# Mathematical web page of Sebastien Martineau

## General Presentation

This web page presents my research, my teaching material, several mathematical links and a few other things.

The text of this web page is fully in English but many documents are only available in French.
This is not the case as far as research is concerned.

If you need to contact me, click here.

## Research

I obtained in December 2014 a PhD in mathematics under the supervision of Vincent Beffara, at the UMPA.

My thesis is entitled: Percolation on groups and directed models.

After 2 years at the Weizmann Institute of Science, as a postdoc hosted by Itai Benjamini and Gady Kozma,

I am now working at the Université Paris-Sud, as a postdoc of Nicolas Curien and Jean-François Le Gall.

## Presentation

Take a porous stone and plunge it into water. We would like to know whether the water penetrates deeply into the stone or only superficially. Percolation theory allows us to answer this question without breaking the stone. First, weigh the stone to determine its porosity $$p$$, that is its proportion of holes. Then, define a probabilistic model of a stone with porosity $$p$$. To do so, divide a non-porous stone into tiny cubes and throw each of them away with probability $$p$$, independently. Cubes that are not thrown away stay at the same position. If the study of this model reveals that the probability of the event of deep infiltration is very close to 0 or 1, we may postulate that our stone agrees with the majority. And it turns out that, in the limit, this is the case: there is a critical parameter $$p_c$$ such that infiltration typically occurs if $$p > p_c$$ and typically does not occur if $$p < p_c$$. By measuring one number, we have solved a complex geometric problem with a low probability of error.

Mathematically, we consider a graph ($$\mathbb{Z}^3$$ for the stone problem) and a real number $$p$$ between 0 and 1. Each edge is then kept with probability $$p$$ and removed otherwise, in an independent way. We want to know if this random subgraph typically contains an infinite connected component or not.

I study this process for a finitely generated group instead of $$\mathbb{Z}^3$$. I want to understand how the critical parameter depends on the graph under study and what can be said about the connected components that appear. More generally, I am interested in statistical mechanics, ergodic theory and the geometry of graphs/groups.

## Teaching

From September 2011 to June 2015, I gave exercise sessions and coures at the mathematics department of the ENS Lyon.

When the level of a teaching activity is unspecified, it corresponds to "third year of University".

Master 1 corresponds to the fourth year of University and agrégation is prepared during the fifth year of University.

From 2011 to 2015, I taught the probability course to the students preparing agrégation. Here are my lecture notes (which unfortunately do not cover all the material taught in class), the exercise sheets and some additional material.

From 2011 to 2015, I supervised agrégation lessons. For the one on combinatorics, I typed these notes.

In winter 2012, I led with François Le Maître a student working group on John Meier's book Groups, graphs and trees.

In fall 2012, Emmanuel Jacob and I gave the TA of the course Introduction to probability, which was taught by Grégory Miermont. Here are the solutions to some of these exercises.

During winter and spring 2014, Emmanuel Jacob, Vincent Tassion and I gave the TA of the course Integration and probability, which was taught by Grégory Miermont.
Here are his lecture notes.

In 2014 and 2015, I gave the TA of the course Brownian motion and stochastic processes, which was taught by Emmanuel Jacob (Master 1).

In January 2014, I was teaching at the snow class for students preparing agrégation. We talked about a few things, including countable connected sets.

During winter and spring 2015, I gave a course for non-mathematicians called Glimpses of mathematics.

At last, during the academic year 2014-2015, Emmanuel Jacob and I were in charge of the preparation to the option A of agrégation (probability and statistics)). Here are a few Scilab exercises.