The so-called "séminaire compréhensible" is a weekly-held seminar, usually in Salle 04, on Wednesday at 17:00.
Speakers are PhD. students, ATER, Post-Doc, ...

It is followed by the compréhensible goûter, in the coffee room.

- 11/03/2020
- Julien Poirier
- Caen
- Magnetohydrodynamics

Magnetohydrodynamics (MHD) is the study of electrically conducting flux (salt water, plasma...) subject to an electromagnetic field. We model it with two coupled partial differential equations: the Navier-Stokes equations and the Maxwell equations. In this lecture, in a first time we will understand the building of this model and applications in different fields, and in a second time we will discuss about the existence and unicity of solutions for different types of boundary conditions (Navier, on the pressure, Dirichlet ...).- 04/03/2020
- Clément Bérat
- IF
- Flat bundles with solvable linear groups

In this talk, we will explain and prove a result of W. Goldman and M. Hirsch about flat bundles with solvable structure group. After explaining basic facts about flat bundles, we will prove that if the structure group is solvable and linear, such bundles are virtually trivial. If time permits, we will talk about a generalization of this to groups that are not linear.- 19/02/2020
- Jean-François Bougron
- IF
- Markovian Random Repeated Interaction Systems

This talk introduces a particular class of quantum evolution called "Markovian Random Repeated Interaction Systems" or MRRIS. MRRIS are a special case of RIS where a chain of quantum systems (called the probes) goes through another one (called the small system). The specificity of Markovian Random RIS is that the nature of the probes follows a homogeneous Markov process, so that the one currently going through the small system depends only on the probe that preceded it.

In this talk, we present some facts concerning quantum channels, quantum dynamical semigroups which are semigroups of quantum channels, how this applies to RIS and MRRIS and finally we give some results concerning the large-time behaviour of the MRRIS.- 12/02/2020
- Arnaud Plessis
- IF
- An introduction to the diophantine geometry

Let \(V\) be an algebraic variety over a number field K. In mathematics, diophantine geometry consists to study the set \(V(K)\) (the set of points on \(V\) with coordinates in \(K\)). Typical questions about \(V(K)\) are :

1) What is its nature (empty, finite, infinite, group ...) ?

2) What about the "size" of its elements ?

Using an intuitive way, I will give a precise definition of the "size", called " height " in the literaure, in the most simplest case, namely when \(V\) is the multiplicative group (it's the group defines by the equation \(x^0 = 1\), that is \(V(K)=K^*\)). This height has numerous good properties, that I will state. Finally, I will finish my talk giving some applications, like the Mordell-Weil theorem.- 29/01/2020
- Ouriel Bloede
- Angers
- An advertisement for singular cohomology

We could summarize the aim of algebraic topology as, trying to find combinatorical or algebraic obstructions to the existence of certain maps between topological spaces. As an example we will describe piece by piece, the different algebraic structures one can put on singular cohomology (over \(F_2\)), and describe topological results that arise from it, building from the abelian group structure to the one of unstable algebra over the Steenrod algebra.- 22/01/2020
- Guillaume Gandolfi
- Caen
- The monoid embedding problem

A monoid is defined in the same way as a group except one do not require the elements to be invertible. Because of this simple difference the class of monoids is more chaotic than the one of groups and therefore when studying a monoid, it can be useful to know whether this monoid can be embedded into a group because the monoid will then behave more nicely than average. This is the monoid embedding problem. In this talk after having described the basic notions linked to this problem, I will talk about some sufficient, necessary and both sufficient and necessary conditions for a monoid to be embeddable into a group before mentioning a different approach, namely the compatibility of embeddability with some algebraic constructions.- 15/01/2020
- Dorian Chanfi
- École Polytechnique
- An introduction to Bruhat-Tits buildings and their compactifications

Bruhat-Tits buildings were introduced by Iwahori, Goldman, Matsumoto, and formalized by Bruhat and Tits as a tool for studying the structure of p-adic groups appearing in number theory, representation theory and harmonic analysis. The general theory associates to a semisimple group (think \(SL_n(Q_p))\) a (poly-)simplicial complex on which the group acts which encodes much of its structure. In this talk, we shall give an introduction to this circle of ideas, the end goal being to explain a compactification procedure for these structures, based on Berkovich geometry.- 18/12/2019
- Arnaud Plessis
- IF
- Axiom of choice is stronger than a mythological monster

With the help of Paris, Kirby killed a mythological monster using the axiom of choice. In this talk, we will see how.

In the introduction, we will organize a run between Rick and many Morty. After that, we will study a famous (and fun) sequence whose the main theorem about this one will surprise you ! Finally, we will kill a mythological monster, who is immune to Peano.- 04/12/2019
- Mokdad Mokdad
- IF
- Mathematical General Relativity part II: Black holes

We will continue our discussion on mathematical general relativity, in particular, on the exact solutions of Einstein's equations that are referred to as black hole solutions. We will start with Minkowski space, serving both as the first ("trivial") solution to the equations, and as a rapid review of the basic notions (so even if you've missed the first talk, this one will be comprehensible). Next we will discuss the first and simplest black hole solution: the Schwarzschild black hole. It is an eternal non-rotating and uncharged black (and white also!) hole. It is part of a more general family --the so called "Kerr-Neumann" family-- of rotating and/or charged black holes that are described by three parameters: the mass, the charge, and the angular momentum. We will look at the spherically symmetric cases of this family, and the effect of the presence of a positive cosmological constant. The rotating case is described by the Kerr solution which we will touch on at the end.- 20/11/2019
- Pascal Millet
- IF
- Introduction to additive combinatorics

The term additive combinatorics refers to the study of the additive structure of a commutative group (sometimes the multiplicative structure of a ring) relying on combinatorial and elementary algebraic tools like graphs, counting methods and Fourier transform on finite group. Let’s illustrate the type of question that arises in this field with two basic examples. Let \(A \subset \mathbb{Z}\) be a finite set. We denote by \(A + A\) the set of all integers of the form \(a+a'\) where \(a, a' \in A\). We have the following elementary fact: \(|A + A| = 2|A| − 1\) if and only if \(A\) is an arithmetic progression. In other words, there is a link between the cardinal of \(A + A\) and the additive structure of \(A\). A generalized (and more robust) version of this fact is the Freiman-Ruzsa theorem, which asserts that if \(|A + A|\) is small, then \(A\) is a large subset of a generalized arithmetic progression. Another interesting question concerning the additive structure is whether some arithmetic progressions of a given length exist in the set \(A\). For example, by a pigeonhole argument, we have that every subset \(A\) of \(\{1, ..., N \}\) of cardinal strictly greater than \(2 ( E( \frac{N}{3} ) + 1 ) \) contains an arithmetic progression with three terms. We deduce that for \(\delta > \frac{2}{3}\) and for \(N\) large enough, every subset A of \(\{1, ..., N \}\) of density \(\delta\) contains an arithmetic progression with three terms. Roth’s theorem asserts that the same statement is true for every \(\delta > 0\). In my presentation, I will begin by proving basic combinatorial facts about the sum of sets to provide familiarity with some important concepts. Then I will introduce the discrete Fourier transform, which is a key tool in additive combinatorics. Finally, I will present the proof of Roth’s theorem and some open related questions.- 13/11/2019
- Pengfei Huang
- Nice
- HODGE THEORY: FROM ABELIAN TO NON-ABELIAN

Roughly speaking, non-abelian Hodge theory is an analogue of abelian Hodge theory by replacing the abelian groups into non-abelian groups, which gives the equivalence between the category of semisimple flat bundles, the category of polystable Higgs bundles with vanishing Chern classes, the category of polystable λ-flat bundles and that of harmonic bundles, these were built based on the work of Donaldson, Corlette, Hitchin, Simpson, Deligne and Mochizuki.

In this talk, first I will give a brief introduction on the origin of the non- abelian Hodge theory arising from abelian Hodge theory (i.e, the classical Hodge theory), then I will try to introduce these results with more details. If time permits, I will introduce the related topics on non-abelian Hodge theory, especially the twistor space construction.- 06/11/2019
- Mokdad Mokdad
- IF
- A Brief Intro to General Relativity and Black Holes Solutions

This is going to be a brief and somewhat elementary introduction to the mathematical theory of general relativity, Einstein's theory of gravity. I will introduce the basic notions and concepts of its mathematical framework: Lorentzian geometry. Then I will discuss the field equations, and finally give a quick overview of some of the important exact solutions, namely, black holes solutions.- 16/10/2019
- Renaud Raquepas
- IF
- Mathematical topics related to the use of entropy in physics

After a brief introduction to the notion of entropy in the context of probability theory on finite sets, I will discuss the role of relative entropy in the mathematical description of irreversibility in physical systems. I will also hint at the mathematical work that needs to be done to extend these notions to the noncommutative setting (quantum mechanics).- 09/10/2019
- Romain Durand
- IF
- An introduction to the problematics around infinite-volume measures of the Ising model

In this comprehensible talk, starting from the description of the nearest-neighbour Ising model - which is one of the most studied model in statistical mechanics - we will introduce the concepts of infinite-volume measures, Gibbs states, and Dobrushin states. In dimension 2, the Aizenmann-Higuchi theorem states that all infinite-volume measures are convex combinations of \(\mu^+\) and \(\mu^-\) and are therefore all translation invariant, whereas in dimension 3 and more, there exists some infinite-volume measures that are not translation invariant (Dobrushin states). We will discuss how the question of the existence of such Dobrushin states can be treated on the specific example of 2D long-range Ising models.- 02/10/2019
- Nora Gabriella Szoke
- IF
- Amenability and topological full groups

The notion of amenable groups is central in the topic of geometric group theory. In this talk we will see a new method for establishing the amenability of groups. This method was introduced by Juschenko and Monod and can be applied to topological full groups of certain group actions.- 25/09/2019
- Nicola Cavallucci
- Roma
- Packing conditions on CAT(0) spaces

The study of metric spaces with synthetic notions of curvature is an important topic in geometry. The aim of the talk is to motivate why they are interesting using the special case of CAT(0) spaces satisfying a uniform packing condition.- 18/09/2019
- Gabriel Lepetit
- IF
- About the irrationnality of \(\zeta(3)\)

We introduce some basic principles of diophantine approximation and then focus on a particular problem of this field : Apéry's celebrated elementary proof of the irrationality of \(\zeta(3)\). We will discuss other proofs by Beukers and Nesterenko if time allows.

## PhD Days : 24-25 October 2019

PhD Days Timetable 8:45 – 9:00 9:00 – 10:00 10:00 – 10:30 10:30 – 11:30 11:30 – 12:00 12:00 – 14:00 14:00 – 15:00 15:00 – 15:30 15:30 – 16:30 Thursday Welcome Marcatel break Vézier Flash Talks lunch Philip break Feld Friday Lepetit break Sané Flash Talks lunch Traore break

Abstracts (PDF)All talks will take place in the IRMA tower, Amphi 1.

The organizers are David Leturcq and Clément Berat.

e-mail: