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.

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

My thesis is entitled:

I currently work at the Weizmann Institute of Science, as a postdoc hosted by Itai Benjamini and Gady Kozma.

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.

In this article, I strengthen the Indistinguishability Theorem of Lyons and Schramm for Bernoulli percolation: it was known that the infinite clusters agree on properties, and I generalise this to asymptotic properties. This is done by investigating connections between percolation theory and orbit equivalence theory. We also study a model of coalescing random walks on the free group with two generators, which satisfies indistinguishability but not strong indistinguishability. The reader familiar with percolation and orbit equivalence may want to have a look at **this old poster**.
*L'Enseignement Mathématique* (2) 61 (2015), 285–319

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.

Large audience

A site to visualise orders of magnitude.

*Euclidea* is a great video game on compass-and-straightedge constructions.

The youtube channel udiprod provides very good short videos about computer science and physics.

In this conference on games, Tadashi Tokieda presents many stunning experiments.

The video *Not knot* (in two parts, here and there) will make you see the amazing three-diemensional geometry of knots.

The video entitled *How to turn a sphere inside out* (in two parts, here and there) will teach you... how to turn a sphere inside out!

The website *Mathematical Etudes* contains a lot of interesting videos.

The movie *Dimensions* will make you see in four dimensions.

What is chaos? A situation is said to be chaotic if knowing the initial state with a slight uncertainty does not allow us to predict what will happen in a few units of time. In this movie, you will see how chaos appears in the pool game, meteorology, the motion of planets... and that being unable to predict the near future does not mean that we cannot say anything about the long term!

The book *Science and hypothesis* (written by Henri Poincaré) is really worth reading. It deals deeply with the connection between science, reality, truth, etc.

As for novels, I advise you to read *Uncle Petros and Goldbach's Conjecture* and *The Carpet Makers*, written respectively by Apostolos Doxiadis and Andreas Eschbach. The story of the first book takes place with mathematics in the landscape; the author has managed to write a good novel that gives a fair image of the mathematical world. The second book has no *direct* link with mathematics, but it is built to provide the reader with a good awareness of orders of magnitude (and the ending is sumptuous).

Science for students

*How to turn a sphere inside out?* This video explains in twenty minutes how to exchange the internal and external faces of a sphere by deforming it. The sphere is not asked to be plunged at every time but it must always be immersed. No prerequisite needed.

The video *Not knot* will allow you, in twenty minutes and without any prerequisite, to visualise the hyperbolic structure of the complement of a knot (or more precisely, of a link, the Borromean one).

Terence Tao's blog contains many enlightening articles. The level of the prerequisites may vary a lot from one article to the other. Here are a few accessible articles.

With this series of conferences of Feynman, you will access the nature of the laws of physics. Within a few hours and without prerequisite exceeding the high school level. If you want to get acquainted in detail with these laws, you may want to read Feynman's lecture notes.

*The Princeton Companion to Mathematics* is an excellent companion to pure mathematics.

*Resonances and small divisors* is an article of Etienne Ghys that can be read by an undergraduate student. It deals with the problem of the stability of the solar system.

Here is a very good series of lectures on the uniformisation theorem, by Etienne Ghys.

Most mathematical entries of David Madore's blog are written in French. But you can at least enjoy a few jokes.

Finally, I recommend to the student in mathematics the links and books of the *Large audience* section.

Here are a few enigmas for undergraduate students.

This is a document where topology is introduced via neighbourhoods instead of open sets.

Here are a few exercises typed when I was young. I was then teaching at the olympic training programme Grésillon-2008.

At last, a few verses of Malherbe that make me think of Gromov:

*Me, the fortune of whom is so close to the sky*

That I see all things under me,

And everything I see seems a dot to my eye.

This is a document where topology is introduced via neighbourhoods instead of open sets.

Here are a few exercises typed when I was young. I was then teaching at the olympic training programme Grésillon-2008.

At last, a few verses of Malherbe that make me think of Gromov:

That I see all things under me,

And everything I see seems a dot to my eye.