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.

Due to Covid, the seminar now takes place on the Internet through Zoom on Thursday at 3:30pm. The Zoom meeting invitation is sent by email, so if you do not receive it please contact us.

- 10/06/2021
- Oussama Hamza
- Western University, Ontario
- Poincaré Series and mild groups

Let \(G\) be a pro-\(p\)-group, which admits a minimal presentation, with \(d\) generators and \(r\) relations. In 1964, Golod and Shafarevich showed that if \(G\) is a \(p\)-group, then it satisfies \(d^2 < 4r\). The original proof of this result use a very subtle study of Poincaré Series. Poincaré Series gives also cohomological information on pro-\(p\)-groups. During the 60's, Lazard and Koch showed that a pro-\(p\)-group has cohomological dimension less than two if and only if its Poincaré Series verifies some equality. Between 1980 and 2000's, Anick and Labute, introduced a sufficient and easy condition on the relations of pro-\(p\)-group \(G\), such that \(G\) is of cohomological dimension less than two. Groups satisfying this sufficient condition are called mild. In this talk, we will present, more precisely, Poincaré series, cohomological consequences, and mild groups. If time permits, we will give some examples in an arithmetic context.- 06/05/2021
- Gabriel Lepetit
- IF
- \(E\) and \(G\)-functions

Siegel's \(E\) and \(G\)-functions are classes of special functions including many classical functions, such as polylogarithms and hypergeometric functions. They have the crucial property to be solutions of linear differential equations with polynomial coefficients.

In this talk, we begin by presenting the known irrationality and transcendence results on the values of \(E\) and \(G\)-functions. Then we will interest ourselves in their differential properties.- 18/03/2021
- Filipe Bellio da Nobrega
- UMPA - ENS Lyon
- Osculating conics of a smooth curve on the projective plane

The Tait-Kneser theorem is a well known result of differential geometry which states that the osculating circles of a plane curve with monotonic curvature and no inflection points are disjoint and nested. Therefore, an arc with no vertex gives rise to an interesting foliation of the region of the plane delimited by its largest and smallest osculating circle. In this talk, we will investigate the analogous result for osculating conics. It is already known that the osculating conics of a curve with no sextactic or inflection points are also disjoint and nested. However, we will show that the relative position of two such conics is actually more restricted than that, they are in some sense “convexly nested”.- 11/03/2021
- Baptiste Cerclé
- Paris Saclay
- Methods of 2d Conformal Field Theories applied to a model of quantum gravity

In 1981, Polyakov presented in a pioneer article a canonical way of defining the notion of random surface in the setting of 2d quantum gravity, usually called Liouville conformal field theory.s A few years later Belavin, Polyakov and Zamolodchikov (BPZ) presented in their 1984 seminal work a systematic procedure to « solve » more generally a certain class of models, the so-called 2d CFTs, which like Liouville theory possess certain conformal symmetries.

The main input of their method was to translate in a representation theoretical language the constraints imposed by conformal symmetry and study the consequences of these constraints from an algebraic viewpoint. The main drawback of this machinery being that in order to go from representation theory to actual models (such as Liouville theory), physicists usually rely on certain axioms and often lack mathematically rigorous formulation.

In this talk we will investigate how to implement the method of BPZ to a mathematically rigorous model : the probabilistic construction of Liouville theory. If time permits we will discuss the generalisation of this method to Toda theories, which are natural extensions of Liouville theories.- 04/03/2021
- Léo Dort
- ENS Lyon
- Introduction to Contact Process on Graphs

First introduced and studied by Harris (1974), the Contact Process (CP) is a stochastic process usually used to model the spread of an epidemic in a population. We model population by a graph, where vertices represent the individuals and edges the possible pathways for infection to spread. Each individual can be "healthy" or "infected". The process evolves according to the following dynamic. Each infected vertex become healthy at rate 1. Simultaneously each infected vertex infects all of its neighbours with rate λ.

In this talk, we will give some general and basic properties of the CP, and we will focus on the following interesting and natural question: does the infection become extinct or not ? We will show that for many graphs, we can exhibit a (at least one) phase transition.- 04/02/2021
- Keyao Peng
- IF
- Homotopy type theory for mathematicians (part 2)

Homotopy type can be regarded as the new foundation of maths, replacing the notion of set. We introduce two basic aspects of type: "Spaces as types" in homotopy theory and "Propositions as types" in logic. Then we show some examples of Proof Assistant(LEAN) based on type theory.- 28/01/2021
- Keyao Peng
- IF
- Homotopy type theory for mathematicians

Homotopy type can be regarded as the new foundation of maths, replacing the notion of set. We introduce two basic aspects of type: "Spaces as types" in homotopy theory and "Propositions as types" in logic. Then we show some examples of Proof Assistant(LEAN) based on type theory.- 17/12/2020
- Tanguy Vernet
- EPFL
- The geometry of indecomposable quiver representations

Quivers are finite graphs with an orientation. Like groups or Lie algebras, they have a representation theory and a natural goal is then to classify their indecomposable representations. In this talk, I will cover foundational results concerning the classification of indecomposable quiver representations and illustrate them on elementary examples. The culminating point is Kac's theorem (1982), which links indecomposable representations to root systems of Lie algebras. The proof relies on a geometric interpretation, in which isomorphism classes of representations are identified to orbits of some affine space under the action of an algebraic group. It involves arithmetic techniques and a polynomial counting of representations over finite fields (Kac's polynomial). It was proved more recently that Kac's polynomial has non-negative coefficients (Hausel, Letellier, Rodriguez-Villegas 2013). I will try to give some intuition for the proof in a particular case (Crawley-Boevey, Van den Bergh 2004). If time allows, I will say a few words about a related polynomial counting over finite quotients of local fields.- 10/12/2020
- Loïs Faisant
- IF
- Rational curves : asymptotic behaviour of moduli spaces

The Grothendieck ring of varieties is a ring defined by generators and relations. Generators are isomorphism classes of varieties over a fixed field. Relations are the well-known scissors relations : if \(U\) is an open subset of a variety \(X\), then the class of \(X\) is the sum of the class of \(U\) and the class of its complement in \(X\). This ring provides a relevant framework to study the asymptotic behaviour of moduli spaces of rational curves on a fixed variety, when the degree of such curves becomes infinitely large. This question is the geometric analogue of the asymptotic study of rational points of bounded height, when the bound tends to infinity, known as the Manin conjecture (1989) refined by Peyre (1995). In this talk, we will start presenting this ring of varieties and providing a bunch of simple and easy examples. Then we will (try to) state the motivic Manin-Peyre conjecture (early 2000s) in a comprehensible way. In a second part, we will check that this conjecture is true for the projective space, and state similar results obtained for compactifications of vector spaces.- 03/12/2020
- Arnaud Plessis
- Beijing
- Lehmer's Problem

Lehmer's problem (1933) asks if there exists a constant \(c > 1\) such that the (logarithmic, absolute) Weil height of every nonzero algebraic number \(x\) is bounded from below by \(c [Q(x) : Q]^{-1}\), except if \(x\) is a root of the unity. In 1979, Dobrowolski proved that it is true up to epsilon. In this talk, we are going to study algebraic fields \(L\) which have the Bogomolov Property, that is when the Weil height of every nonzero element of \(L\) which is not a root of the unity can be bounded from below by an universal constant. Next, we will give other kind of "Lehmer's problem" according with these results. Finally, we will state a Lehmer's problem (due to David) for abelian varieties and I will explain the difficulties about this one.- 26/11/2020
- Thomas Mietton
- IF
- Optimal transport in sub-Riemannian geometry

Following the works of Lott, Villani and Sturm, the theory of optimal transport in Riemannian spaces has led to the definition of curvature-dimension bounds \(CD(K,N)\), linking curvature and dimension with some geometric inequalities, applicable to more general metric spaces. In the case of sub-Riemannian spaces, which consist of manifolds where we only consider paths following a bracket-generating distribution in the tangent space, \(CD(K,N)\) conditions don't apply, however we can consider the alternate condition \(MCP(K,N)\). While some important results have been made, there still lacks a general theory for proving said geometric inequalities in this setting, in particular due to the possible existence of singular geodesics. In this talk, I will present the basics of sub-Riemannian geometry and optimal transport, and then try to shed light on some of the challenges in proving existence and unicity of optimal transport maps and interpolation inequalities in sub-Riemannian manifolds.- 19/11/2020
- Félix Lequen
- Cergy
- K3 surfaces, a gluing construction and ergodic theory

K3 surfaces constitute a remarkable family of complex surfaces. Examples can be built by standard geometric constructions in complex geometry or algebraic geometry. Here I will describe a beautiful (and more original) construction due to Koike and Uehara, where such surfaces are obtained by gluing along a real hypersurfaces, which is said to be linear Levi-flat. This leads to the question of which K3 surfaces contain such linear Levi-flat hypersurfaces. This question might seem artificial, but we will see that we can give a very partial answer by using the construction by Koike and Uehara and beautiful ideas of Verbitsky. We will thus obtain an existence result by using ingredients from complex geometry, algebraic geometry, dynamical systems, ergodic theory and homogeneous spaces for Lie groups.- 12/11/2020
- Bingyu Zhang
- IF
- What is "larger" in symplectic geometry?

In this talk, I will start with the basic notions of symplectic geometry and review basic results on the symplectic embedding problem. In particular, I will focus on the numerical constraints of symplectic embeddings between symplectic ellipsoids. In the second part, I would like to review what is the microlocal theory of sheaves and how to apply it to the symplectic embedding problem.- 05/11/2020
- Fardad Pouran
- IF
- RSW theory without FKG

In percolation-type statistical models with FKG property, one can prove RSW inequality and "Box crossing property", which are very useful tools to study many further problems in critical percolation, e.g. studying "arm events", extendability of arms, etc. Therefore one may ask what happens to RSW if we lose FKG, while adding some plausible assumptions, what we will call "good-behavior". In this presentation, I'd like to talk about these assumptions and how they can help us achieving back the so-called "Box crossing property", after providing the main concepts of classical percolation which seem to be general enough to be valid for a wider class of models.- 15/10/2020
- Rufus Lawrence
- IF
- Measure theory with locales

In this talk I will sketch some of the ideas laid out in Alex Simpson's article Measure, randomness and sublocales. We define locales and measures on locales, and give some examples. We then use this formalism to extend the Lebesgue measure to all subsets of \(\mathbb{R}^n\). We then show how this does not contradict the apparent existence of "non-measurable" sets, and "resolve" Tarski's paradox.- 08/10/2020
- Corentin Le Bihan
- Lyon
- How to model a gas in a box

A simple model of gas is the hard spheres model. It is a billiard of little particles which can interact very strongly at very small distance (think for example of real billiards with a lot of balls). Because understanding such system is an outstanding problem, people tried to find a limiting process. A first equation governing the density of one particle was given by Boltzmann : \( \partial_t f + v \cdot \nabla_x f=Q(f,f) \). In its formal derivation Boltzmann supposed that two different particles are almost independent, so the probability of having two particles at the same place is the the product of probability. The validity of such equation is a priori not clear since it adds some irreversibly that does not exist in the hard sphere model.

Lanford solved the problem in its ‘75 paper: Boltzmann’s equation is true, up to a time independent of the number of particles (however each particle will have in mean less than one collision).

Now comes the question of the boundary. We expect to find some “Lanfords” theorem even if we add some boundary condition. A first example are the specular reflections, for a deterministic law. An other example, which would be very important in physics, is the evolution of a gas between two hot plaques. Then the reflection condition is stochastic. I am interested in a third type of reflection, also stochastic, which is a modeling of a rough boundary.

During my talk I will present some ideas of the proof of Boltzmann in the full space and the adaptation in the case with boundary.- 01/10/2020
- Vivek Dewan
- IF
- Percolation of Gaussian fields and Bernoulli percolation

In this talk, we will give a brief introduction to the very new and active research area that is percolation of Gaussian fields. It pertains to the large-scale behaviour of sub-level sets of random smooth functions. We will focus on the 2-dimensional phase transition for fields with fast decorrelation and its many analogies with the well-known phase transition in Bernoulli percolation. We will then survey the differences in the main techniques which are used to establish these results in the two models.- 24/09/2020
- Siarhei Finski
- IF
- Gauss-Bonnet theorem with a look towards local index theory

Gauss-Bonnet theorem states that the mean curvature of a real surface is related to its Euler characteristic. This theorem, hence, relates the geometry of the surface to its topology. By using Hodge theory, the theorem can be rephrased as a calculation of the index of a certain differential operator, which interprets the Gauss-Bonnet theorem as an instance of an index theorem. In this talk, I will explain the above interpretation and show some consequences of that point of view.- 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 Vivek Dewan, Rémi Reboulet and Clément Berat.

e-mail: