First meeting of the network: a three days mini-workshop. The list of participants is:
Thierry Barbot Université d´Avignon France
Jairo Bochi Pontificia Universidad Católica de Chile
Joaquín Brum Universidad de la República Uruguay
Léo Brunswic Université d´Avignon France
Mario Jorge Dias Carneiro Universidade Federal de Minas Gerais Brazil
León Carvajales Universidad de la República Uruguay
Francois Fillastre Université de Cergy-Pontoise France
Carlos Maquera Universidade de Sao Paulo Brazil
Andrés Navas USACH Chile
Viviane Pardini Valerio Universidade Federal de Sao Joao del-Rei Brazil
Cristóbal Rivas USACH Chile
Graham Smith Universidade Federal do Rio de Janeiro Brazil
Richard Urzúa Universidad Católica del Norte Chile
Cristobal Rivas USACH Chile
Julio Sánchez UCN Chile

Start |
End |
Title |
Speaker |

Wed 05 July 2017 |

09:30 |
10:30 |
Anosov representations and conformally flat spacetimes |
Thierry Barbot |

11:00 |
12:00 |
Representaciones de Anosov y descomposiciones dominadas |
Jairo Bochi |

14:00 |
15:00 |
Constant curvature -1 3d spaces and Teichmüller theory |
François Fillastre |

16:00 |
17:00 |
Acciones libres afines de \(\mathbb Z^p\) sobre toro \(\mathbb T^q \) |
Richard Urzua |

Thu 06 July 2017 |

09:30 |
10:30 |
Singularities in spacetimes and piecewise transformations of the circle |
Thierry Barbot |

11:00 |
12:00 |
Singular (2+1) -dimensional GHMC Minkowski spacetimes |
Léo Brunswic |

14:00 |
15:00 |
Some questions concerning groups of piecewise affine and piecewise projective diffeomorphisms |
Andres Navas Flores |

15:30 |
16:30 |
On the Anosov character of the Pappus-Schwartz representations |
Viviane Pardini Valerio |

Fri 07 July 2017 |

10:00 |
11:00 |
Some insights in Anosov actions |
Carlos Alberto Maquera Apaza |

11:30 |
12:30 |
Constructing constant scalar curvature time functions in (3+1) -dimensional GHMC Minkowski spacetime |
Graham Andrew Craig Smith |

#### "Anosov representations and conformally flat spacetimes"

I will present the connections between Anosov representations and conformally flat spacetimes. The talk will be based on the material of the recent survey:
https://arxiv.org/pdf/1609.03863.pdf

#### "Representaciones de Anosov y descomposiciones dominadas"

##### by Jairo Bochi on Wed 05 July 2017

The concept of dominated splitting comes from ODE and differentiable dynamics. It turns out that Anosov representations are a manifestation of domination. I will discuss these relations. I will also sketch our proof of a result of Kapovich, Leeb, and Porti stating that only Gromov-hyperbolic groups admit Anosov representations. This talk is based on my joint work with Rafael Potrie and Andrés Sambarino.

#### "Constant curvature -1 3d spaces and Teichmüller theory"

We briefly review the 3d model spaces of curvature -1, which are hyperbolic space, anti-de Sitter space and co-Minkowski (or half-pipe) space. We then give some examples of relations with Teichmüller space of compact surfaces, mainly focusing on co-Minkowski space. This talk is coming from the surveys arxiv.org/1605.04563 and arxiv.org/1611.01065

#### "Acciones libres afines de \(\mathbb Z^p\) sobre toro \(\mathbb T^q \)"

Toda acción de \(\mathbb Z^p \) sobre \( \mathbb Z^q \) que actúa por automorfismos de \(\mathbb Z^q\), con conjunto de puntos fijos diferente de cero, induce una acción unipotente máxima de \(\mathbb Z^p\) sobre \( \mathbb Z^q\)´, que determina si la acción original es la parte lineal de una acción afín libre de \( \mathbb Z^p \) sobre el toro \(\mathbb T^q\).

#### "Singularities in spacetimes and piecewise transformations of the circle"

I will present the connexion between particles in (2+1)-dimensional spacetimes and the theory of projective transformations on a projective circle. The main objective is to formulate the classification problem of collisions of tachyons, gravitons and other black hole. The content of the talk will be greatly extracted from sections 3 and 5 of https://arxiv.org/pdf/1010.3602.pdf

#### "Singular (2+1) -dimensional GHMC Minkowski spacetimes"

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.

#### "Some questions concerning groups of piecewise affine and piecewise projective diffeomorphisms"

In this talk we will review some recent results concerning obstructions for C^1 actions on the circle of certain groups of homeomorphisms and stress that these questions remain open for certain groups of piecewise-projective homeomorphisms.

#### "On the Anosov character of the Pappus-Schwartz representations"

The talk will be devoted to the Pappus-Schwartz representation and their recent generalization. The reference for this talk is https://arxiv.org/pdf/1610.04049.pdf

#### "Some insights in Anosov actions"

In this talk we will discuss the problem of classification of Anosov actions of abelian (or nilpotent) Lie groups. Our emphasis will be on the case where the action is of codimension one.

#### "Constructing constant scalar curvature time functions in (3+1) -dimensional GHMC Minkowski spacetime"

We prove that every (3+1)-dimensional flat GHMC Minkowski spacetime carries a unique foliation by spacelike hypersurfaces of constant scalar curvature. In other words, we prove that every such spacetime carries a unique time function with isochrones of constant scalar curvature. Furthermore, this time function is smooth.