## Construction and visualization of domains of dependence

**Goal:**
There are several models for the shape of the universe, but the simplest
are globally hyperbolic constant curvature Lorentzian spaces. They are the
quotient by a discrete group of what is known as a domain of dependence.
The main goal of the project will be to construct and visualize in
several different ways such domains of dependence in positive, zero or
negative curvature, and in dimension 3 and 4.

There will be several stages to this project, and completing in a
satisfactory any of those stages will already be interesting.

- Computing the holonomy representations of hyperbolic structures
on surfaces, in terms of natural parameters describing those structures
(the so-called Fenchel-Nielsen coordinates)
- Computing holonomy representations of globally hyperbolic constant
curvature manifolds in dimension 3 from similar coordinates
- Visualizing (in one of several possible ways) the universal covers
of those globally hyperbolic spaces
- computing holonomy representations of a few examples of
hyperbolic 3-manifolds
- computing the holonomy representations of globally hyperbolic
4-dimensional constant curvature spaces
- representing the domains of dependence constructed in this manner.

Before going into computational aspects, it will be necessary to learn and
understand geometric notions beyond what is taught at the bachelor level.
For this reason the project can only be conducted as a 6th semester "memoire"
or at the master level.

**Schedule:** to be determined.

**Persons involved:**
Advised by Jean-Marc Schlenker, NN.

**Difficulty level:**
Advanced.

**Tools:**
programs in python/sage.

**Expected results:**
Pictures of domains of dependence in dimension 2+1, possibly 3d models.