller space. I'll discuss an array of applications to Weil-Petersson geometry as well as some new results. This is joint work with Ken Bromberg, with new work adding Martin Bridgeman to the collaboration.

We present two constructions of polyhedral Cauchy-surfaces in flat globally hyperbolic Cauchy-compact spacetimes. The first construction is inspired by a 1987 paper of Penner on so-called decorated Teichmüller space : the convex hull method. We give a new interpretation of this construction in the context of Cauchy-compact flat spacetimes with BTZ-like singularities giving a bijective map from the moduli space of a marked Cauchy-compact spacetimes with BTZ of linear holonomy to the moduli space of marked closed polyhedral surface. The starting point of the second construction is the inverse of this map ; essentially described by Penner, we give a generalization with the prospect of extending this correspondance to spacetimes with massive particles and BTZ singularities.

The wave function of a quantum “test particle” in a (2+1)-dimensional classical gravitational background must respect the geometric structure of the spacetime. The wave function of a (2+1)-dimensional quantum spacetime that includes a point particle (represented as a puncture) must be invariant under the mapping class group. For reason that I do not fully understand, these conditions are exactly equivalent at lowest order. Higher order corrections may allow us to describe the backreaction of a quantum particle on a quantum spacetime.

We prove that any right-angled Coxeter group $\Gamma$ admits a properly discontinuous action by affine transformations on some real vector space of dimension $N$. The construction contains "Margulis spacetimes" ($\Gamma$ free, $N=3$) as a special case. Time permitting, we will discuss the connection to a projective notion of convex cocompactness. This is joint work with Jeff Danciger and Fanny Kassel.

We shall describe a mathematical definition of Polyakov partition function in 2D bosonic string theory using the Liouville quantum field theory, a conformal field theory quantizing the classical Liouville action. This follows the David-Distler-Kawai formalism. Mathematically, we use probabilistic methods to define the "path integral" and we show the convergence of the partition function on the moduli space of Riemann surface when the central charge of the liouville theory is c_L=25 (correspoonding to d=1 boson in Polyakov partition function). Joint work with Rhodes and Vargas.

I will describe how 3D gravity (with non-zero cosmological constant) can be reformulated as a theory of (spin) connection rather than the metric. In this formalism, the metric is reconstructed from the curvature of the spin connection, and the action functional is just the total volume of the space computed using the metric defined by the connection. Interestingly, this connection formulation can be seen as arising via a dimensional reduction of a certain theory of 3-forms in 6 dimensions. Thus, the action functional of the connection formulation coincides with the Hitchin 6-dimensional functional evaluated on the Chern-Simons 3-form in the total space of the principal SU(2) (or SL(2,R) - depending on the signature) bundle over our 3-dimensional manifold.

We use a unified description of 3d hyperbolic space, anti de Sitter space and half-pipe space to obtain a unified description of ideal tetrahedra in these spaces and a unified formula for their volumes. We also incorporate 3d de Sitter, Minkowski, anti de Sitter space in this picture, describe the duality between these spaces and the first three and discuss dual ideal tetrahedra in these spaces.

This is ongoing work with Carlos Scarinci.

Given a convex cocompact group of isometries of the hyperbolic space, its critical exponent (a number measuring the growth of orbits) is equal to the Hausdorff dimension of its limit set. This equality provides a link between the asymptotic behavior of the dynamics on the hyperbolic space and the local geometry of the boundary. When the limit set is a topological circle, this number is at least 1. A theorem of Bowen states that this lower bound can only be reached if the limit set is a geometric circle.

We will see that this relationship between asymptotic dynamics and local geometry of the boundary also exists for groups acting on Anti de Sitter space (the Lorentzian analogue of hyperbolic space). We will also discuss a Lorentzian version of Bowen's Theorem.

This is a joint work with O. Glorieux (University of Luxembourg).

I will introduce a set of coordinates on the moduli spaces of globally hyperbolic cusped 3 dimensional spacetimes generalising the shear coordinates on Teichmüller space of cusped Riemann surfaces. I will then discuss the Poisson structure and the mapping class group action on the moduli spaces and describe some of the ideas behind their quantisation. This talk is based on joint work with Catherine Meusburger and ongoing work with Hyun Kyu Kim.

abstract (pdf)

In this talk I will revisit the asymptotic structure of the SL(2,C) character variety of a closed surface group. Recent work of Taubes and Mazzeo, et.al. describes the large scale behavior of solutions to the Hitchin equations in terms of certain limiting configurations. I will show how these correspond, via harmonic maps, exactly to Bonahon's parametrization of pleated surfaces in hyperbolic 3-space by transverse and bending cocycles for a geodesic lamination. This is joint work with Andreas Ott, Jan Swoboda, and Mike Wolf.