Séminaires doctorant en 2024

We're interested in image generation of the Rauzy fractal. There are different ways of drawing this fractal, each involving specific mathematical tools. In this talk, we present the different computer data structures adapted to implement these mathematical objects. We also discuss various way of generating these pictures. The tools we have developed let us explore and manipulate more easily similar fractals.

TBA

The idea of -category owes, to a much extent, to a certain dissatisfaction with the classical language of derived categories developed by Grothendieck. An -category should have, apart of objects and morphisms, a sort of “higher morphisms” between the morphisms, as well as morphisms between these “higher morphisms”, and so on. The works of Brown, Joyal, Jardine, Toen, Vezzosi and Lurie have provided the mathematical tools for the formalisation of this notion, providing a solid theory for the “working mathematician”. The aim of this talk will be to give the motivations behind the notion of the -category and to demystify its abstract character through examples from different areas of pure mathematics where this notion has been shown to be an extremely useful tool.

We consider a defocusing quasilinear nonlinear Schrödinger equation in dimension one with nonzero conditions at infinity. The talk aims to present the classification of the traveling-wave solutions of this equation in terms of two parameters: the strength of the quasilinear term and the speed of the wave. These traveling waves are thought to be fundamental excitations for nonlinear dispersive equations i.e. they play a role similar to the eigenvectors for linear equations With access to the theory of ODEs, we found multiple branches of solutions coexisting in the same region of parameters: A branch of smooth localized solutions common to other nonlinear dispersive PDEs arising in fluid mechanics; but also branches of localized solutions with lower regularity, on which we will provide some properties and illustrations. In a second time, we will address some variational properties of the regular solutions which are important to show the stability of these structures upon small perturbations.

Consider a scalar hyperbolic conservation law with convex flux perturbed by saturating diffusion and linear dispersion of the form , where and are small positive parameters. First, we study the existence of particular solutions that are travelling waves. In the diffusive case , the existence of smooth travelling waves depends on . We also prove their convergence when to the shock waves of the hyperbolic equation, provided . When and , the existence of travelling waves is established, and their behaviour depends on the parameters and . Give conjectures for the convergence of the general solutions to the entropy solution of the hyperbolic equation.

Smoluchowski’s type equations are under a suitable change of variables non-local (integro-differential) conservation laws. They are intended to describe the aggregation phenomena (coagulation) ruled by binary collisions of clusters in a homogeneous diluted system. In some special cases (integral kernels of homogeneity one) instability and strong oscillation behaviours of some particular solutions were numerically observed. We use approximations of the non-local equation by quasilinear dissipative-dispersive equations to study those behaviours.

In this talk we’ll discover the notion of cardinality of a groupoïd with the simplest tools of category theory possible. I will try to convince you that this very notion fills in the gaps left by traditional cardinality theory, especially when it comes to quotienting. We will then continue our journey by defining the Hall algebra of some category of modules and try to link this with our previous notion in order to simplify some results. At last we will briefly see what the cardinality of an infinity groupoïd is and how it may help extending the definition and properties of hall algebras to triangulated categories. (Fear not young one we will avoid abstract nonesense as much as it is possible)

Deciding if a system of polynomial equations has a solution over is one of the oldest questions in number theory and over the years it has proved to be a real challenge. One of the first approaches to this question is to prove existence of solutions over bigger and ''simpler'' fields (i.e. , ) and then to try to ''restrict'' the solutions, this is known as the local-global principle. In this talk, I will present the local-global principle, some counter examples to this principle, and further approaches to finding rational solutions.

A large variety of micro-swimmers in nature use active appendages-like organelles such as cilia or flagella to swim inside fluids at a low Reynolds number. These cilia and flagella all have a characteristic periodic motion, which naturally enables them to swim as efficiently as possible in viscous fluids. However, when studying a simple elastic filament, it turns out that the wave propagating along the swimmer is attenuated very quickly, contrarily to the behaviors observed in the tails of swimming micro-organisms in biology. Some form of activation along the swimmer is thus necessary. We aim to explain how to take into account the complex structure of the biological flagellum and in particular its influence on the tail’s oscillating pattern. We focus on these flagellar activation mechanisms, from a mathematical point of view, both theoretically and numerically.

The Kawahara equation is a higher-request Korteweg-de Vries (KdV) equation with an extra fifth order derivative term. It was inferred by Hasimoto as a model of the gravity waves in a vastly long channel over a flat bottom in a long wave with surface tension. We derived the Kawahara-type equation over an uneven bottom and we prove its consistency with the Euler system. After that, we propose a regularized-approximate version up to the good order (i.e. order of derivation of the Kawahara equation) and then we prove its unconditional well-posedness. Finally, we perform a numerical simulation on the Kawahara equation and its approximation equation and we verify numerically the coherence of the order of approximation.

Les q-analogues de nombres sont issus d'une déformation des nombres entiers qui consiste à introduire une variable formelle "q", en remplaçant chaque nombre par un polynôme de telle sorte qu'on retrouve le nombre initial en faisant tendre q vers 1. Cette idée sous-tend par exemple la notion de série génératrice, déjà utilisée par Euler pour aborder des problèmes combinatoires. Depuis le XVIIIème siècle, les q-nombres ont fait leur apparition dans de nombreuses branches des mathématiques, des formes modulaires aux groupes quantiques, en passant par l'analyse des séries hypergéométriques. Malgré ces succès, il a fallu attendre les années 2020 pour avoir une bonne déformation des nombres rationnels, qui généralise de manière satisfaisante les propriétés combinatoires des q-nombres. On verra comment définir ces q-rationnels, en donnant trois points de vue équivalent sur cette construction, via une action du groupe modulaire, via des fractions continues et via le pavage de Farey. Ensuite, on donnera des interprétations combinatoires de ces q-rationnels, par analogie avec les modèles combinatoires décrit par les q-nombres entiers. On illustrera enfin la transversalité des q-rationnels en décrivant leurs liens avec la topologie algébrique et la théorie des noeuds.

Quantum machine learning (QML) is a promising area of research that combines the principles of quantum computing with traditional machine learning techniques. This fusion opens new perspectives for solving com- plex problems and accelerating intensive calculations. This talk aims to provide a brief introduction to QML. Starting with the basics of traditional machine learning, we will introduce key concepts such as supervised and unsupervised learning algorithms. We will also discuss the limitations of classic machine learning when faced with large-scale problems. Next, we’ll dive into the world of quantum computing by explaining the fundamentals of quantum mechanics, focusing on concepts such as the qubit, quantum superposition, quantum entanglement, and quantum logic gates. Once the basics of machine learning and quantum computing have been established, we will give a brief introduction to existing QML approaches and algorithms. Above all, we will highlight the potential benefits of QML, such as speeding up calculations, the ability to work with large amounts of data and solving complex optimization problems. Finally, we will discuss the current challenges and prospects of QML. We will discuss the hardware constraints related to the implementation of quantum computers, as well as recent advances in the development of more efficient and accessible quantum platforms.

First of all, an introduction to some general concepts in Ergodic theory will be given with two important notions : The Birkhoff ergodic theorem and the recurrence Poincaré theorem. Second, a class of dynamical system called self-Induced systems and the links with the generalised substitution shifts will be presented.

Nous nous intéressons à la résolution numérique de problèmes de diffraction dans des guides d’ondes électromagnétiques fermés au moyen de méthodes d’Éléments Finis. Pour ce faire, il est nécessaire de tronquer le domaine et de créer une condition transparente adaptée sur la frontière artificielle pour éviter les réflexions parasites. Nous montrons ici comment étendre à ce cas plusieurs techniques développées en acoustique ou en élasticité. Pour écrire la condition transparente, on utilise une décomposition modale. Numériquement, il est nécessaire de tronquer la série correspondante. Pour justifier la convergence de notre approche, on montre que le problème en domaine tronqué est bien posé, et que l’erreur avec la solution exacte décroît exponentiellement avec le rang de troncature. Nous montrerons des résultats numériques obtenus à l’aide de la librairie XliFE++, qui illustrent la résolution des équations de Maxwell 3D en utilisant les éléments finis de Nédélec.

The mathematical description of Water Waves is still a challenging problem. Indeed, the intricacies of the underlying physical phenomenon (due to, e.g., the interface between a gas and a liquid, the salinity, the presence of foam or the coupling between different geophysical scales) motivate thorough mathematical analysis in order to extract scientific knowledge. Among the different topics related to water waves, wave breaking is by far one of the most popular, as well as one of the most challenging. In this presentation, we shall provide an introduction to the Water Waves problem, derived from Euler's equation with extra assumptions that will be discussed. These well-known equations fail at describing wave breaking but we will provide a natural way of bypassing this difficulty. At the end of the talk, we will come back to the main assumptions made throughout the derivation of the Water Waves equations, i.e. the absence of both viscosity and vorticity, as some of our numerical results show that these are not exactly justified.

The Hecke algebra is a one-parameter deformation of the group algebra of a Coxeter group W , with a standard basis indexed by elements in the group. At least in the case where W is a Weyl group, the Hecke algebra controls many aspects of representation theory: the category O of semisimple Lie algebras, representations of quantum groups, representations of reductive groups in positive characteristic, and more. In 1979, Kazhdan and Lusztig introduced a new basis for the Hecke algebra. Its coefficients with respect to the standard basis, called Kazhdan–Lusztig polynomials, provide answers to many problems in representation theory; they also encode the local intersection cohomology of Schubert varieties. A significant portion of the talk will be dedicated to explaining an algorithm to compute them. In the case where W is an affine Weyl group, we have a spherical Hecke algebra embedded in the affine Hecke algebra, which plays an important role: in particular, the corresponding Kazhdan–Lusztig polynomials give Lusztig’s q-analogue of the Weyl character formula. In 2022, Libedinsky, Patimo, and Plaza introduced the pre-canonical bases to the spherical Hecke algebra. These bases interpolate between the spherical Kazhdan–Lusztig basis and the spherical standard basis, providing a new way to understand the Kazhdan–Lusztig basis and its consequences. The talk will primarily emphasize examples of the aforementioned structures.

Perhaps one of the most fruitful ideas in the field of topological dynamics was the introduction, in the 1960s, of an object called the enveloping semigroup of a system. The algebraic and topological properties of this semigroup are deeply connected to the dynamical properties of the original system. This object presents a surprising dichotomy: it is either small in cardinality and topologically simple, or pathologically large and complicated. In this talk, we aim to introduce the enveloping semigroup and explore its connection to the desirable properties of the underlying dynamical system. Special focus will be given to understanding the dynamics of tame systems, which are those whose enveloping semigroups fall into the first category of this dichotomy.

In recent years, the study of nilsystems has drawn much interest because of their applications in the structural theory of dynamical systems, number theory, and combinatorics. In particular, the algebraic properties of the enveloping semigroups of this class of systems have been studied. In this talk, the concepts of pro-nilsystem and the regionally proximal relation of order d will be introduced. We will explore the relationship between the proximal extensions of these systems and their enveloping semigroups. Finally, an algebraic description of the regionally proximal relation of order d in a certain class of systems will be presented.

Under a certain threshold constraint, viscoplastic fluids are rigid and behave like a solid and only start to flow when the constraint is above this threshold. In this talk we will see how to handle this phenomenom from a numerical point of view and which methods are mainly used. We will show an application through a Shallow-water type equation for 3D viscoplastic. We perform numerical simulation compared with experiments lead in INRAE Grenoble.

In an unpublished note, Sejong Park and Radu Stancu states the following theorem: for a -linear category with a finite-dimensional endomorphism ring, we can parametrize simple representations by pairs , where is an objet of and is a primitive idempotent in the idempotent completion of . The goal of this talk is to explain the notions involved and generalize a bit this theorem.

Local class field theory studies and classifies abelian extensions of local fields, which describes the Galois group of the maximal abelian extension of a local field via Artin's local reiprocity map. Let be a finite extension of the -adic number field with its ring of integers . In 1964, Lubin and Tate constructed a 1-dimensional formal group, popularly known as the Lubin-Tate formal group, over and used it to generate a totally ramified maximal abelian extension of the ground field. Moreover, Lubin and Tate offered a new proof of the main theorem (local Artin reciprocity theorem) of local class field theory and thus provided a parallel interpretation of local class field theory. Despite the remarkable applications of the 1-dimensional Lubin-Tate formal group, there has been no suitable generalization of the 1-dimensional to the upper dimension. In this talk, I would like to discuss how one can construct a class of -dimensional formal groups over the ring of -adic integers that provide an actual higher-dimensional analogue of the usual -dimensional Lubin-Tate formal groups. Then, we will see that the -torsion points of such a formal group generate an abelian extension over a certain unramified extension of , and some ramification properties of these abelian extensions.

In 1971, James Serrin proved that if there exists a positive and bounded solution to the problem in a bounded domain , where Dirichlet 's () and Neumann's () conditions are imposed on the boundary, then is a ball.
In this talk, I will study this problem in the case where is an epigraph of a function , i.e, Thanks to a new monotonicity result in epigraph and a Poincaré-geometric's type inequality, I will show news flattening results in small dimensions, for general non-linearities .

Gentle algebras are a kind of algebras generated through quivers and relations with a strong link with combinatorics. The modules over a gentle algebra are in bijection with dissected surfaces with marked points which gives an easy way to do most of the computations. The description of all the the resolving subcategories of the category of the modules of this algebra is not complete up to now but can be done through the combinatorics. The aim of this talk is to give the tools to understand gentle algebras over examples and to describe what is a resolving subcategory.

Given a gentle quiver, resolving subcategories are subcategories of representations containing all the projective objects and closed under sums, summands, extensions, and kernels of epimorphisms. Combinatorially, those subcategories can be described thanks to their nonprojective indecomposable representations that are closed under some handlable rules. Our main goal is to determine explicitly all the resolving subcategories. To do so, by restricting ourselves to gentle trees, we construct an algorithm derived from step-by-step simplifications of a well-known one and applied to the geometric model. Using the resolving order of the nonprojective indecomposable objects, this algorithm will extract a collection of maximal objects that generates the same resolving subcategory as any given collection of nonprojective indecomposable representations. In this talk, after recalling some notions introduced by Michael last week and introducing useful tools, I aim to explain how this algorithm works with a combinatorial perspective. If time allows, I will discuss some ideas to update this algorithm for some generalized family of gentle quivers. This is a joint work in progress with Michael Schoonheere.

Depuis quelques mois, je me pose des questions au sujet des enjeux environnementaux (qui ne se limitent pas au dérèglement climatique). Le but de cet exposé est d'initier des discussions autour de ces défis et de s'interroger sur l'impact que peuvent avoir les scientifiques. Je commencerai par rappeler brièvement les problèmes auxquels nous faisons face. On s'intéressera ensuite aux solutions faisant appel à de nouvelles technologies et à leur pertinence. Ce sera l'occasion de s'interroger sur notre dépendance à certaines technologies et à leur coût. Je terminerai la présentation par quelques pistes de réflexions pour aller plus loin.

One of the main themes of my thesis is to study the complex braid group using Garside theory. This last group is particular among complex braid groups, as it is not known to admit a Garside group structure, but rather a Garside groupoid structure: the Springer groupoid. Springer groupoids are defined in general for centralizers of regular braids in well-generated complex braid groups, which applies to in particular. In the first part, I will show how we can derive presentations of using its attached Springer groupoid. I will also show how the Springer groupoid can be used to study the center of , along with that of the pure braid group .
In the second part, I will study parabolic subgroups of regular centralizers. Parabolic subgroups of complex braid groups were recently introduced by González-Meneses and Marin. In particular, they proved general results on these parabolic subgroups in every case except that of . I will explain how to deduce a complete description of the parabolic subgroups of a regular centralizer in an arbitrary complex braid group. This description uses in particular the concept of shoal of standard parabolic subgroupoids of Springer groupoids. Lastly, I will show how this description allows us to extend the results of González-Meneses and Marin to thus completing the proof of these results for all complex braid groups.

This talk will introduce the notion of spectrum in algebraic topology, a central concept that provides a framework for studying stable homotopy theory in a more algebraic context. A spectrum can be considered as the homotopy-coherent version of an abelian group, capturing the behavior of different phenomena in topology. We will explore how spectra serve as a tool to understand stable homotopy groups and their interactions, offering a systematic approach to problems in homotopy theory. By discussing their algebraic nature and key applications, this talk will highlight how spectra bridge geometry, algebra, and topology. If time permits, we will mention the place of the category of spectra in the context of the stable -categories.

In this talk, I will introduce bisets and construct the biset category. I will discuss the remarkable factorization properties of morphisms, which are analogous to the factorization of group homomorphisms. If time allows, I will also talk briefly about biset functors and the parametrization of simple functors.

In this work, two families of dynamical systems are studied by using their S-adic representations. First we provide an explicit S-adic representation of rank-one subshifts with bounded spacers. We compute their topological rank and their strong and weak orbit equivalence classes. We observe that they have an induced system that is a Toeplitz subshift having discrete spectrum. We also characterize continuous and non-continuous eigenvalues. Then we revisit a well-known result on Toeplitz subshifts due to Jacobs--Keane giving a sufficient combinatorial condition to ensure discrete spectrum. We characterize spectral properties of the factor maps onto the maximal equicontinuous topological factors by means of coincidences density.

Dans la première partie, nous étudions l'amortissement de la solution de KdV où l'opérateur d'amortissement s'écrit sous la forme d'un filtre de fréquence. Puis, nous introduisons la méthode SAM (Symbol Approximation Method) pour reconstruire le symbole de l'opérateur d'amortissement quand on connaît la solution de KdV amortie comme une suite discrète. Dans la deuxième partie, nous introduisons un système de Boussinesq obtenu en ajoutant un opérateur pseudo-différentiel non local. Nous présentons une approche double: D'abord, nous prouvons théoriquement l'existence d'un développement asymptotique pour la solution du problème de Cauchy associée à ce système de Boussinesq régularisé par rapport au paramètre de régularisation. Ensuite, nous calculons numériquement les coefficients de ce développement et vérifions la validité de ce développement jusqu'à l'ordre 2. D'autre part, nous utilisons la méthode SAM pour approcher le symbole de l'opérateur d'amortissement dans le système de Boussinesq. Dans la troisième partie, nous dérivons l'équation de type Kawahara sur un fond non plat et nous prouvons sa consistance avec le système d'Euler. Après cela, nous proposons une version régularisée jusqu'au bon ordre. Ensuite, nous prouvons l'existence de la solution de cette équation régularisée. Enfin, nous effectuons une simulation numérique pour montrer la convergence de la solution de l'équation de Kawahara vers la solution de sa version régularisée.