Evénements passés

Cardiac arrhythmia is one of the world's leading causes of death. The electrical rhythm of the heart is disturbed, and one treatment is cardiac ablation, which aims to electrically isolate certain parts of the heart. The bidomain model is a very classical mathematical model for cardiac electrophysiology. However it turns out to be unsuitable to describe the application of short and intense electric pulses as used in pulsed electric field ablation (PFA) - a therapeutical innovation in the context of cardiac ablation. We propose a macroscopic model designed to account for PFA and be compatible with cardiac electrophysiology. After deriving it from the cell-scale equations of electrophysiology using two-scale convergence, we present some numerical simulations, followed by an overview of the perspectives from the mathematical point of view (proof of the convergence, analysis of the PDE system), from the modeling point of view (ionic term, fibers orientation) and from the simulation point of view (sensitivity analysis, data assimilation). Particular emphasis will be made on the mathematical homogenization process (two-scale convergence).

In many applications in signal/image processing and statistical problem, we wish to recover information from a limited amount of linear measurements, i.e solve an underdetermined system. Surprisingly, in the mid-2000 E. Candès, J. Romberg and T. Tao showed that under the assumption of an underlying sparsity, we could recover a signal with a small amount of measurement, largely surpassing the previous assumptions based upon the Shannon-Nyquist Sampling theorem. This is can be viewed as the birth of compressive sensing, a rich topic of mathematics that uses a wide array of branches of mathematics, e.g linear algebra, random matrices, convex analysis, optimization. In this talk, we shall discuss some of the main topics regarding compressive sensing and provide an overview of the conditions under which sparse and low-rank vectors/matrices may be recovered from a measurement. Finally, we shall see how this may relate to image decomposition.

This talk is devoted to the study of Schrödinger equations in the presence of resonant interactions that can lead to energy transfer. When the domain is a Diophantine torus we prove that, over very long time scales, the majority of small solutions in high regularity Sobolev spaces do not exchange energy from low to high frequencies. We first provide context on Birkhoff normal form approaches to study of the long-time dynamics of the solutions to Hamiltonian partial differential equations. Then, we introduce the induction on scales normal form, central to our proof. Throughout the iteration, we ensure appropriate non-resonance properties while modulating the frequencies (of the linearized system) with the amplitude of the Fourier coefficients of the initial data. Our main challenge is then to address very small divisor problems and to describe the set of admissible initial data.The results are based on a joint work with Joackim Bernier, and an ongoing joint work with Gigliola Staffilani.

$$abla$$

Chebyshev's bias is the phenomenon stating that the number of prime numbers $p \leq x$ that are congruent to a non-square $a \mod q$, denoted $\pi(x;q,a)$, has strong tendency for these to to be larger than those congruent to a square $b \mod q$, $\pi(x;q,b)$. This bias was quantitatively proven by Rubinstein and Sarnak, in 1994, under some hypotheses, including the Generalized Riemann Hypothesis. In our talk, we will explore their results and extend the discussion to the equivalent concept of Chebyshev's bias in Number Fields.

Decomposing an image into meaningful components is a challenging inverse problem in image processing and has been widely applied to cartooning, texture removal, denoising, soft shadow/spotlight removal, detail enhancement etc. In this talk, I will review the different approaches and models proposed during the years to tackle this problem, focusing on the crucial role played by parameter selection. Then, I will present a two-stage variational model for the additive decomposition of images into piecewise constant, smooth, textured and white noise components and show numerical results of decomposition of textured images corrupted by several kinds of additive white noises.

L'ordre du jour sera le suivant :

1. Approbation du compte rendu de la réunion du conseil scientifique du 20 février (vote)

2. Exposé scientifique de Raphaël Loubère (CSM) : "Chaire PROVE et équipe projet MONADE"

3. Examen de demandes d'ADT/HDR

4. Informations de la direction

5. Questions diverses.

In this talk we are going to explore some parts of the concept of fractal! What are they? How can they possibly have a non-integer dimension? Is this useful? (of course not) The tools we will use come from undergraduate-level measure theory, and a bit of topology. Stay until the end and you will be able to enjoy some nice pictures o:

Cf. https://plmbox.math.cnrs.fr/f/136ed3186ea241e8b980/

En théorie des invariants, on est parfois amené à s'intéresser à la polynomialité de l'algèbre des invariants ${\mathbb C}[V]^G$ des fonctions polynomiales sur un espace vectoriel complexe $V$ de dimension finie, par l'action d'un groupe linéaire algébrique $G$.Par exemple si $G$ est connexe, semi-simple agissant par l'action adjointe (ou coadjointe) sur son algèbre de Lie $V=g$ (isomorphe à son dual), un théorème célèbre de Chevalley permet de conclure que l'algèbre des invariants ${\mathbb C}[V]^G$ est une algèbre de polynômes. D'autre part, un théorème de Kostant permet d'établir un isomorphisme d'algèbres entre ${\mathbb C}[g]^G$ et l'algèbre des fonctions polynomiales sur une "tranche de Kostant", par restriction des fonctions à cette tranche : cela donne ce que l'on peut nommer aussi une "section de Weierstrass" pour ${\mathbb C}[g]^G$.Je passerai d'abord en revue quelques exemples ou contre-exemples de polynomialité de certaines algèbres d'invariants obtenues en faisant agir $G$ sur le dual de son algèbre de Lie par l'action coadjointe, et donnerai quelques exemples de sections de Weierstrass obtenues dans le cas de certaines sous-algèbres paraboliques.Je définirai ensuite la contraction d'Inönü-Wigner d'une sous-algèbre parabolique $p$ d'une algèbre de Lie simple, que l'on peut voir comme une certaine dégénérescence de $p$.En m'appuyant sur des techniques employées pour les sous-algèbres paraboliques, je tenterai d'expliquer comment on peut obtenir des (semi)-invariants pour le cas où $V$ est le dual de la contraction d'Inönü-Wigner d'une sous-algèbre parabolique sur lequel agit le groupe adjoint de la contraction.En particulier, pour les contractions d'Inönü-Wigner de certaines sous-algèbres paraboliques maximales (notamment en type B), je donnerai des sections de Weierstrass pour les algèbres de semi-invariants correspondantes, ce qui prouvera en particulier la polynomialité de ces algèbres de semi-invariants.Ceci est un travail en cours, dont une partie se trouve sur arXiv :

https://arxiv.org/abs/2310.06761

In this talk I am interested in formulas describing the low-lying eigenvalues of the Witten Laplacian $\Delta_V = -h^2\Delta + | V^{\prime} |^2 - h V^{\prime \prime}$. The case where $V$ is a Morse function has been largely studied and here I try to obtain similar results when $V$ has some degeneracy. In the end of the presentation I will also give an example of new behaviors that were not observed in the Morse case.

Generally, polynomial systems that arise in algebraic cryptanalysis have extra structure compared to generic systems, which comes from the algebraic modelling of the cryptographic scheme. Sometimes, part of this extra structure can be caught with polynomial rings with non-standard grading. For example, in the Kipnis-Shamir modelling of MinRank one considers the system over a bi-graded polynomial ring instead. This allows for better approximations of the solving degree of such systems when using Gröbner basis algorithms.

In this talk, I will present ongoing work in which this idea is extended to multi-graded polynomial rings. Furthermore, I will show how we can use this grading to tailor existing algorithms to use this structure and speed up computation. 

Les surfaces del Pezzo et leurs groupes d'automorphismes jouent un rôle important dans l'étude des sous-groupes algébriques du groupe de Cremona du plan projectif.

Sur un corps algébriquement clos, il est classique qu’une surface del Pezzo est soit isomorphe à $\mathbb{P}^{1} \times \mathbb{P}^{1}$ soit à l’éclatement de $\mathbb{P}^{2}$ en au plus $8$ points en position générale, et dans ce cas, les automorphismes des surfaces del Pezzo (de tout degré) ont été décrits. En particulier, il existe une unique classe d'isomorphismes de surfaces del Pezzo de degré $5$ sur un corps algébriquement clos. Dans cet exposé, nous nous intéresserons aux surfaces del Pezzo de degré $5$ définies sur un corps parfait. Dans ce cas, il y a beaucoup de surfaces supplémentaires (comme on peut déjà le voir si le corps de base est le corps des nombres réels), et la classification ainsi que la description du groupe d’automorphismes de ces surfaces sur un corps parfait $\mathbf{k}$ se ramènent à comprendre les actions du groupe de Galois $\operatorname{Gal}(\overline{\mathbf{k}}/\mathbf{k})$ sur le graphe des $(-1)$-courbes.

Consider a control system 𝛛t f + Af = Bu. Assume that 𝛱 is

a projection and that you can control both the systems

 𝛛t f + 𝛱Af = 𝛱Bu,

 𝛛t f + (1-𝛱)Af = (1-𝛱)Bu.

Can you conclude that the first system itself is controllable ? We

cannot expect it in general. But in a joint work with Andreas Hartmann,

we managed to do it for the half-heat equation. It turns out that the

property we need for our case is:

 If 𝛺 satisfies some cone condition, the set {f+g, f∈L²(𝛺), g∈L²(𝛺),

f is holomorphic, g is anti-holomorphic} is closed in L²(𝛺).

The first proof by Friedrichs consists of long computations, and is

very "complex analysis". But a later proof by Shapiro uses quite

general coercivity estimates proved by Smith, whose proof uses some

tools from algebra : Hilbert's nullstellensatz and/or primary ideal

decomposition.

In this first talk, we will introduce the algebraic tools needed and

present Smith's coercivity inequalities. In a second talk, we will

explain how useful these inequalities are to study the control

properties of the half-heat equation.

We consider the standard Ginzburg-Landau system for N-dimensional maps defined in the unit ball for some parameter eps>0. For a boundary data corresponding to a vortex of topological degree one, the aim is to prove the (radial) symmetry of the ground state of the system. We show this conjecture in any dimension N≥7 and for every eps>0, and then, we also prove it in dimension N=4,5,6 provided that the admissible maps are curl-free. This is part of joint works with L. Nguyen, M. Rus, V. Slastikov and A. Zarnescu.

Lien zoom

We will make a tour of related concepts whose motivation lies in

quantum information theory. We consider the detection of entanglement

in unitarily-invariant states, a class of positive (but not completely

positive) multilinear maps, and the construction of tensor polynomial

identities. The results are established through the use of commutative

and noncommutative Positivstellensätze and the representation theory of

the symmetric group.

We discuss a new swarm-based gradient descent (SBGD) method for non-convex optimization. The swarm consists of agents, each is identified with position $x$ and mass $m$. There are three key aspects to the SBGD dynamics: (i) persistent transition of mass from agents at high to lower ground; (ii) a random marching direction, aligned with the steepest gradient descent; and (iii) a time stepping protocol which decreases with $m$.

The interplay between positions and masses leads to dynamic distinction between `heavier leaders’ near local minima, and `lighter explorers’ which explore for improved position with large(r) time steps. Convergence analysis and numerical simulations demonstrate the effectiveness of SBGD method as a global optimizer. 

We give a light talk on optimality of shapes in geometry and physics. First, we recollect classical geometric results that the disk has the largest area (respectively, the smallest perimeter) among all domains of a given perimeter (respectively, area). Second, we recall that the circular drum has the lowest fundamental tone among all drums of a given area or perimeter and reinterpret the result in a quantum-mechanical language of nanostructures. In parallel, we discuss the analogous optimality of square among all rectangles in geometry and physics. As the main body of the talk, we present our recent attempts to prove the same spectral-geometric properties in relativistic quantum mechanics, where the mathematical model is a matrix-differential (Dirac) operator with complex (infinite-mass) boundary conditions. It is frustrating that such an illusively simple and expected result remains unproved and apparently out of the reach of current mathematical tools.

Neural style transfer (NST) is a deep learning technique that produces an unprecedentedly rich style transfer from a style image to a content image. It is particularly impressive when it comes to transferring style from a painting to an image. NST was originally achieved by solving an optimization problem to match the global statistics of the style image while preserving the local geometric features of the content image. The two main drawbacks of this original approach is that it is computationally expensive and that the resolution of the output images is limited by high GPU memory requirements. Many solutions have been proposed to both accelerate NST and produce images with larger size. However, our investigation shows that these accelerated methods all compromise the quality of the produced images in the context of painting style transfer. Indeed, transferring the style of a painting is a complex task involving features at different scales, from the color palette and compositional style to the fine brushstrokes and texture of the canvas. This paper provides a solution to solve the original global optimization for ultra-high resolution (UHR) images, enabling multiscale NST at unprecedented image sizes. This is achieved by spatially localizing the computation of each forward and backward passes through the VGG network. Extensive qualitative and quantitative comparisons, as well as a user study, show that our method produces style transfer of unmatched quality for such high-resolution painting styles. By a careful comparison, we show that state of the art fast methods are still prone to artifacts, thus suggesting that fast painting style transfer remains an open problem.

Joint work with Lara Raad, José Lezama and Jean-Michel Morel.

The Beurling--Selberg extremal approximation problems aim to find optimal unisided bandlimited approximations of a target function of bounded variation. We present an extension of the Beurling--Selberg problems, which we call “of higher-order,” where the approximation residual is constrained to faster decay rates in the asymptotic, ensuring the smoothness of their Fourier transforms. Furthermore, we harness the solution’s properties to bound the extremal singular values of confluent Vandermonde matrices with nodes on the unit circle. As an application to sparse super-resolution, this enables the derivation of a simple minimal resolvable distance, which depends only on the properties of the point-spread function, above which stability of super-resolution can be guaranteed.

The coupling of coastal wave models, such as Boussinesq-type (BT) and Saint-Venant (SV) equations, has been explored since the 1990s. Despite numerous models and coupling examples, the literature exhibits significant disagreement regarding induced artifacts and methods for their analysis. This work aims to elucidate these issues, proposing explanations and a method for evaluating and comparing coupling techniques. We ground our explanation in the mathematical properties of each model's Cauchy and half-line problems, highlighting the sensitivity of these models to numerical artifacts. Additionally, we demonstrate how one-way models provide insights into expected physical effects, unexpected artifacts, and errors relative to 3D models. We demonstrate this analysis with linearized models, where we establish the well-posedness of a popular coupling, characterize analytically the "coupling error" in terms of wave reflections, and prove its asymptotic behavior in shallow water. We will discuss how these insights can be applied to other linear/nonlinear models, providing a foundation for the evaluation and comparison of new coupled coastal wave models.

Les assistants de preuves sont des logiciels permettant de rédiger des énoncés mathématiques et leur démonstration, la compilation du tout garantissant (modulo d'infimes détails) la correction de l'ensemble. Après avoir été surtout promu par la communauté informatique, ils font l'objet d'un engouement croissant chez les mathématicien·nes.

Il y a quelques mois, j'ai formalisé au sein du logiciel Lean/mathlib une démonstration d'un théorème classique, élémentaire, de théorie des groupes : la simplicité du groupe alterné sur au moins 5 lettres, via un critère d'Iwasawa généralement utilisé pour démontrer la simplicité des groupes géométriques.

Je présenterai ce travail, son contexte, et quelques perspectives. (Aucune familiarité avec les assistants de preuve n'est requise.)

Nous considérerons l'interaction entre une molécule diatomique et un pulse laser et verrons comment calculer semi-classiquement la probabilité pour qu'elle change d'état rotationnel. Nous nous concentrerons en particulier sur le calcul de l'indexe de phase, crucial pour une prise en compte précise des interférences quantiques. 

Une singularité de dimension $d$ est quasi-ordinaire par rapport à une projection finie $X$ -----> ${\mathbb C}^d$ si le discriminant de la projection est un diviseur à croisements normaux. Les singularités quasi-ordinaires sont au cœur de l'approche de Jung de la résolution des singularités en caractéristique zéro. En caractéristiques positives, elles ne sont pas très utiles du point de vue de la résolution des singularités, le problème de leurs résolutions étant presque aussi compliqué que le problème de résolution des singularités en général. En utilisant une version pondérée du polyèdre caractéristique de Hironaka (ou tout simplement la géométrie des équations) et des plongements successifs dans des espaces affines de "grandes" dimensions, nous introduisons la notion de singularités Teissier qui coïncide avec les singularités quasi-ordinaires en caractéristiques zéro, mais qui en est différente en caractéristiques positives. Nous démontrons qu'une singularité Teissier définie sur un corps de caractéristique positive est la fibre spéciale d'une famille équisingulière sur une courbe de caractéristique mixte dont la fibre générique (en caractéristique zéro donc) a des singularités quasi-ordinaires. Ici, L'équisingularité de la famille correspond à l'existence d'une résolution plongée simultanée.

Travail en collaboration avec Bernd Schober.

The regular model of a curve is a key object in the study of the arithmetic of the curve, as information about the special fiber of a regular model provides information about its generic fiber (such as rational points through the Chabauty-Coleman method, index, Tamagawa number of the Jacobian, etc). Every curve has a somewhat canonical regular model obtained from the quotient of a regular semistable model by resolving only singularities of a special type called quotient singularities. We will discuss in this talk what is known about the resolution graphs of $Z/pZ$-quotient singularities in the wild case, when $p$ is also the residue characteristic. The possible singularities that can arise in this process are not yet completely understood, even in the case of elliptic curves in residue characteristic 2.

Dans cet exposé nous discuterons des résonances pour un graphe quantique dont sa partie compacte est attachée en un sommet à une arête infinie. Les conditions de transmission à ce sommet dépendent d’un petit paramètre et nous démontrons sous certaines hypothèses sur la géométrie du graphe l’existence d’une famille de résonances dont la partie imaginaire tend vers l’infini.

Ce travail est motivé par une question issue de la physique expérimentale où de telles familles de résonances ont été observées. Je montrerai comment avec des outils mathématiques élémentaires il est possible de montrer l’existence et la localisation de ces résonances.

Il s’agit d’un travail interdisciplinaire en collaboration avec Maxime Ingremeau, Ulrich Kuhl, Olivier Legrand, Junjie Lu (Univ. Nice).

Monsters populate mathematics : topologist's sine, Vitali set, Weierstrass function… These counter-examples to naive intuitions often have in common that they are defined either in a convoluted way, either with an oscillating function like the sine. The o-minimal paradigm allows us to forget those oddness and to make our first intuitions true, by considering only objects that have a "reasonable" definition in a way. What is an o-minimal structure? What examples do we know of? What is happening there? How do they act in complex geometry, in number theory, in optimization?

In this talk, we will introduce a new exact algorithm to solve two-stage stochastic linear programs. Based on the multicut Benders reformulation of such problems, with one subproblem for each scenario, this method relies on a partition of the subproblems into batches. The key idea is to solve at most iterations only a small proportion of the subproblems by detecting as soon as possible that a first-stage candidate solution cannot be proven optimal. We also propose a general framework to stabilize our algorithm, and show its finite convergence and exact behavior. We report an extensive computational study on large-scale instances of stochastic optimization literature that shows the efficiency of the proposed algorithm compared to nine alternative algorithms from the literature. We also obtain significant additional computational time savings using the primal stabilization schemes.

In this talk, I will present the results of a collaboration with Benjamin McKenna on the injective norm of large random Gaussian tensors and uniform random quantum states and, time allowing, describe some of the context underlying this work. The injective norm is a natural generalization to tensors of the operator norm of a matrix and appears in multiple fields. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, known as geometric entanglement. In our recent preprint, we provide high-probability upper bounds in different regimes on the injective norm of real and complex Gaussian random tensors, which corresponds to lower bounds on the geometric entanglement of random quantum states, and to bounds on the ground-state energy of a particular multispecies spherical spin glass model. Our result represents a first step towards solving an important question in quantum information that has been part of folklore.

Simuler numériquement de manière précise l'évolution des interfaces séparant différents milieux est un enjeu crucial dans de nombreuses applications (multi-fluides, fluide-structure, etc). La méthode MOF (moment-of-fluid), extension de la méthode VOF (volume-of-fluid), utilise une reconstruction affine des interfaces par cellule basée sur les fractions volumiques et les centroïdes de chaque phase. Cette reconstruction d'interface est solution d'un problème de minimisation sous contrainte de volume. Ce problème est résolu dans la littérature par des calculs géométriques sur des polyèdres qui ont un coût important en 3D. On propose dans cet exposé une nouvelle approche du calcul de la fonction objectif et de ses dérivées de manière complètement analytique dans le cas de cellules hexaédriques rectangulaires et tétraédriques en 3D. Les résultats numériques montrent un gain important en temps de calcul.

L'existence de métriques kählériennes canoniques (Kähler-Einstein, à courbure scalaire constante, etc...) dans une classe de cohomologie donnée d'une variété kählérienne compacte admet une formulation variationnelle comme équation d'Euler-Lagrange de certaines fonctionnelles. Grâce aux travaux profonds de Darvas-Rubinstein et Chen-Cheng, on sait que de plus qu'elles admettent des points critiques (donc des métriques canoniques) ssi elles satisfont une condition de croissance linéaire. Après avoir passé en revue ces objets fondamentaux, j'expliquerai comment cette caractérisation permet de généraliser des travaux d'Arezzo-Pacard et Seyyedali-Szekelyhidi portant sur la stabilité de telles métriques par éclatement de la variété. Il s'agit d'un travail en collaboration avec Mattias Jonsson et Antonio Trusiani.

Algebraic curves over a finite field $\mathbb{F}_q$ have been a source of great fascination, ever since the seminal work of Hasse and Weil in the 1930s and 1940s. Many fruitful ideas have arisen out of this area, where number theory and algebraic geometry meet, and many applications of the theory of algebraic curves have been discovered during the last decades. 

A very important example of such application was provided in 1977-1982 by Goppa, who found a way to use algebraic curves in coding theory. The key point of Goppa's construction is that the code parameters are essentially expressed in terms of the features of the curve, such as the number $N_q$ of $\mathbb{F}_q$-rational points and the genus $g$. In this light, Goppa codes with good parameters are constructed from curves with large $N_q$ with respect to their genus $g$.

Given a smooth projective, algebraic curve of genus $g$ over $\mathbb{F}_q$, an upper bound for $N_q$ is a corollary to the celebrated Hasse-Weil Theorem,

$$N_q \leq q+ 1 + 2g\sqrt{q}.$$

Curves attaining this bound are called $\mathbb{F}_q$-maximal. The Hermitian curve is a key example of an $\mathbb{F}_q$-maximal curve, as it is the unique curve, up to isomorphism, attaining the maximum possible genus of an $\mathbb{F}_q$-maximal curve. 

It is a result commonly attributed to Serre that any curve which is $\mathbb{F}_q$-covered by an $\mathbb{F}_q$-maximal curve is still $\mathbb{F}_q$-maximal. In particular, quotient curves of $\mathbb{F}_q$-maximal curves are $\mathbb{F}_q$-maximal. Many examples of $\mathbb{F}_q$-maximal curves have been constructed as quotient curves of the Hermitian curve by choosing a subgroup of its very large automorphism group.

It is a challenging problem to construct maximal curves that cannot be obtained in this way, as well as to construct maximal curves with many automorphisms (in order to use the machinery described above). A natural question arises also: given two maximal curves over the same finite field, how can one decide whether they are isomorphic or not? A way to try to give an answer to this question is to look at the birational invariants of the two curves, that is, their properties that are invariant under isomorphism. 

In this talk, we will describe our main contributions to the theory of maximal curves over finite fields and their applications to coding theory. In relation with the question described before, during the talk, the behaviour of the birational invariant of maximal curves will also be discussed.

Gröbner bases lie at the forefront of the algorithmic treatment of polynomial systems and ideals in symbolic computation. They are

defined as special generating sets of polynomial ideals which allow to decide the ideal membership problem via a multivariate version of

polynomial long division. Given a Gröbner basis for a polynomial ideal, a lot of geometric and algebraic information about the

polynomial ideal at hand can be extracted, such as the degree, dimension or Hilbert function.

Notably, Gröbner bases depend on two parameters: The polynomial ideal which they generate and a monomial order, i.e. a certain kind

of total order on the set of monomials of the underlying polynomial ring. Then, the geometric and ideal-theoretic information that can be

extracted from a Gröbner basis depends on the chosen monomial order. In particular, the lexicographic one allows us to solve a polynomial system.

Such a lexicographic Gröbner basis is usually computed through a change of order algorithm, for instance the seminal FGLM algorithm. In this talk,

I will present progress made to change of order algorithms: faster variants in the generic case, complexity estimates for system of critical values, computation

of colon ideals or of generic fibers.

This is based on different joint works with A. Bostan, Ch. Eder, A. Ferguson, R. Mohr, V. Neiger and M. Safey El Din.

In this talk, we present results on the eigenvalue distribution for perturbed magnetic Dirac operators in two dimensions. We derive third-order asymptotic formulas that incorporate a geometric property of the perturbation's support. Notably, our approach allows us to consider some perturbations that do not necessarily have fixed sign, which is one the main novelties of our work.

This is part of a joint work together with Vincent Bruneau.

Adjustable robust optimization problems, as a subclass of multi-stage optimization under uncertainty problems, constitute a class of problems that are very difficult to solve in practice. Although the exact solution of these problems under certain special cases may be possible, for the general case, there are no known exact solution algorithms. Instead, approximate solution methods have been developed, often restricting the functional form of recourse actions, these are generally referred to as “decision rules“. In this talk, we will present a review of existing decision rule approximations including affine and extended affine decision rules, uncertainty set partitioning schemes and finite-adaptability. We will discuss the reformulations and solution algorithms that result from these approximations. We will point out existing challenges in practical use of these decision rules, and identify current and future research directions. When possible we will emphasize the connections to multi-stage stochastic programming literature.

We study the growth of the resolvent of a Toeplitz operator $T_b$, defined on the Hardy space, in terms of the distance to its spectrum $\sigma(T_b)$. We are primarily interested in the case when the symbol $b$ is a Laurent polynomial (\emph{i.e., } the matrix $T_b$ is banded). We show that for an arbitrary such symbol the growth of the resolvent is quadratic, and under certain additional assumption it is linear. We also prove the quadratic growth of the resolvent for a certain class of non-rational symbols.

This is a joint work with S. Kupin and A. Vishnyakova.

A une surface algébrique S on associe son groupe des transformations birationnelles Bir(S). Ces groupes et leurs structures algébriques et dynamiques ont fait l'objet d'études approfondies dans les dernières décennies. Dans cet exposé on verra une réponse positive à une question de Charles Favre concernant des sous-groupes dont tous les éléments sont d'un certain type, dit algébrique. J'expliquerai pourquoi ce résultat technique est intéressant et je l'utiliserai pour décrire des propriétés dynamiques des sous-groupes de type fini de Bir(S). Il s'agit d'un travail commun avec Anne Lonjou et Piotr Przytycki.

La conjecture de Birch et Swinnerton-Dyer prédit un lien entre les points rationnels d'une variété abélienne et les valeurs spéciales de sa fonction L. Cette conjecture est réputée difficile, nous commencerons donc par voire comment l'attaquer à l'aide d'une conjecture intermédiaire où l'on se focalise en un nombre premier $p$. Ensuite, nous verrons comment dans le cas des surfaces abéliennes on peut obtenir une preuve de cette conjecture (la conjecture intermédiaire) en faisant varier $p$-adiquement une classe de cohomologie galoisienne obtenue à partir de la cohomologie de la variété de Shimura de GSp(4).

In this talk, I will present some introductory facts on Hardy-Toeplitz and Bergman-Toeplitz operators. I will also discuss the presence (or absence) of discrete spectrum for a Bergman-Toeplitz operator; this part of the talk will be based on works of Zhao- Zheng et al., 2010- 2020.

Witsenhausen's problem asks for the maximum fraction α_n of the n-dimensional unit sphere that can be covered by a measurable set containing no pairs of orthogonal points. We extended well known optimization hierarchies based on the Lovász theta number, like the Lasserre hierarchy, to Witsenhausen's problem and similar problems. We then showed that these hierarchies converge to α_n, and used them to compute the best upper bounds known for α_n in low dimensions.

Ordre du jour :

1) Informations diverses

2) Listes de diffusion

3) Retours sur le sondage "missions" et propositions

4) Questions diverses.    

Optimal Transport is a useful metric to compare probability distributions and to compute a pairing given a ground cost. Its entropic regularization variant (eOT) is crucial to have fast algorithms and reflect fuzzy/noisy matchings. This work focuses on Inverse Optimal Transport (iOT), the problem of inferring the ground cost from samples drawn from a coupling that solves an eOT problem. It is a relevant problem that can be used to infer unobserved/missing links, and to obtain meaningful information about the structure of the ground cost yielding the pairing. On one side, iOT benefits from convexity, but on the other side, being ill-posed, it requires regularization to handle the sampling noise. This work presents an in-depth theoretical study of the l1 regularization to model for instance Euclidean costs with sparse interactions between features. Specifically, we derive a sufficient condition for the robust recovery of the sparsity of the ground cost that can be seen as a far reaching generalization of the Lasso's celebrated Irrepresentability Condition. To provide additional insight into this condition, we work out in detail the Gaussian case. We show that as the entropic penalty varies, the iOT problem interpolates between a graphical Lasso and a classical Lasso, thereby stablishing a connection between iOT and graph estimation, an important problem in ML.

Two-stage stochastic programs (TSSP) are a classic model where a decision must be made before the realization of a random event, allowing recourse actions to be performed after observing the random values. For example, many classic optimization problems, like network flows or facility location problems, became TSSP if we consider, for example, a random demand. 

Benders decomposition is one of the most applied methods to solve TSSP with a large number of scenarios. The main idea behind the Benders decomposition is to solve a large problem by replacing the values of the second-stage subproblems with individual variables, and progressively forcing those variables to reach the optimal value of the subproblems, dynamically inserting additional valid constraints, known as Benders cuts. Most traditional implementations add a cut for each scenario (multi-cut) or a single-cut that includes all scenarios. 

In this paper we present a novel Benders adaptive-cuts method, where the Benders cuts are aggregated according to a partition of the scenarios, which is dynamically refined using the LP-dual information of the subproblems. This scenario aggregation/disaggregation is based on the Generalized Adaptive Partitioning Method (GAPM). We formalize this hybridization of Benders decomposition and the GAPM, by providing sufficient conditions under which an optimal solution of the deterministic equivalent can be obtained in a finite number of iterations. Our new method can be interpreted as a compromise between the Benders single-cuts and multi-cuts methods, drawing on the advantages of both sides, by rendering the initial iterations faster (as for the single-cuts Benders) and ensuring the overall faster convergence (as for the multi-cuts Benders). 

Computational experiments on three TSSPs validate these statements, showing that the new method outperforms the other implementations of Benders method, as well as other standard methods for solving TSSPs, in particular when the number of scenarios is very large.

Given an inner function $\Theta \in H^\infty(\mathbb D)$ and $[g]$ in the quotient algebra $H^\infty/ \Theta H^\infty$,

its quotient norm is 

$\|[g]\|:= \inf \left\{ \|g+\Theta h\|_\infty, h \in H^\infty \right\}$. We show that 

when $g$ is normalized so that $\|[g]\|=1$, the quotient norm of its inverse can be made 

arbitrarily close to $1$ by imposing $|g(z)|\ge 1- \delta$ when $\Theta(z)=0$, with $\delta>0$ small enough, 

(call this property SIP)

 if and only if the function $\Theta$ satisfies the following growth property:

$$

\lim_{t\to 1} \inf\left\{ |\Theta(z)|: z \in \mathbb D, \rho(z, \Theta^{-1} \{0\} ) \ge t \right\} =1,

$$

where $\rho$ is the usual pseudohyperbolic distance in the disc, $\rho(z,w):= \left| \frac{z-w}{1-z\bar w}\right|$.

We prove that an inner function is SIP if and only if for any $\eps>0$, the set $\{ z: 0< |\Theta (z) | < 1-\eps\}$

cannot contain hyperbolic disks of arbitrarily large radius.

Thin Blaschke products provide an example of such functions. Some SIP Blaschke products fail to be interpolating

(and thus aren't thin), while there exist Blaschke products which are interpolating and fail to be SIP. 

We also study the functions which can be divisors of SIP inner functions.

Sur une variété riemannienne (possiblement singulière), pour chaque classe d'homologie la norme stable mesure la longueur du plus court représentant possible de cette classe. C'est un raffinement naturel du concept de systole, et on s'attend à ce que la norme stable contienne beaucoup d'information géométrique: en contrepartie, la norme stable est généralement très difficile à calculer, si bien qu'il existe très peu d'exemples explicites. 

Dans cet exposé je m'intéresserai à la norme stable des surfaces plates. Plus précisément, je montrerai qu'il est possible de calculer la norme stable des tores plats fendus avec la suite de Farey. Ensuite, en recollant des tores fendus je montrerai que l'on obtient des surfaces de demi-translation sur lesquelles la norme stable est connue. Enfin, je montrerai que sur ces surfaces le nombre de classes d'homologie minimisées par des courbes simples de longueur inférieure à un réel x croît sous-quadratiquement en x.

The main tool of soliton theory (aka completely integrable systems) is the inverse scattering transform (IST) which relies on solving the Faddeev-Marchenko integral equation. The latter amounts to inverting the I+Hankel operator which historically was done by classical techniques of integral operators and the theory of Hankel operations was not used. In the recent decade however the interest in the soliton community has started shifting from classical initial conditions of integrable PDEs to more general ones (aka none classical initial data) for which the classical IST no longer works. In this talk, on the prototypical example of the Cauchy problem for the Korteweg-de Vries (KdV) equation, we show how the classical IST can be extended to serve a broad range of physically interesting initial data. Our approach is essentially based on the theory of Hankel operators.

Les anneaux de déformation potentiellement Barsotti–Tate sont un outil essentiel pour l’obtention de résultats profonds en arithmétique, comme la conjecture de Shimura–Taniyama–Weil ou la conjecture de Breuil–Mézard. Néanmoins leur géométrie n’est pas encore bien comprise, et présente de comportement variés avec la parution de points irréguliers ou non-normaux (comme montré par des exemples et conjectures de Caruso–David–Mézard). Dans cet exposé nous discuterons comment les champs de modules de Breuil–Kisin peuvent être utilisés pour décrire la géométrie des champs des représentations potentiellement et modérément Barsotti–Tate (en rang 2, pour des extension non ramifiées de $\mathbf{Q}_p$), en utilisant la théorie des modèles locaux des groupes des lacets en caractéristique mixte. L’outil technique principal est une analyse de la p-torsion d’un complexe tangent pour relever des cartes affines pour des images schématiques entre champs de Breuil–Kisin et des représentations Galoisiennes. Avec ce procédé, nous obtenons un algorithme pour calculer des présentations explicites des anneaux de déformation potentiellement modérément Barsotti–Tate pour les représentations Galoisiennes de dimension 2 pour des extensions non-ramifiées de $\mathbf{Q}_p$. Ceci est un travail en commun avec B. Le Hung et A. Mézard.

Page de l'événement : https://indico.math.cnrs.fr/event/11353/overview

Nous présentons plusieurs exemples de fonctions différentiables ayant des propriétés pathologiques. Nous démontrerons en particulier le résultat suivant, obtenu en collaboration avec A. Daniilidis et S. Tapia. Pour tout N≥1, il existe une fonction f de R ^N dans R, localement Lipschitzienne et différentiable en tout point, telle que pour tout compact connexe d'intérieur non vide, il existe x dans R^N tel que K={ lim Df(x_n); (x_n) converge vers x}.

We are interested in standing waves for the nonlinear Schrodinger equation with double power nonlinearities, whose typical example is the cubic-quintic nonlinearity in $R^3$.

The cubic term is focusing and the quintic term can be chosen to be both focusing and defocusing.

I will introduce my recent results on the existence, uniqueness and non-degeneracy of ground state solutions based on the variational method and the shooting method

We will discuss computable descriptions of isomorphism classes in a fixed isogeny class of both polarised abelian varieties over finite fields (joint work with Bergström-Marseglia) and Drinfeld modules over finite fields (joint work with Katen-Papikian). 

More precisely, in the first part of the talk we will describe all polarisations of all abelian varieties over a finite field in a fixed isogeny class corresponding to a squarefree Weil polynomial, when one variety in the isogeny class admits a canonical lifting to characteristic zero. The computability of the description relies on applying categorical equivalences, due to Deligne and Centeleghe-Stix, between abelian varieties over finite fields and fractional ideals in étale algebras. 

In the second part, we will use an action of fractional ideals, inspired by work of Hayes, to compute isomorphism classes of Drinfeld modules. As a first step and a problem of independent interest, we prove that an isogeny class contains a Drinfeld module whose endomorphism ring is minimal if and only if the class is either ordinary or defined over the prime field. We obtain full descriptions in these cases, that can be compared to the Drinfeld analogues of those of Deligne and Centeleghe-Stix, respectively.

L’ordre du jour sera le suivant :

1) Adoption du Compte-Rendu du Conseil de Laboratoire du 2 avril 2024 (vote)

2) Informations générales

) Premières discussions sur le Plan de Gestion des Emplois des enseignants-chercheurs 2025

4) Questions diverses

We present a new model for heat transfer in compressible fluid flows. The model is derived from Hamilton’s principle of stationary action in Eulerian coordinates, in a setting where the entropy conservation is recovered as an Euler–Lagrange equation. A sufficient criterion for the hyperbolicity of the model is formulated. The governing equations are asymptotically consistent with the Euler equations for compressible heat conducting fluids, provided the addition of suitable relaxation terms. A study of the Rankine–Hugoniot conditions and Clausius–Duhem inequality is performed for a specific choice of the equation of state. In particular, this reveals that contact discontinuities cannot exist while expansion waves and compression fans are possible solutions to the governing equations. Evidence of these properties is provided on a set of numerical test cases.

In this talk, I will discuss the self-adjointness of the two-dimensional Dirac operator coupled with a singular combination of electrostatic and Lorentz scalar $\delta$-interaction, supported on a closed Lipschitz curve. The main new ingredients are an explicit use of the Cauchy transform on non-smooth curves and a direct link with the Fredholmness of a singular boundary integral operator. This results in a proof of self-adjointness for a new range of coupling constants, which includes and extends all previous results for this class of problems. The study is particularly precise for the case of curvilinear polygons, as the angles can be taken into account in an explicit way. In particular, if the curve is a curvilinear polygon with obtuse angles, then there is a unique self-adjoint realization with domain contained in $H^{1/2}$ for the full range of non-critical coefficients in the transmission condition. The results are based on a joint work with Badreddine Benhellal and Konstantin Pankrashkin.

Translation surfaces arise naturally in many different contexts, for example when unfolding billard trajectories or when equipping a Riemann surface with an abelian differential. Most visually, they can be described by (finitely or infinitely many) polygons that are glued along edges which are parallel and have the same length.

In this talk, we will be interested in the Veech groups of translation surfaces, that is, the stabilizer of the natural GL(2,R) action on the moduli space for a given translation surface. Although Veech groups have been studied for several decades, they are in itself not fully understood yet. In particular, it is not known in general whether a given abstract group can be realized as the Veech group of a translation surface.

After introducing the realization problem for Veech groups, I will speak about some recent progress in this direction for infinite translation surfaces. This is joint work with Mauro Artigiani, Chandrika Sadanand, Ferrán Valdez, and Gabriela Weitze-Schmithuesen.

Certain sets of germs at $+ \infty$ of monotone bijections between neighborhoods of $+ \infty$ form groups under composition. This is the case for germs of functions definable in an o-minimal structure, for certain germs lying in Hardy fields, as well as for more abstract functions defined on fields of formal series, such as transseries. 

In this talk I will describe properties of the resulting ordered groups, and show that they can be studied using valuation-theoretic tools adapted to this non-commutative context.

Une partition d'un entier n est une suite décroissante d'entiers positifs de somme n. Cette définition est étroitement liée au groupe symétrique et à sa théorie des représentations. Notamment, pour étudier les représentations sur un corps de caractéristique p on peut utiliser le procédé de p-régularisation, introduit par James, qui à une partition associe une partition p-régulière, c'est-à-dire une partition dont aucune part ne se répète p fois ou plus.Une mesure de probabilité classique sur l'ensemble des partitions de n est la mesure de Plancherel. Un résultat spectaculaire de Kerov–Vershik et Logan–Shepp (1977) donne une forme limite asymptotique pour les grandes partitions tirées selon la mesure de Plancherel. Dans cet exposé, nous montrerons ce que devient ce résultat pour la p-régularisation de grandes partitions. Notamment, il y a toujours existence d'une forme limite, qui est donnée par le « secouage » (shaking) de la courbe de Kerov-Vershik-Logan-Shepp.

I will present recent joint work with Magnus Carlson, where we provide formulas for 3-fold Massey products in the étale cohomology of the ring of integers of a number field. Using these formulas, we identify the first known examples of imaginary quadratic fields with a class group of p-rank two possessing an infinite p-class field tower, where p is an odd prime. Furthermore, we establish a necessary and sufficient condition, in terms of class groups of p-extensions, for the vanishing of 3-fold Massey products. As a consequence, we offer an elementary and sufficient condition for the infinitude of class field towers of imaginary quadratic fields. Additionally, we disprove McLeman’s (3,3)-conjecture.

Reduced order models (ROMs) are parametric mathematical models derived from PDEs using previously computed solutions. In many applications, the solution space turns out to be low dimensional, so that one can trade a minimal loss of accuracy for speed and scalability of the numerical model. ROMs counteract the curse of dimensionality by significantly reducing the computational complexity. Overall, reduced order models have reached a certain level of maturity in the last decade, allowing their implementation in large-scale industrial codes, mainly in structural mechanics. Nevertheless, some hard points remain. Parametric problems governed by advection fields or solutions with a substantial compact support such as shock waves suffer from a limited possibility of dimensional reduction and, at the same time, from an insufficient generalization of the model (out-of-sample solutions). The main reason is that the solution space is usually approximated by an affine or linear representation. In this thesis, we aim to contribute to the use of non-intrusive model reduction methods by working on three axes: (i) Application to unsteady computations with non-intrusive interpolation methods; (ii) Use of hybrid models linking reduced models and numerical simulation models with a domain decomposition type approach; (iii) Application to complex industrial problems The flutter problem on a fin will be used as a first complex application case. Indeed, this fluid-structure problem presents very different behaviors according to the flow regimes and is very expensive to simulate without simplifying assumptions. Thus, a hybrid model could accelerate the computation time while remaining accurate in the complex areas. This CIFRE thesis financed by Ingeliance is part of the chaire PROVE financed by ONERA and the Nouvelle Aquitaine region.

Holomorphic dynamics studies the evolution of complex manifolds under the iteration of holomorphic maps.

While significant progress has been made in understanding the theory of one-dimensional holomorphic dynamics, the transition to higher dimensions still presents difficult challenges since the situation is vastly different from the one-dimensional case.

Even only the study of the dynamics of automorphisms (i.e. holomorphic maps injective and surjective) in two dimensions already poses deep difficulties, and the construction of significant examples is an active area of research.

In this talk, we provide an overview of the dynamics in several complex variables, focusing particularly on the stable dynamics of automorphisms of C^2. We introduce concepts such as Fatou sets, polynomial and transcendental Hénon maps, and limit functions. Finally, we address two recently resolved questions that refer to the current state of my research  (a joint work with A. M. Benini and A. Saracco):

Can limit sets for (non-recurrent) Fatou components be hyperbolic?

Can limit sets be distinct?

This paper explores strategic optimization in updating essential medical kits crucial for humanitarian emergencies. Recognizing the perishable nature of medical components, the study emphasizes the need for regular updates involving complex recovery, substitution and disposal processes with the associated costs. The goal is to minimize costs over an unpredictable time horizon. The introduction of the kit-update problem considers both deterministic and adversarial scenarios. Key performance indicators (KPIs), including updating time and destruction costs, are integrated into a comprehensive economic measure, emphasizing a strategic and cost-effective approach.

The paper proposes an innovative online algorithm utilizing available information at each time period, demonstrating its 2-competitivity. Comparative analyses include a stochastic multi-stage approach and four other algorithms representing former and current MSF policies, a greedy improvement of the MSF policy, and the perfect information approach.

Analytics results on various instances show that the online algorithm is competitive in terms of cost with the stochastic formulation, with differences primarily in computation time. This research contributes to a nuanced understanding of the kit-update problem, providing a practical and efficient online algorithmic solution within the realm of humanitarian logistics.

This talk presents a family of algebraically constrained finite element schemes for hyperbolic conservation laws. The validity of generalized discrete maximum principles is enforced using monolithic convex limiting (MCL), a new flux correction procedure based on representation of spatial semi-discretizations in terms of admissible intermediate states. Semi-discrete entropy stability is enforced using a limiter-based fix. Time integration is performed using explicit or implicit Runge-Kutta methods, which can also be equipped with property-preserving flux limiters. In MCL schemes for nonlinear systems, problem-dependent inequality constraints are imposed on scalar functions of conserved variables to ensure physical and numerical admissibility of approximate solutions. After explaining the design philosophy behind our flux-corrected finite element approximations and showing some numerical examples, we turn to the analysis of consistency and convergence. For the Euler equations of gas dynamics, we prove weak convergence to a dissipative weak solution. The convergence analysis to be presented in this talk is joint work with Maria Lukáčová-Medvid’ová and Philipp Öffner.

We consider the moduli space of Abelian differentials on compact Riemann surfaces. It is stratified by the degree of the zeros of the differential and each stratum has a linear structure coming from period coordinates. Each stratum admits an action by GL(2,R) and this action is relevant in the study of billiard dynamics. I aim to discuss works in collaboration with Julian Rüth and Kai Fu in which we design computer programs to guess and certify GL(2,R)-orbit closures.

For threefolds over the complex numbers, much is understood about the rationality problem, i.e. the property of being birational to projective space. However, much less is known over fields that are not algebraically closed. For example, a threefold defined over the real numbers could become rational after base changing to C, but in general, the complex rationality construction may not descend to R. In this talk, we study this question for real threefolds with a conic bundle structure. This talk is based on joint work with S. Frei, S. Sankar, B. Viray, and I. Vogt, and joint work with M. Ji.

Une partie de la Cellule Informatique participe à la semaine de travail de l'équipe de la PLM au CIRM du 24 au 28 juin 2024 pouvant impacter des délais de traitements des demandes plus longs que d'habitude.

Pensez à anticiper les retraits et les réservations de matériel par exemple.

SQIsign is an isogeny-based signature scheme in Round 1 of NIST’s recent alternate call for signature schemes. In this talk, we will take a closer look at SQIsign verification and demonstrate that it can be performed completely on Kummer surfaces. In this way, one-dimensional SQIsign verification can be viewed as a two-dimensional isogeny between products of elliptic curves. Curiously, the isogeny is then defined over Fp rather than Fp2. Furthermore, we will introduce new techniques that enable verification for compression signatures using Kummer surfaces, in turn creating a toolbox for isogeny-based cryptography in dimension 2.This is based on joint work with Krijn Reijnders.

We are interested here in questions related to the maximal regularity of solutions of elliptic problems div $(A abla\, u) = f$ in $\Omega$ with Dirichlet boundary condition. For the last 40 years, many works have been concerned with questions when $A$ is a matrix or a function and when $\Omega$ is a Lipschitz domain. Some of them contain incorrect results that are corrected in the present work.

We give here new proofs and some complements for the case of the Laplacian, the Bilaplacian and the operator $\mathrm{div}\, (A abla)$, when ${\bf A}$ is a matrix or a function. And we extend this study to obtain other regularity results for domains having an adequate regularity. We give also new results for the {Dirichlet-to-Neumann operator for Laplacian and Bilaplacian.

Using the duality method, we can then revisit the work of Lions-Magenes, concerning the so-called very weak solutions, when the data are less regular.

Thanks to the interpolation theory, it permits us to extend the classes of solutions and then to obtain new results of regularity.

Dans cette présentation, nous exposons un problème bi-niveaux, multi leader- single follower, de tarification sur le marché de l'électricité. Les meneurs correspondent aux sociétés productrices d'énergie qui doivent soumettre une offre à un agent centralisateur (ISO). L'ISO sélectionne les offres et distribue la demande sur le réseaux. Les générateurs sont composés de plusieurs technologies de production, avec différents coûts et quantités de pollution produite. Nous exposerons les particularités de ce problème ainsi que les différents algorithmes qui permettent de trouver un ou plusieurs équilibres de Nash.

In the first part of the talk, we introduce the concept: Controllability of Differential Equations. Then we give some examples in finite (ODE) and infinite dimensional(PDE) contexts. We recall the controllability results of the Transport and Heat equation.

In the second part of the talk, we consider compressible Navier-Stokes equations in one dimension, linearized around a positive constant steady state . It is a Coupled system of Transport (for density) and Heat type (for velocity) equations.  We study the boundary null-controllability  of this linearized system in an interval when a Dirichlet control function is acting either only on the density or only on the velocity component at one end of the interval. In this setup, we state some new control results which we have obtained. We see that these controllability results are optimal/sharp concerning the regularity of initial states (in the velocity case) and time (in the density case). The proof is based on a spectral analysis and on solving a mixed parabolic-hyperbolic moments problem and a parabolic hyperbolic joint Ingham-type inequality.  This is a joint work with Kuntal Bhandari, Rajib Dutta and Jiten Kumbhakar. Finally, the talk ends with some ongoing and future directions of research.

Un sous-ensemble $A$ de $\mathbf{N}$ est dit dense s’il est de densité asymptotique supérieure positive, et épars s’il est de densité nulle. Un théorème classique de Furstenberg et Sarközy dit que si $A$ est dense, alors il existe des éléments distincts $a, a'$ dans $A$ tels que $a-a' = n^2$ pour un certain entier $n$. Un ensemble $H$ d'entiers positifs est dit intersectif si l'on peut remplacer l'ensemble des carrés par $H$ dans le théorème de Furstenberg-Sarközy, autrement dit si $(A-A) \cap H$ est non vide. L'étude des ensembles intersectifs se trouve à l'intersection de plusieurs domaines de mathématiques, y compris la théorie des nombres, la combinatoire et la théorie ergodique.

Dans cet exposé, je discuterai dans quelle mesure ce phénomène est toujours valable, lorsque $A$ est un sous-ensemble dense de l'ensemble des nombres premiers, ou plus généralement d'un ensemble épars quelconque $E$ (à la place de $\mathbf{N}$). Il s'agit d'un travail en commun avec J. T. Griesmer, P.-Y. Bienvenu et A. Le.

We discuss the computational problem of finding pairs of consecutive smooth integers, also known as smooth twins. Such twins have had some relevance in isogeny-based cryptography and reducing the smoothness bound of these twins aids the performance of these cryptosystems. However searching for such twins with a small smoothness bound is the most challenging aspect of this problem especially since the set of smooth twins with a fixed smoothness bound is finite. This talk presents new large smooth twins which have a smaller smoothness bound compared to twins found with prior approaches.

L'ordre du jour sera le suivant :

1) Adoption du Compte-Rendu du conseil du 11 juin (vote)

2) Informations générales

3) Plan de Gestion des Emplois des enseignants-chercheurs 2025

4) Questions diverses

In this talk we show that spectral shift function can be expressed via (regularised) determinant of Birman-Schwinger operator in the setting suitable for higher order differential operators. We then use this expression to show that the spectral shift function for massless Dirac operator is continuous everywhere except possibly at zero. Behaviour of the spectral shift function at zero is influenced by the presence of zero eigenvalue and/or resonance of the perturbed Dirac operator.

On montrera comment résoudre le problème de F. John sur les objets flottants dans le cas d'un objet fixe. Il s'agit de comprendre comment des vagues linéaires se comportent en présence d'un objet partiellement immergé. La difficulté principale vient du fait que le domaine du fluide présente des singularités aux points de contact entre l'objet et la surface de l'eau.

Mercredi 10/07

14h00 Jurgen Angst (Univ. Rennes)

Title :  TLC in total variation for beta-ensembles

Abstract : In this talk, we study the fluctuations of linear statistics associated with beta-ensembles, which are statistical physics models generalizing random matrix spectra. In the context of random matrices precisely (e.g. GOE, GUE), the "law of large numbers" is Wigner's theorem, which states that the empirical measure of eigenvalues converges to the semicircle law, and fluctuations around equilibrium can be shown to be Gaussian. We will describe how this result generalizes to beta-ensembles and how it is possible to quantify the speed of convergence to the normal distribution. We obtain optimal rates of convergence for the total variation distance and the Wasserstein distances. To do this, we introduce a variant of Stein's method for a generator $L$ that is not necessarily invertible, and which allows us to establish the asymptotic normality of observables that are not in the image of $L$. Time permitting, we will also look at the phenomenon of super-convergence, which ensures that convergence to the normal law takes place for very strong metrics, typically the $C^{\infty}$-convergence of densities. The talk is based on recent works with R. Herry, D. Malicet and G. Poly.

15h00 Nicolas Juillet (Univ. Haute-Alsace)

Title :  Exact interpolation of 1-marginals

Abstract : I shall present a new type of martingales that exactly interpolates any given family of 1-dimensional marginals on R1 (satisfying the suitable necessary assumption). The construction makes use of ideas from the (martingale) optimal transportation theory and relies on different stochastic orders. I shall discuss of related constructions and open questions (joint work with Brückerhoff and Huesmann).

16h00 Kolehe Coulibaly-Pasquier (Inst. Ellie Cartan)

Title :  On the separation cut-off phenomenon for Brownian motions on high dimensional rotationally

symmetric compact manifolds.

Abstract : Given a family of compact, rotationally symmetric manifolds indexed by the dimension and a weighted function, we will study the cut-off phenomena for the Brownian motion on this family.

Our proof is based on the construction of an intertwined process, a strong stationary time, an estimation of the moments of the covering time of the dual process, and on the phenomena of concentration of the measure.

We will see a phase transition concerning the existence or not of cut-off phenomena, which depend on the shape of the weighted function.

Jeudi 11/07

11h00 Masha Gordina (Univ. of Connecticut)

Title :   Dimension-independent functional inequalities on sub-Riemannian manifolds

Abstract : The talk will review recent results on gradient estimates, log Sobolev inequalities, reverse Poincare and log Sobolev inequalities on a class of sub-Riemannian manifolds. As for many of such setting curvature bounds are not available, we use different techniques including tensorization and taking quotients. Joint work with F. Baudoin, L. Luo and R. Sarkar.

In this thesis we study couplings of subelliptic Brownian motions in several subRiemannian manifolds: the free, step $2$ Carnot groups, including the Heisenberg group, as well as the groups of matrices $SU(2)$ et $SL(2,\mathbb{R})$.

Taking inspiration from previous works on the Heisenberg group we obtain successful non co-adapted couplings on $SU(2)$, $SL(2,\mathbb{R})$ (under strong hypothesis) and also on the free step $2$ Carnot groups with rank $n\geq 3$. In particular we obtain estimates of the coupling rate, leading to gradient inequalities for the heat semi-group and for harmonic functions. We also describe the explicit construction of a co-adapted successful coupling on $SU(2)$.

Finally, we develop a new coupling model "in one sweep" for any free, step $2$ Carnot groups. In particular, this method allows us to obtain relations similar to the Bismut-Elworthy-Li formula for the gradient of the semi-group by studying a change of probability on the Gaussian space.

L'ordre du jour sera le suivant :

1) Adoption du Compte-Rendu du conseil du 2 juillet (vote)

2) Informations générales

3) Approbation du Document Unique d'Évaluation des Risques (DUER) (vote)

Retour sur l'enquête sur les propos racistes à l'IMB

4) Approbation de la demande DIALOG de l'IMB (vote)

5) Approbation du texte relatif aux préconisations de l'IMB concernant les

déplacements en avion dans le cadre des missions (vote)

6) Exposé scientifique de Yann Traonmilin : l'IA à l'IMB

La direction propose de nommer Y. Traonmilin responsable de la thématique IA à

l'IMB (vote)

7) Questions diverses

8) Uniquement pour le Conseil Scientifique : examen des demandes d'HDR

La motivation principale de cet exposé est de trouver des temps forts de stationnarité pour des processus de Markov (X_t), c'est à dire des temps d'arrêt T tels que X_T soit à l'équilibre, T et X_T soient indépendants. Pour trouver des temps fort de stationnarité T, il est naturel et très facile dans certains cas d'utiliser des processus duaux (D_t), tels que T soit un temps d'atteinte d'un certain état pour le processus dual. On étudiera l'entrelacement entre (X_t) et (D_t). On donnera des exemples pour des chaînes de Markov à espace d'états finis, puis on s'intéressera au mouvement brownien avec des processus duaux à valeur ensemble. L'étonnant théorème "2M-X" de Pitman donne un exemple d'entrelacement du mouvement brownien dans le cercle. On généralisera ce théorème aux variétés riemanniennes compactes, et on construira des temps forts de stationnarité. On étudiera la transition de phase en grande dimension. Finalement, on s'intéressera à des duaux à valeur mesure."

In 1987, Coleman submitted a certain conjecture for curves of genus greater than one over complete discrete valuation fields of mixed characteristics. Roughly speaking, this conjecture asserts that the residue fields of the torsion points of the Jacobian lying on the curve are unramified over the base field. As an application, (the already proven part of) this conjecture gives another proof of the Manin-Mumford conjecture (Raynaud's theorem) on the finiteness of torsion points on curves. In this talk, after overviewing some known results on the Coleman conjecture by Coleman, Tamagawa, Hoshi, et al., I explain my recent approach to the conjecture using Raynaud's classification of vector space schemes and discuss “quasi-supersingular group schemes'', which I introduced in another possible approach to the conjecture.

Our work aims to quantify the benefit of storage flexibilities such as a battery on several short term electricity markets. We especially focus on two different markets, the intraday market (ID) and the activation market of the automatic Frequency Restoration Reserve (aFRR), also known as the secondary reserve. We propose algorithms to optimize the management of a small battery (<= 5 MWh) on these markets. In this talk, we first present the modeling of the problem, then we show some theoretical results and numerical simulations. We conclude by listing some limitations of the method.

The existence of Kaehler-Einstein metrics on Fano 3-folds can be determined by studying lower bounds of stability thresholds. An effective way to verify such bounds is to construct flags of point-curve-surface inside the Fano 3-folds. This approach was initiated by Abban-Zhuang, and allows us to restrict the computation of bounds for stability thresholds only on flags. We employ this machinery to prove K-stability of terminal quasi-smooth Fano 3-fold hypersurfaces. This is deeply intertwined with the geometry of the hypersurfaces: in fact, birational rigidity and superrigidity play a crucial role. The superrigid case had been attacked by Kim-Okada-Won. In this talk, I will discuss the K-stability of strictly rigid Fano hypersurfaces via Abban-Zhuang Theory. This is a joint work with Takuzo Okada.

The Benjamin-Ono (BO) equation is a nonlocal asymptotic model for the unidirectional propagation of weakly nonlinear, long internal waves in a two-layer fluid. The equation was introduced formally by Benjamin in the '60s and has been a source of active research since. For instance, the study of the long-time behavior of solutions, stability of traveling waves, and the low regularity well-posedness of the initial value problem. However, despite the rich theory for the BO equation, it is still an open question whether its solutions are close to the ones of the original physical system.

In this talk, I will explain the main steps involved in the rigorous derivation of the BO equation.

In this talk, we present a new construction of quantum codes that enables the integration of a broader class of classical codes into the mathematical framework of quantum stabilizer codes. Next, we discuss new connections between twisted codes and linear cyclic codes and provide novel bounds for the minimum distance of twisted codes. We demonstrate that classical tools, such as the Hartmann-Tzeng minimum distance bound, are applicable to twisted codes. This has led to the discovery of five new infinite families and many other examples of record-breaking, and sometimes optimal, binary quantum codes. Additionally, we explore the role of the $\gamma$ value on the parameters of twisted codes and present new findings regarding the construction of twisted codes with different $\gamma$ values but identical parameters.

In this talk we present an information discovery framework in optimization under uncertainty. In this framework, uncertain parameters are assumed to be “discoverable” by the decision-maker under a given discovery (query) budget. The decision-maker therefore decides which parameters to discover (query) in a first stage then solves an optimization problem with the discovered uncertain parameters at hand in a second stage. We model this problem as a two-stage stochastic program and develop decomposition algorithms for its solution. Notably, one of the algorithms we propose reposes on a Dantzig-Wolfe reformulation of the recourse problem. This allows to treat the cases where the recourse problem involves binary decisions without relying on integer L-Shaped cuts. In its implementation, this approach requires to couple Benders’ and Dantzig-Wolfe reformulation with the subproblems of the Benders’ algorithm being solved using the column generation algorithm. We present some preliminary results on the kidney exchange problem under uncertainty of the compatibility information.

Cet exposé se veut une introduction par l'exemple à une théorie de la causalité développée depuis la fin des années 90 par Judea Pearl. Elle lui a valu une partie de son prix ACM Turing en 2011, l'équivalent en informatique du prix Abel. Nous considérerons un modèle classique dont des hypothèses sont formulées par un graphe de cause. Il contient notamment une cause commune inobservable et une variable éthiquement non contrôlable. Adoptant ici un vocabulaire informatique, nous traiterons en détail une requête sur les traces d'exécution d'un programme inexécutable à l'aide de statistiques sur les traces d'un autre programme lui exécutable. Les éléments rencontrés lors de cette analyse seront alors utilisés dans une présentation de l'architecture globale de la démarche de Pearl. Si le temps le permet, nous discuterons quelques éléments sur les calculs probabilistes dans ce contexte qui s'avèrent souvent reformulable uniquement en terme de théorie des graphes.

Networks of hyperbolic PDEs arise in different applications, e.g. modeling water- or gas-networks or road traffic. In the first part of this talk we discuss modeling aspects of coupling conditions for hyperbolic PDEs.

Starting from an kinetic description we derive coupling conditions for the associated macroscopic equations. For this process a detailed description of the boundary layer is important. In the second part appropriate numerical methods are considered.

Different high order approaches are compared and applications to district heating or water networks are discussed.

It is known that the partition function $p(n)$ obeys Benford's law in any integer base $b\ge 2$. In a recent paper, Douglass and Ono asked for an explicit version of this result. In my talk, I will show that for any string of digits of length $f$ in base $b$, there is $n\le N(b,f)$, where 

$$N(b,f):=\exp\left(10^{32} (f+11)^2(\log b)^3\right)$$

such that $p(n)$ starts with the given string of digits in base $b$. The proof uses a lower bound for a nonzero linear form in logarithms of algebraic numbers with algebraic coefficients due to  Philippon and Waldschmidt. A similar result holds for the plane partition function. 

Several algorithmic problems on supersingular elliptic curves are

currently under close scrutiny. When analysing algorithms or reductions

in this context, one often runs into the following type of question:

given a supersingular elliptic curve E and an object x attached to E, if

we consider a random large degree isogeny f : E -> E' and carry the

object x along f, how is the resulting f(x) distributed among the

possible objects attached to E'? We propose a general framework to

formulate this type of question precisely, and prove a general

equidistribution theorem under a condition that is easy to check in

practice. The proof goes from elliptic curves to quaternionic

automorphic forms via an augmented Deuring correspondence, and then to

classical modular forms via the Jacquet-Langlands correspondence. This

is joint work with Benjamin Wesolowski.

After a quick overview of the general principles of Life Cycle Assessment (LCA), we will investigate how such a tool can be helpful to compare the environmental impact of different architectures of computer systems used for teaching purposes in higher education. In particular, we will see how to perform the life cycle inventory of the systems under studies from a practical standpoint. We will then review the main results from the life cycle impact assessment and discuss them as well as the limitations of this study.   

Multidimensional simulations of magnetohydrodynamic phenomena occurring in stellar interiors are essential for understanding how stars evolve and die. The highly subsonic flow regimes found in the regions deep inside stars pose severe challenges to conventional methods of computational MHD, such as the popular "high-resolution shock-capturing'' schemes. After giving a brief overview of work on astrophysical simulations (including also supernova explosions and common-envelope evolution) in our group at Heidelberg, we summarize the challenges and present suitable numerical solvers optimized for magnetized, low-Mach-number stellar flows, implemented in our Seven-League Hydro code. We show how the choice of the numerical method can drastically affect both the performance of the code and its accuracy in real astrophysical simulations.

Présentation des membres de l'équipe

Un pavage de Penrose est formé de deux tuiles polygonales dont le ratio des fréquences est égal au nombre d'or. De même, les pavages par la monotuile apériodique découverte en 2023 par David Smith sont tels que le ratio des fréquences des deux orientations de la monotuile est égal à la quatrième puissance du nombre d'or. Aussi, la structure des pavages de Jeandel-Rao est expliquée par le nombre d'or. On connait des pavages apériodiques qui ne sont pas reliés au nombre d'or. Toutefois, la caractérisation des nombres possibles pour de tels ratios est une question, posée dès 1992 par Ammann, Grünbaum et Shephard, qui est toujours ouverte aujourd'hui.

Pour chaque entier positif $n$, nous introduisons un ensemble $\mathcal{T}_n$ composé de $(n+3)^2$ tuiles de Wang (carrés unitaires avec des bords étiquetés). Nous représentons un pavage par des translations de ces tuiles comme une fonction $\mathbb{Z}^2\to\mathcal{T}_n$ appelée configuration. Une configuration est valide si le bord commun des tuiles adjacentes a la même étiquette. Pour chaque entier $n\geq1$, nous considérons le sous-décalage de Wang $\Omega_n$ défini comme l'ensemble des configurations valides pour les tuiles $\mathcal{T}_n$.

La famille $\{\Omega_n\}_{n\geq1}$ élargit la relation entre les entiers quadratiques et les tuiles apériodiques au-delà de l'omniprésent nombre d'or, car la dynamique de $\Omega_n$ implique la racine positive $\beta$ du polynôme $x^2-nx-1$. Cette racine est parfois appelée $n$-ième nombre métallique (https://fr.wikipedia.org/wiki/Nombre_métallique), et en particulier, le nombre d'or lorsque $n=1$ et le nombre d'argent lorsque $n=2$.

L'ensemble $\Omega_n$ est auto-similaire, apériodique et minimal pour l'action de décalage. De plus, il existe une partition polygonale de $\mathbb{T}^2$ qui est une partition de Markov pour une $\mathbb{Z}^2$-action sur le tore. La partition et les ensembles de tuiles de Wang sont symétriques, ce qui les rend, comme les tuiles de Penrose, dignes d'intérêt.

Les détails peuvent être trouvés dans les prépublications disponibles à

https://arxiv.org/abs/2312.03652 (partie I) et

https://arxiv.org/abs/2403.03197 (partie II).

L'exposé présentera une vue d'ensemble des principaux résultats.

Freiman's $3k-4$ Theorem states that if a subset $A$ of $k$ integers has a Minkowski sum $A+A$ of size at most $3k-4$, then it must be contained in a short arithmetic progression. We prove a function field analogue that is also a generalisation: it states that if $K$ is a perfect field and if $S\supset K$ is a vector space of dimension $k$ inside an extension $F/K$ in which $K$ is algebraically closed, and if the $K$-vector space generated by all products of pairs of elements of $S$ has dimension at most $3k-4$, then $K(S)$ is a function field of small genus, and $S$ is of small codimension inside a Riemann-Roch space of $K(S)$. Joint work with Alain Couvreur. 

Discussion autour de l’après thèse et les carrières académiques (concours, candidatures), en priorité à destination des doctorantes et doctorants, post-doctorant·e·s et ATER à l’IMB. 

Je discuterai un travail récent avec Yann Chaubet et Daniel Han-Kwan (Nantes). Nous nous sommes intéressés à la dynamique en temps long de l'équation de Vlasov non-linéaire sur une variété à courbure négative lorsque le noyau d'interaction est lisse. J'expliquerai que, pour des petites données initiales lisses et supportées loin de la section nulle, les solutions de cette équation convergent à vitesse exponentielle vers un état d'équilibre du problème linéaire. Pour obtenir un tel résultat, on fait appel à des outils d'analyse microlocale développés initialement dans le contexte de l'étude des systèmes dynamiques chaotiques (Baladi, Dyatlov, Faure, Sjöstrand, Tsujii, Zworski).

The classical modular polynomial phi_N parametrizes pairs of elliptic curves connected by an isogeny of degree N. They play an important role in algorithmic number theory, and are used in many applications, for example in the SEA point counting algorithm.

This talk is about a new method for computing modular polynomials. It has the same asymptotic time complexity as the currently best known algorithms, but does not rely on any heuristics. The main ideas of our algorithm are: the embedding of N-isogenies in smooth-degree isogenies in higher dimension, and the computation of deformations of isogenies.

The talk is based on a joint work with Damien Robert.

In recent months, the proliferation of conversational agents such as ChatGPT has had a major impact on Artificial Intelligence research, but also on the way AI is perceived by the public. Because of some rather bluff results, some people wonder if this agent is as intelligent as us, if it can replace us, or even if it has a conscience. But also because of the rather crude mistakes it makes, people wonder whether its use should not be prohibited under certain conditions. To answer such questions, it might be useful to know more about how ChatGPT works. After that, we'll be able to discuss the potential of such tools and the uses to which they can be put.

We study non-conservative hyperbolic systems of balance laws and are interested in development of well-balanced (WB) numerical methods for such systems. One of the ways to enforce the balance between the flux terms and source and non-conservative product terms is to rewrite the studied system in a quasi-conservative form by incorporating the latter terms into the modified global flux. The resulting system can be quite easily solved by Riemann-problem-solver-free central-upwind (CU) schemes. This approach, however, does not allow to accurately treat non-conservative products. We therefore apply a path-conservative (PC) integration technique and develop a very robust and accurate path-conservative central-upwind schemes (PCCU) based on flux globalization. I will demonstrate the performance of the WB PCCU schemes on a wide variety of examples.

Séminaire IOP banalisé

https://www.math.u-bordeaux.fr/~skupin/conf-pthomas-2024.html

Une variété est dite PSC si elle admet une métrique riemannienne complète à courbure scalaire positive. Vers la fin des années 1970, des résultats de Schoen et Yau reposant sur la théorie des surfaces minimales et, en parallèle, des méthodes basées sur la théorie de l’indice développées par Gromov et Lawson, ont permis de classifier les 3-variétés fermées PSC : ce sont exactement celles qui se décomposent en sommes connexes de variétés sphériques et de produits S2xS1. Dans cet exposé, nous présenterons un résultat de décomposition des 3-variétés PSC non compactes : si sa courbure scalaire décroît assez lentement, alors la variété se décompose en somme connexe (possiblement infinie) de variétés sphériques et S2xS1. Ce résultat fait suite à des travaux récents de Gromov et de Wang.

Il s'agit d'un travail en collaboration avec F. Balacheff et S. Sabourau.

Résumé. L'exposé porte sur l'analogue du Théorème d’Artin-Furtwängler sur la capitulation des groupes de classes dans le corps de Hilbert obtenu en transposant aux groupes de classes logarithmiques des corps de nombres la preuve algébrique classique du Théorème de l’idéal principal.

Abstract. We establish a logarithmic version of the classical result of Artin-Furwängler on the principalization of ideal classes in the Hilbert class-field by applying the group theoretic description of the transfert map in the logarithmic context.

Exceptionnellement, l'accueil de la Cellule Informatique au bureau 225

• ne sera ouvert qu'à partir de 10h
• et sera fermé vendredi
• et il n'y aura pas d'accueil au bureau 270.

en raison de la participation d'une partie de l'équipe informatique à l'Action Nationale de Formation Mathrice au CIRM à Marseille.

• Pensez à anticiper les retraits de matériel de prêt.

We investigate the connection between the propagation of smallness in two dimensions and one-dimensional spectral estimates. The phenomenon of smallness propagation in the plane, originally obtained by Yuzhe Zhu, reveals how the value of solutions in a small region extends to a larger domain. By revisiting Zhu’s proof, we obtain a quantitative version that includes an explicit dependence on key parameters. This refinement enables us to establish spectral inequalities for one-dimensional Schrödinger operators.

Dans le sillage d'une éolienne ou d'un hélicoptère se créent naturellement des filaments de tourbillon en forme d'hélice. Le mouvement des filaments de tourbillon fait l'objet d'une conjecture importante : lorsque le diamètre du filament tend vers 0 (en conservant son intensité), son mouvement devrait suivre en première approximation le flot par courbure binormale. Cette conjecture n'est prouvée que pour les filaments rectilignes et pour les anneaux de tourbillon. Nous montrons, dans le contexte des équations d'Euler 3D incompressibles en symétrie hélicoïdale que les filaments hélicoïdaux suffisamment concentrés suivent également le flot par courbure binormal.

The slides are in english but the talk will be in french.

In a category enriched in a closed symmetric monoidal category, the power

object construction, if it is representable, gives a contravariant monoidal

action. We first survey the construction, due to Serre, of the power object

by (projective) Hermitian modules on abelian varieties. The resulting

action, when applied to a primitively oriented elliptic curve, gives a

contravariant equivalence of category (Jordan, Keeton, Poonen, Rains,

Shepherd-Barron and Tate).

We then give several applications of this module action:

1) We first explain how it allows to describe purely algebraically the

ideal class group action on an elliptic curve or the Shimura class group

action on a CM abelian variety over a finite field, without lifting to

characteristic 0.

2) We then extend the usual algorithms for the ideal action to the case of

modules, and use it to explore isogeny graphs of powers of an elliptic

curve in dimension up to 4. This allows us to find new examples of curves

with many points. (This is a joint work with Kirschmer, Narbonne and

Ritzenthaler)

3) Finally, we give new applications for isogeny based cryptography. We

explain how, via the Weil restriction, the supersingular isogeny path

problem can be recast as a rank 2 module action inversion problem. We also

propose ⊗-MIKE a novel NIKE (non interactive isogeny key exchange) that only

needs to send j-invariants of supersingular curves, and compute a dimension

4 abelian variety as the shared secret.

Les algorithmes quantiques sont une piste majeure d'accélération pour certains calculs. Dans cet exposé, nous présenterons les principaux problèmes susceptibles d'en bénéficier. Nous développerons également quelques grands principes sous-jacents à ces algorithmes.

This talk focuses on models for multivariate count data, with emphasis on species abundance data. Two approaches emerge in this framework: the Poisson log-normal (PLN) and the Tree Dirichlet multinomial (TDM) models. The first uses a latent gaussian vector to model dependencies between species whereas the second models dependencies directly on observed abundances. The TDM model makes the assumption that the total abundance is fixed, and is then often used for microbiome data since the sequencing depth (in RNA seq) varies from one observation to another leading to a total abundance that is not really interpretable. We propose to generalize TDM model in two ways: by relaxing the fixed total abundance and by using Polya distribution instead of Dirichlet multinomial. This family of models corresponds to Polya urn models with a random number of draws and will be named Polya splitting distributions. In a first part I will present the probabilistic properties of such models, with focus on marginals and probabilistic graphical model. Then it will be shown that these models emerge as stationary distributions of multivariate birth death process under simple parametric assumption on birth-death rates. These assumptions are related to the neutral theory of biodiversity that assumes no biological interaction between species. Finally, the statistical aspects of Polya splitting models will be presented: the regression framework, the inference, the consideration of a partition tree structure and two applications on real data.

Seminaire joint avec OptimAI.

This talk focuses on models for multivariate count data, with emphasis on species abundance data. Two approaches emerge in this framework: the Poisson log-normal (PLN) and the Tree Dirichlet multinomial (TDM) models. The first uses a latent gaussian vector to model dependencies between species whereas the second models dependencies directly on observed abundances. The TDM model makes the assumption that the total abundance is fixed, and is then often used for microbiome data since the sequencing depth (in RNA seq) varies from one observation to another leading to a total abundance that is not really interpretable. We propose to generalize TDM model in two ways: by relaxing the fixed total abundance and by using Polya distribution instead of Dirichlet multinomial. This family of models corresponds to Polya urn models with a random number of draws and will be named Polya splitting distributions. In a first part I will present the probabilistic properties of such models, with focus on marginals and probabilistic graphical model. Then it will be shown that these models emerge as stationary distributions of multivariate birth death process under simple parametric assumption on birth-death rates. These assumptions are related to the neutral theory of biodiversity that assumes no biological interaction between species. Finally, the statistical aspects of Polya splitting models will be presented: the regression framework, the inference, the consideration of a partition tree structure and two applications on real data.

Les métriques Lorentziennes à courbure constante ayant un nombre fini de singularités coniques offrent de nouveaux exemples naturels de structures géométriques sur le tore. Des travaux de Troyanov sur leur analogue Riemannien ont montré que la donnée de la structure conforme et des angles aux singularités classifient entièrement les métriques Riemanniennes à singularités coniques. Dans cet exposé nous nous intéresserons aux tores de-Sitter singuliers, en construirons des exemples, et présenterons un phénomène de rigidité rappelant celui de Troyanov : les tores de-Sitter à une singularité d'angle fixé sont déterminés par la classe d'équivalence topologique de leur bi-feuilletage lumière. Nous verrons que cette question géométrique est intimement liée à un problème de dynamique sur les difféomorphismes par morceaux du cercles.

In this talk, we investigate intersecting codes. In the Hamming metric, these are codes where two nonzero codewords always share a coordinate in which they are both nonzero. Based on a new geometric interpretation of intersecting codes, we are able to provide some new lower and upper bounds on the minimum length $i(k, q)$ of intersecting codes of dimension k over $\mathbb{F}_q$, together with some explicit constructions of asymptotically good intersecting codes. We relate the theory of intersecting codes over $\mathbb{F}_q$ with the theory of $2$-wise weighted Davenport constants of certain groups, and to nonunique factorization theory. Finally, we will present intersecting codes in the rank metric.

L'équation de Gross-Pitaevskii décrit le mouvement de superfluides, et possède entre autres des solutions stationnaires en forme de vortex. Si deux vortex sont présents, ils se déplacent ensemble à une vitesse constante.

Dans cet exposé, on montrera un résultat de stabilité orbitale dans un espace métrique pour cette paire. On expliquera comment adapter le schéma de preuve de stabilité à un tel espace, et pourquoi on ne peut pas prouver le résultat dans un espace plus simple. Ce travail a été fait en collaboration avec Philippe Gravejat et Frédéric Valet.

Dans le contexte du changement climatique, de nombreuses études prospectives, englobant généralement tous les domaines de la société, imaginent des futurs possibles pour faire émerger des nouveaux récits et/ou guider les prises de décisions. Dans cette présentation, nous analyserons les technologies numériques envisagées dans un monde qui a atténué le changement climatique ou s'y est adapté. Pour cela, les variables d'analyse utilisées dans notre étude seront décrites. Elles ont été appliquées pour étudier 14 études prospectives et les 35 scénarios futurs correspondants. Nous constatons que tous les scénarios considèrent la technologie numérique comme présente dans le futur et peu d'entre eux interrogent notre rapport au numérique et sa matérialité. Notre analyse montre l'absence d'une vision systémique des technologies de l'information et de la communication dans les scénarios prospectifs. Nous conclurons la présentation en discutant d'alternative pour imaginer les scénarios alternatifs pour le numérique.

In the Kidney Exchange Problem (KEP), we consider a pool of altruistic donors and incompatible patient-donor pairs. 

Kidney exchanges can be modelled in a directed weighted graph as circuits starting and ending in an incompatible pair or as paths starting at an altruistic donor.

The weights on the arcs represent the medical benefit which measures the quality of the associated transplantation.

For medical reasons, circuits and paths are of limited length and are associated with a medical benefit to perform the transplants.

The aim of the KEP is to determine a set of disjoint kidney exchanges of maximal medical benefit or maximal cardinality (all weights equal to one).

In this work, we consider two types of uncertainty in the KEP which stem from the estimation of the medical benefit (weights of the arcs) and from the failure of a transplantation (existence of the arcs).

Both uncertainty are modelled via uncertainty sets with constant budget.

The robust approach entails finding the best KEP solution in the worst-case scenario within the uncertainty set.

We modelled the robust counter-part by means of a max-min formulation which is defined on exponentially-many variables associated with the circuits and paths.

We propose different exact approaches to solve it: either based on the result of Bertsimas and Sim or on a reformulation to a single-level problem.

In both cases, the core algorithm is based on a Branch-Price-and-Cut approach where the exponentially-many variables are dynamically generated.

The computational experiments prove the efficiency of our approach.  

This talk explores two advanced numerical methods for solving compressible two-phase flows modelled using the conservative Symmetric Hyperbolic Thermodynamically Compatible (SHTC) model proposed by Romenski et al. I first address the weak hyperbolicity of the original model in multidimensional cases by restoring strong hyperbolicity through two distinct approaches: the explicit symmetrization of the system and the hyperbolic Generalized Lagrangian Multiplier (GLM) curl-cleaning approach. Then, I will present two numerical methods to solve the proposed problem: a high-order ADER Discontinuous Galerkin (ADER-DG) scheme with an a posteriori sub-cell finite volume limiter and an exactly curl-free finite volume scheme to handle the curl involution in the relative velocity field. The latter method uses a staggered grid discretization and defines a proper compatible gradient and a curl operator to achieve a curl-free discrete solution. Extensive numerical test cases in one and multiple dimensions validate both methods' accuracy and stability. 

Many phenomena in the life sciences, ranging from the microscopic to macroscopic level, exhibit surprisingly similar structures. Behaviour at the microscopic level, including ion channel transport, chemotaxis, and angiogenesis, and behaviour at the macroscopic level, including herding of animal populations, motion of human crowds, and bacteria orientation, are both largely driven by long-range attractive forces, due to electrical, chemical or social interactions, and short-range repulsion, due to dissipation or finite size effects. Various modelling approaches at the agent-based level, from cellular automata to Brownian particles, have been used to describe these phenomena. An alternative way to pass from microscopic models to continuum descriptions requires the analysis of the mean-field limit, as the number of agents becomes large. All these approaches lead to a continuum kinematic equation for the evolution of the density of individuals known as the aggregation-diffusion equation. This equation models the evolution of the density of individuals of a population, that move driven by the balances of forces: on one hand, the diffusive term models diffusion of the population, where individuals escape high concentration of individuals, and on the other hand, the aggregation forces due to the drifts modelling attraction/repulsion at a distance. The aggregation-diffusion equation can also be understood as the steepest-descent curve (gradient flow) of free energies coming from statistical physics. Significant effort has been devoted to the subtle mechanism of balance between aggregation and diffusion. In some extreme cases, the minimisation of the free energy leads to partial concentration of the mass. Aggregation-diffusion equations are present in a wealth of applications across science and engineering. Of particular relevance is mathematical biology, with an emphasis on cell population models. The aggregation terms, either in scalar or in system form, is often used to model the motion of cells as they concentrate or separate from a target or interact through chemical cues. The diffusion effects described above are consistent with population pressure effects, whereby groups of cells naturally spread away from areas of high concentration. This talk will give an overview of the state of the art in the understanding of aggregation-diffusion equations, and their applications in mathematical biology.

Le problème de Manin-Mumford dynamique est un problème en dynamique algébrique inspiré par des résultats classiques de géométrie arithmétique.

Étant donné un système dynamique algébrique $(X,f)$, où $X$ est une variété projective et $f$ est un endomorphisme polarisé de $X$, on veut déterminer sous quelles conditions une sous-variété $Y$ qui contient une quantité Zariski-dense de points à orbite finie, doit avoir elle-même une orbite finie.

Dans un travail en commun avec Romain Dujardin et Charles Favre, on montre que cette propriété est vérifiée quand $f$ est un endomorphisme régulier du plan projectif provenant d'un endomorphisme polynomial de ${\mathbf C}^2$ (de degré $d \ge 2$), sous la condition supplémentaire que l'action de $f$ à l'infini n'a pas de points critiques périodiques.

La preuve se base sur des techniques provenant de la géométrie arithmétique et de la dynamique analytique, à la fois sur ${\mathbf C}$ et sur des corps non-archimédiens.

Joint work with Bas Edixhoven. 

We present a generalization of Chabauty's method, that allows to compute the rational points on curves /$\mathbf{Q}$ when the Mordell-Weil rank is strictly smaller than $g1$, where $g$ is the genus of the curve and $s$ is the rank of the Néron-Severi group of the Jacobian.

The idea is to enlarge the Jacobian by talking a $\mathbf{G}_m$-torsor over it and the algorithm ultimately consists in intersecting the integral points on the $\mathbf{G}_m$-torsor with (an image of) the $\mathbf{Z}_p$-points on the curve.

We can also view the method as a way of rephrasing the quadratic Chabauty method by Balakrishnan, Dogra, Muller, Tuitman and Vonk.

Due to the complexity of real-world planning processes addressed by major transportation companies, decisions are often made considering subsequent problems at the strategic, tactical, and operational planning phases. However, these problems still prove to be individually very challenging. This talk will present two examples of tactical transportation problems motivated by industrial applications: the Train Timetabling Problem (TTP) and the Service Network Scheduling Problem (SNSP). The TTP aims at scheduling a set of trains, months to years before actual operations, at every station of their path through a given railway network while respecting safety headways. The SNSP determines the number of vehicles and their departure times on each arc of a middle-mile network while minimizing the sum of vehicle and late commodity delivery costs. For these two problems, the consideration of capacity and uncertainty in travel times are discussed. We present models and solution approaches including MILP formulations, Tabu search, Constraint Programming techniques, and a Progressive Hedging metaheuristic.

Despite the supreme importance of fluid flow models, the well-posedness of three-dimensional viscous and inviscid flow equations remains unsolved. Promising efforts have recently evolved around the concept of statistical solutions. In this talk, we present stochastic lattice Boltzmann methods for efficiently approximating statistical solutions to the incompressible Navier–Stokes equations in three spatial dimensions. Space-time adaptive kinetic relaxation frequencies are used to find stable and consistent numerical solutions along the inviscid limit toward the Euler equations. With single level Monte Carlo and stochastic Galerkin methods, we approximate responses, e.g., from initial random perturbations of the flow field. The novel combinations of schemes are implemented in the parallel C++ data structure OpenLB and executed on heterogeneous high-performance computing machinery. Based on exploratory computations, we search for scaling of the energy spectra and structure functions in terms of Kolmogorov’s K41 theory. For the first time, we numerically approximate the limit of statistical solutions of the incompressible Navier–Stokes solutions toward weak-strong unique statistical solutions of the incompressible Euler equations in three dimensions. Applications to wall-bounded turbulence and the potential to provide training data for generative artificial intelligence algorithms are discussed.

Isogeny-based cryptography is founded on the assumption that the Isogeny problem—finding an isogeny between two given elliptic curves—is a hard problem, even for quantum computers.

In the security analysis of isogeny-based schemes, various related problems naturally arise, such as computing the endomorphism ring of an elliptic curve or determining a maximal quaternion order isomorphic to it.

These problems have been shown to be equivalent to the Isogeny problem, first under some heuristics and subsequently under the Generalized Riemann Hypothesis.

In this talk, we present ongoing joint work with Benjamin Wesolowski, where we unconditionally prove these equivalences, notably using the new tools provided by isogenies in higher dimensions.

Additionally, we show that these problems are also equivalent to finding the lattice of all isogenies between two elliptic curves.

Finally, we demonstrate that if there exist hard instances of the Isogeny problem then all the previously mentioned problems are hard on average.

L'ordre du jour sera le suivant :

1) Adoption du Compte-Rendu du conseil du 10 septembre (vote)

2) Informations générales

3) Élection d'un nouveau membre du conseil scientifique (vote)

4) Présentation du projet de nouveau site web de l'IMB. Discussions sur la présentation et les couleurs à adopter.

5) Discussion autour des comités de sélection sur la base des propositions apparaissant dans la lettre ouverte

6) Questions diverses

Pensez à donner votre procuration

Column elimination is an exact algorithm to solve discrete optimization problems via a 'column' formulation in which variables represent a combinatorial structure such as a machine schedule or truck route. Column elimination works with a relaxed set of columns, from which infeasible ones are iteratively eliminated. As the total number of columns can typically be exponentially large, we use relaxed decision diagrams to compactly represent and manipulate the set of columns. In this talk, we provide an introduction to the column elimination method and present recent algorithmic improvements resulting in competitive performance on large-scale vehicle routing problems. Specifically, our approach closes a large vehicle routing benchmark instance with 1,000 locations for the first time.

Le projet européen SimCardioTest essaye de montrer qu'il est possible et utile de réaliser des essais cliniques in-silico pour des médicaments ou des dispositifs médicaux cardiaques. Pour cela, une plateforme internet à été créée, à travers laquelle il est possible d'exécuter des simulations numériques de modèles représentant trois usages possibles en cardiologie. Garantir la crédibilité des simulations est alors un point clé pour un usage industriel de cette plateforme. Cela repose sur des procédures standardisées de vérification et de validation pour chaque usage. À l'université de Bordeaux, au sein de l'IHU LIRYC, nous avons construit, vérifié et travaillons à la validation d'un modèle qui permet d'étudier l'efficacité énergétique d'un stimulateur cardiaque. J'expliquerais ce travail et des difficultés auxquelles nous avons été confrontées.

Deep learning has revolutionised image processing and is often considered to outperform classical approaches based on accurate modelling of the image formation process. In this presentation, we will discuss the interplay between model-based and learning-based paradigms, and show that hybrid approaches show great promises for scientific imaging, where interpretation and robustness to real-world degradation is important. We will present two applications on super-resolution and high-dynamic range imaging, and exoplanet detection from direct imaging at high contrast.

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A Coulter counter is an impedance measurement system widely used in blood analyzers to count and size red blood cells, thus providing information about the most numerous cells of the body. In Coulter counters, cells flow through a detection zone where an electric field is imposed, which is disturbed when a cell passes through. The number of these impedance signals yield the red blood cell count, while the cell volume is supposed to be proportional to the amplitude of the signals. However, in real systems, the red blood cells trajectories in the system does not allow to verify the assumptions necessary to provide an accurate volume measurement. For a few years, IMAG has been developing the YALES2BIO solver for the prediction of red blood cell dynamics under flow. In this presentation, I will describe the fluid-structure problem and the numerical method used, then share how numerical simulation has been used to understand the signals in industrial Coulter counters and to improve the measurements of red blood cell volumes rendered by such systems. In addition, I will discuss how the mechanical properties of RBCs impact the measurements. This work has been performed during the PhD theses of Pierre Taraconat and Pierre Pottier (Horiba Medical & IMAG).

Soit $K$ un corps algébriquement clos de caractéristique quelconque. Soit $f \in K[[x,y]]$ une série réduite et $r(f)$ le nombre de ses facteurs irréductibles. Soit $\mathcal{O}=K[[x,y]]/(f)$ et $\overline{\mathcal{O}}$ sa cloture intégrale. On note $\delta(f)=\dim_K \overline{\mathcal{O}}/\mathcal{O}$ et $\mu(f)=\dim_K K[[x,y]]/(f'_x,f'_y)$, le nombre de Milnor. Milnor a montré en 1968 que si $K=\mathbb{C}$,

$$\mu(f)=2\delta(f)-r(f)+1.$$

En 1973, Deligne a montré que si la caractérisque de $K$ est arbitraire

$$\mu(f)\geq 2\delta(f)-r(f)+1.$$

Le but de cet exposé est d'énoncer une conjecture sur la caractéristique de $K$ pour avoir l'égalité.

La conjecture standard de type Hodge porte sur les nombres d'intersections de sous-variétés d'une variété projective. Elle a de nombreuses conséquences en arithmétique, dans cet exposé on construira des variétés abéliennes A qui satisfont à cette conjecture. L'outil principal permettant la construction de variétés abéliennes A est la théorie de Honda-Tate, qui relie ces dernières à des objets de théorie algébrique des nombres. On sera ensuite amené à étudier l'algèbre des classes de Tate de A, qui est un invariant plus manipulable que l'ensemble des sous-variétés de A.

We will focus on the formation of extreme waves in the open sea, adopting a probabilistic point of view. We will first identify the first term of the asymptotic development of the probability of occurrence of such a wave when the wave height tends to infinity. If an extreme wave occurs, what is the most likely mechanism that produced it? We will answer this question using two toy models. In the case of an integrable system, we will show that a linear superposition mechanism is the most likely. In the case of a strongly resonant system, the main formation mechanism is a nonlinear focusing effect, which induces an increase in the probability of occurrence of large waves.

A pair elliptic curves E/Q and E’/Q are isogenous if and only if they have the same number of points mod p for every (good) prime p. A conjecture of Frey and Mazur predicts that E and E’ are isogenous if and only if they are N-congruent for any sufficiently large integer N > N_0 (i.e., #E(F_p) = #E’(F_p) mod N for all good p).

Congruences appear quite naturally in applications, for example:

- In isogeny-based cryptography (an abelian surface being (N,N)-split implies that the corresponding pair of elliptic curves are N-congruent).

- In Diophantine problems (e.g., Fermat’s last theorem),

- In descent obstructions (via Mazur’s notion of “visible elements of Sha”).

Despite the Frey—Mazur conjecture, it is not known for which integers there exist non-isogenous N-congruent elliptic curves: what is N_0? I will discuss progress towards refining the Frey—Mazur conjecture by studying the geometry of “Humbert surfaces” which parametrise N-congruences.

Orateur: Simon Kristensen (Aarhus)

Titre: Partitions and Fibonacci numbers.

Résumé: The asymptotic theory of partitions is classical, going back to (at least) Hardy and Ramanujan who found an asymptotic formula for the number of unrestricted partitions of n. If the partitions are restricted, the problem becomes a different one, and as opposed to the unrestricted case, the asymptotic behaviour can be a more difficult oscillatory function. In this talk, we will find the asymptotic behaviour of the function p_F(n), which counts the number of representations of n as the sum of possibly non-distinct Fibonacci numbers. This function has this oscillatory behaviour. We will also show how this generalises to a variety of linear recurrence sequences. The work is joint with Michael Coons and Mathias L. Laursen.

Oratrice: Maiken Gravgaard (Aarhus)

Titre: Twisted inhomogeneous Dirichlet improbability.

Résumé: For a real valued non-increasing function psi a pair (A, b) consisting of a real valued matrix A with m rows and n columns and a real vector b is called psi-Dirichlet improvable if the system ||Aq+b||^m < psi(T) and |q|^n < T has a solution q in Z^m for all sufficiently large T. Kleinbock and Wadleigh proved a zero-full law for the Lebesgue measure of the set of psi-Dirichlet pairs. 

Kim and Kim established results for the Hausdorff measure of the set of psi-Dirichlet non-improvable pairs as well as for the b-fixed singly metric case. 

Previous proofs have used homogeneous dynamics. In this talk, we show full Hausdorff dimension in the A-fixed singly metric case for all A and certain psi using a different approach to the problem. This is joint with Simon Kristensen.

Dans cette conférence, nous explorerons les évolutions récentes du secteur spatial dans le cadre du mouvement NewSpace, qui révolutionne l'accès à l'espace par une approche plus agile et commerciale. Nous aborderons également le rôle croissant du spatial dans la surveillance et la lutte contre les changements climatiques, avec un accent particulier sur les technologies permettant de recueillir des données environnementales cruciales. Enfin, nous illustrerons ces avancées à travers le cas de LEOBLUE, une société innovante qui développe des solutions de communication directe entre satellites en orbite basse et smartphones, permettant de nouvelles applications à large échelle.

We are interested in the numerical discretization of the growth-fragmentation equation. This type of models typically describes the time evolution of a mass-structured population from the growth and division of its individuals. Integral properties of said population provide macroscopic quantities that may help calibrating the model on experimental data. More precisely, we are interested in the 0th and 1st moments of the solution, corresponding respectively to the total population density or of the total population mass.

However, the numerical resolution of such models may often encounters consistency issues with respect to one or several of its integral properties. In this talk, I will introduce a new finite volume scheme corrected with weights, which allows to simultaneously retrieve the model’s solution and its two first moments.

In statistical learning, many analyses and methods rely on optimization, including its stochastic versions introduced for example, to overcome an intractability of the objective function or to reduce the computational cost of the deterministic optimization step.

In 1951, H. Robbins and S. Monro introduced a novel iterative algorithm, named "Stochastic Approximation", for the computation of the zeros of a function defined by an expectation with no closed-form expression. This algorithm produces a sequence of iterates, by replacing at each iteration the unknown expectation with a Monte Carlo approximation based on one sample. Then, this method was generalized: it is a stochastic algorithm designed to find the zeros of a vector field when only stochastic oracles of this vector field are available.

Stochastic Gradient Descent algorithms are the most popular examples of Stochastic Approximation : oracles come from a Monte Carlo approximation of a large sum. Possibly less popular are examples named "beyond the gradient case" for at least two reasons. First, they rely on oracles that are biased approximation of the vector field, as it occurs when biased Monte Carlo sampling is used for the definition of the oracles. Second, the vector field is not necessarily a gradient vector field. Many examples in Statistics and more

generally in statistical learning are "beyond the gradient case": among examples, let us cite compressed stochastic gradient descent, stochastic Majorize-Minimization methods such as the Expectation-Maximization algorithm, or the Temporal Difference algorithm in reinforcement learning.

In this talk, we will show that these "beyond the gradient case" Stochastic Approximation algorithms still converge, even when the oracles are biased, as soon as some parameters of the algorithm are tuned enough. We will discuss what 'tuned enough' means when the quality criterion relies on epsilon-approximate stationarity. We will also comment the efficiency of the

algorithm through sample complexity. Such analyses are based on non-asymptotic convergence bounds in expectation: we will present a unified method to obtain such bounds for a large class of Stochastic Approximation methods including both the gradient case and the beyond the gradient case. Finally, a Variance Reduction technique will be described and its efficiency illustrated.

Characterizing topological electronic states in crystals is a formidable goal in condensed matter physics. Such states offer substantial breakthroughs toward a deeper understanding of matter and potential applications ranging from electronics to quantum computing. Their unique electronic features – robust boundary states and quantized bulk electromagnetic responses – derive from topological properties of the wave functions in reciprocal space as a refinement of Bloch band theory. Direct evidence of the band topology traditionally relies on the macroscopic response of the electrons to external electromagnetic fields in ultra-clean samples.

In this seminar, I will introduce an alternative approach to identify topological systems. I will present two experiments (from a theoretical physicist perspective) that image the local response of the electrons to a boundary in the local density of states (LDOS). The first experiment focuses on graphene — a 2D semimetal — imaged by scanning tunneling microscopy. The second experiment emulates a 1D insulator with dielectric resonators inside a microwave cavity. Remarkably, both systems exhibit wavefront dislocations in the LDOS near the boundary. I will show that the dislocation charge is a real-space measure of the reciprocal-space band topology.

Can we understand the nature of the singularities that have to be admitted after a blow-up sequence that preserves the normal crossings locus of an algebraic (or complex-analytic) variety X? For example, every surface can be transformed by blowings-up preserving normal crossings to a surface with at most additional Whitney umbrella singularities. We will discuss general conjectures in arbitrary dimensions, and partial solutions. The techniques involve circulant matrices, elementary Galois theory and Newton-Puiseux expansion in several variables. We will discuss results in collaboration with Edward Bierstone and Ramon Ronzon Lavie.

Soit E une courbe elliptique définie sur un corps de nombres K, et S un point d'ordre infini dans E(K). Une conjecture de Lang et Trotter prédit que l'ensemble des premiers p de K pour lesquels la réduction de S engendre le groupe des points rationnels sur E mod p a une densité qui peut être exprimée en termes de la représentation galoisienne associée aux « corps de division » du point S. Cette conjecture généralise aux courbes elliptiques la plus classique conjecture de la racine primitive d'Artin formulée pour le groupe multiplicatif d’un corps de nombres.

Dans cet exposé, j'examinerai les cas où la densité conjecturale de Lang-Trotter est nulle, c'est-à-dire lorsque S n'est presque jamais localement primitif. En m'appuyant sur l'analogie avec la conjecture d'Artin, je présenterai un projet en cours – en collaboration avec Nathan Jones, Francesco Pappalardi et Peter Stevenhagen – visant à classifier certains de ces phénomènes d'imprimitivité pour les courbes elliptiques sur le corps des rationnels.

We provide a characterization of universal and multiplier interpolating sequences for de Branges-Rovnyak spaces where the defining function is a non-extreme, rational function. Previous work on this topic examined interpolation in de Branges-Rovnyak spaces specifically in cases where the space coincides with a local Dirichlet space under norm equivalence. The more general setting we are interested in here corresponds to higher order local Dirichlet spaces recently investigated by Gu, Luo and Richter. In this general setting we characterize universal interpolating sequences and show that they coincide with multiplier interpolating sequences. We also explore random interpolation through the use of Steinhaus sequences.

This is work in progress with A. Hartmann.

The aim of the talk is to explain an unexpected link between a class of molecules composed of carbon and hydrogen atoms, and the theory of elliptic curves over finite fields. The correspondence is of topological nature and doesn't include, so far, any of the crucial geometric features of the cycloalkanes. We will nevertheless explain how modular curves help making this connection, the role of modular polynomials, give details about explicit computations we performed, and give several examples. The talk is based on joint work with Henry Bambury and Francesco Campagna. 

En chirurgie du foie assistée par ordinateur, des systèmes de réalité augmentée ont été développés pour aider le personnel du bloc opératoire à mieux visualiser la déformation des organes du patient. En effet, dans un cadre laparoscopique, le chirurgien interagit avec les organes indirectement à l’aide d’outils passés à travers la peau du patient, et cela rend la navigation difficile. L’affichage d’une vue tri-dimensionnelle de l’organe déformé nécessite au préalable d'estimer sa déformation en résolvant un problème de recalage élastique : un maillage représentant le foie dans sa configuration initiale est déformé pour épouser un nuage de point représentant une observation partielle de sa surface. Nous utilisons le formalisme du contrôle optimal pour traiter ce problème de recalage, car une telle formulation permet de décrire plus précisément la physique du problème, ce qui mène à des recalages de meilleure qualité.

Dans cet exposé, nous présentons une méthode d’adjoint mise en place pour résoudre numériquement le problème, et nous évaluons sa performance sur un problème d'estimation d'une force ponctuelle. En particulier, lorsqu’on modélise le foie au moyen d'une loi élastique non-linéaire, la résolution du problème direct (élasticité) par une méthode de Newton est longue et peu commode. Impossible, dans ce cadre, de faire du recalage en temps réel.

Pour cette raison, nous avons mis au point une version accélérée de la méthode d’adjoint, dans lequel le problème d’élasticité est résolu au moyen d’un réseau de neurones. Le réseau, de type perceptron multi-couches, a été entraîné sur un ensemble de simulations par éléments finis utilisant un modèle néo-hookéen.

Nous donnerons quelques informations sur le réseau et son intégration dans la méthode d’adjoint, puis nous montrerons quelques résultats obtenus avec la méthode d’adjoint accélérée. Bien qu’il reste quelques axes de progression, la méthode d’adjoint avec le réseau constitue une avancée non-négligeable vers un recalage de bonne qualité en temps réel.

Dans une première partie, nous discuterons du mécanisme d'attention qui est au cœur des Transformers (l'architecture la plus classique pour les LLMs). Ensuite, nous discuterons des différentes techniques pour enrichir des LLMs avec des connaissances extérieures : Retrieval augmented generation (RAG) et Agents.

On s'intéresse au problème d'optimiser une fonction objectif g(W x) + c^T x pour x entier, où chaque coordonnée de x est contrainte dans un intervalle. On suppose que la matrice W est à coefficient entiers de valeur absolue bornée par Delta, et qu'elle projette x sur un espace de petite dimension m << n. Ce problème est une généralisation du résultat de Hunkenschröder et al. dans lequel g est séparable convexe, et x est dans un 0-1 hypercube.

On présentera un algorithme en complexité n^m (m Delta)^O(m^2), sous la supposition que l'on sache résoudre efficacement le problème lorsque n = m. Cet algorithme utilise les travaux d'Eisenbrand et Weismantel sur la programmation linéaire entière avec peu de contraintes.

L'algorithme présenté peut être employé théoriquement dans plusieurs problèmes notamment la programmation mixte linéaire avec peu de contraintes, ou encore le problème du sac à dos où l'on doit acheter son sac.

Stochastic optimization naturally appear in many application areas, including machine learning. Our goal is to go further in the analysis of the Stochastic Average Gradient Accelerated (SAGA) algorithm. To achieve this, we introduce a new $\lambda$-SAGA algorithm which interpolates between the Stochastic Gradient Descent ($\lambda=0$) and the SAGA algorithm ($\lambda=1$). Firstly, we investigate the almost sure convergence of this new algorithm with decreasing step which allows us to avoid the restrictive strong convexity and Lipschitz gradient hypotheses associated to the objective function. Secondly, we establish a central limit theorem for the $\lambda$-SAGA algorithm. Finally, we provide the non-asymptotic $L^p$ rates of convergence.

In this talk, we consider a bounded domain in the Euclidean plane and examine the Laplacian eigenvalue problem supplemented with specific boundary conditions. A famous conjecture by Berry proposes that in chaotic systems, eigenfunctions resemble random monochromatic waves; however, this behavior is generally not expected in integrable systems. In this talk, we explore the behavior of high-energy eigenfunctions and their connection to Berry’s random wave model. We do so by studying a related property called Inverse Localization, which describes how eigenfunctions can approximate monochromatic waves in small regions of the domain.

Une notion simple de complexité topologique d'une variété lisse est donnée par la nombre minimal de simplexes dans une triangulation. Pour une variété riemannienne fermée à courbures sectionnelles normalisées il est naturel de comparer cet invariant au volume riemannien. Gelander a conjecturé au début du siècle que pour les variétés localement symétriques irréductbles de dimension $d \ge 4$ le rapport de ces deux quantités devrait être borné dans les deux sens (par une constante ne dépendant que de d). Je présenterai un travail en commun avec Mikolaj Fraczyk et Sebastian Hurtado où nous démontrons cette conjecture dans le cas des variétés arithmétiques.

L'objet de cet exposé est d'établir un lien entre les formes automorphes en caractéristique positive et le champ des G-zips introduit par Pink-Wedhorn-Ziegler. Dans le cas des variétés modulaires de Siegel, j'expliquerai comment les poids des formes automorphes sont entièrement contrôlés par ce champ.

L'équipe Lambda vous donne rendez-vous le samedi 23 novembre 2024 à partir de 14h pour une journée d'intégration ! L'évènement est ouvert à l'ensemble des doctorant.e.s et post-doctorant.e.s du laboratoire, anciens comme nouveaux. L'occasion de se rencontrer autour de diverses activités sportives ou ludiques.

Le planning de la journée sera :

14h-14h30 : Arrivée

14h30 - 15h : Présentation de l'asso, des membres et présentation des participants

15h - 16h : Balle au prisonnier

16h-16h30 : Goûter

16h30 - 17h30 : Mini-jeux (puzzle, blindtest, quizz)

17h30 - 18h : Remise des prix

18h : Bar

Pour participer, il suffit de s'inscrire sur ce sondage. Le point de rendez-vous est le COSEC Rocquencourt (8 Av. Jean Babin, 33600 Pessac).

Au plaisir de vous voir,

L'équipe Lambda

The Lambda team invites you on Saturday, November 23rd, 2024 from 2:00 pm for an integration day! This event is open to all PhD students and postdocs in the lab, both old and new members. It's a great opportunity to meet around various sports and fun activities. 

14h-14h30 : Arrival

14h30 - 15h : Presentation of the association, its members and participants

15h - 16h : Dodgeball

16h-16h30 : Snack

16h30 - 17h30 : Mini games (puzzle, blindtest, quiz)

17h30 - 18h : Prize-giving ceremony

18h : Bar

To participate, simply sign up through this survey. The meeting point is COSEC Rocquencourt (8 Av. Jean Babin, 33600 Pessac).

We look forward to seeing you there,

The Lambda Team

La problématique générale des espaces atteignables peut être résumée de la manière suivante pour un système contrôlé donné: étant donné un état initial $u_i$ et un temps $T \gt 0$, décrire l'espace $R(u_i,T)$ des états finaux $u_f$ que l'on peut atteindre à partir de $u_i$ au temps $T$. Déterminer l'espace atteignable des systèmes contrôlés est l'un des principaux problèmes de la théorie du contrôle. Donner une caractérisation précise des états qui peuvent être atteints en un certain temps fixé est une question encore largement ouverte pour les systèmes paraboliques: même pour l'équation de la chaleur à coefficients constants en une dimension et contrôlée depuis la frontière, la caractérisation complète de l'espace atteignable, en termes d'espaces de Bergman, n'a été obtenue que très récemment. Basé sur un travail en commun avec Sylvain Ervedoza, je présenterai des résultats sur l’espace atteignable pour l’équation de la chaleur avec des perturbations d’ordre inférieur ou des semi-linéaire en dimension $d\geq 1$.

In 2022, Ducas et al. introduced the signature scheme Hawk, based of the presumed hardness of a new problem in lattice-based cryptography: the Lattice Isomorphism Problem for the module-lattice O_L^2, where L is a cyclotomic number field. Last year we presented a polynomial time algorithm solving this problem when L is a totally real number field (thus not affecting the security of Hawk). More recently, we provided a reduction of the same problem when L is now a CM field (thus containing Hawk's instance) to the problem of finding a generator of a principal quaternionic ideal. 

In this talk we give a framework containing both the totally real and the CM case, and we will discuss the differences. This is based on a joint work with C. Chevignard, P-A. Fouque, A. Pellet-Mary, H. Pliatsok and A. Wallet.

This talk is concerned with asymptotic persistence, extinction and spreading properties for structured population models resulting in non-cooperative Fisher-KPP systems with space-time periodic coefficients, motivated by a wide class of models in population biology. Results are formulated in terms of a family of generalized principal eigenvalues associated with the linearized problem. When the maximal generalized principal eigenvalue is negative, all solutions to the Cauchy problem become locally uniformly positive in long-time, at least one space-time periodic uniformly positive entire solution exists, and solutions with compactly supported initial condition asymptotically spread in space at a speed given by a Freidlin-Gärtner-type formula. When another, possibly smaller, generalized principal eigenvalue is nonnegative, then on the contrary all solutions to the Cauchy problem vanish uniformly and the zero solution is the unique space-time periodic nonnegative entire solution. When the two generalized principal eigenvalues differ and zero is in between, the long-time behavior depends on the decay at infinity of the initial condition. The proofs rely upon double-sided controls by solutions of cooperative systems. The control from below is new for such systems and makes it possible to shorten the proofs and extend the generality of the system simultaneously.

Les impacts environnementaux de l'apprentissage machine sont de plus en plus visibles et posent question jusque dans la presse généraliste. Face à cette problématique, de nombreux.ses chercheurs.ses et ingénieur.es, dans la recherche publique comme dans l'industrie, développent des approches, méthodes et outils pour mieux documenter et réduire les impacts environnementaux associés aux modèles d'apprentissage machine. De nombreuses pistes d'optimisations sont explorées, telles des optimisations logicielles ainsi que des changements fréquent de matériel pour profiter de la meilleure efficacité énergétique. Ou encore, au travers du déplacement géographique des calculs vers des zones avec une électricité moins carbonée. Tout ce travail d'optimisation ne doit pas passer à côté de deux facteurs cruciaux que sont les déplacements d'impacts ainsi que les tendances de croissance du numérique et de l'IA que ces optimisations participent à continuer. De plus, même si les problématiques environnementales associées à la production et l'utilisation de modèles d'apprentissage machine venaient à être résolue par ces méthodes, de nombreuses autres problématiques, notamment éthiques et sociales persisteraient. Ce séminaire vise à donner un état des lieux des impacts environnementaux et sociaux de l'IA ainsi que des stratégies pensées pour y faire face.

This talk focuses on the mathematical properties of solutions to partial differential equations (PDEs) in the presence of dissipation. We approach this study from two complementary perspectives:

– The passive analysis of the evolution properties observed in response to given data (initial conditions). This corresponds to the Cauchy theory of PDEs.

– The active analysis of the evolution properties, where the goal is to influence the evolution by selecting and synthesizing appropriate data (referred to as control functions) from a predefined class to achieve a desired outcome (e.g., specific initial and final states). This represents the perspective of the control theory of PDEs.

I will illustrate this framework with examples of PDEs studied during my thesis.

Separable states are multipartite quantum states that can be written as a convex combination of product states. Product states are multipartite quantum states that can be written as a tensor product of states in each space. Quantum state separable problem is an NP-hard problem but fundamental for quantum information theory. We propose two relaxation techniques for this problem. In the view of commutative optimization, we treat the states as matrices of multilinear complex polynomials. Our relaxation technique is found similar to that for complex bilinear polynomials arising in the Alternating Current Optimal Power Flow problem. In the view of non-commutative optimization, we treat the states as tensor products of bounded Positive Semi-definite variables. We propose a generalized McCormick relaxations using linear matrix inequalities. These two relaxations will be the key component to drive an exact branch-and-cut algorithm.

Dans cet exposé, je vais m'intéresser au mouvement Brownien dans des cadres simples de géométrie sous riemannienne: le groupe de Heisenberg et les groupes de Carnot de rang 2. Nous proposons une construction d'un couplage de deux mouvement Browniens à un temps fixe. Cette construction est basée sur une décomposition de Legendre du mouvement Brownien standard et de son aire de Lévy. Nous déduisons alors des estimées précises de la décroissance en variation totale entre les lois des mouvements Browniens 

et par une technique de changement de probabilité une formule d'intégration par partie de type Bismut ainsi des estimées de régularisation de type Poincaré inverse pour le semi-groupe associé. Travail en commun avec Marc Arnaudon, Magalie Bénéfice et Delphine Féral

This talk is concerned with representation issues associated with the numerical solution of a unified mathematical model of continuum mechanics, due to Godunov, Peshkov, and Romenski, which can describe ideal fluids, viscous fluids and elastoplastic solids as special cases of a general continuum. The different regimes are characterized solely by the choice of material parameters and the resulting PDE system is of hyperbolic nature, with clearly defined finite wave speeds, in contrast to the standard formulation of viscous fluids via the Navier–Stokes equations. The description of such a general continuum hinges on the evolution of a matrix-valued field called distortion, which is a generalization of the inverse deformation gradient in solid mechanics. In the fluid regime, this quantity can no longer be recovered as a gradient of displacements and encodes very rich information, in particular due to the different orientations that ideal fluid parcels can be found in. The fine features of the distortion field can be challenging (or outright impossible) to resolve with standard well-tested Finite Volume methods. Degenerate situations are routinely encountered where unphysical states are generated simply as a result of taking a convex combination of two data points. We show how changing to an alternative representation of the same distortion field, obtained via polar decomposition, can be used to solve such discretization issues. Instead of the original PDE system, one can instead evolve the rotational and stretch components of the distortion matrix separately, which allows the description of the rotational components through a quaternion-valued partial differential equation. We discuss the peculiarities of quaternion PDEs and some of the discretization strategies that they enable. We present numerical examples of high-Reynolds number simulations which could not be carried out with the previous formulation of the model.

Soit $f$ un endomorphisme du plan projectif complexe de degré $d>1$. Son entropie topologique est $2\log(d)$ (Misiurewicz-Przytycki, Gromov) et $f$ possède une unique mesure d’entropie maximale notée $\mu$. Cette mesure est ergodique et possède deux exposants de Lyapunov qui sont minorés par $(\log d)/2$ (Briend-Duval). La mesure $\mu$ est aussi l’auto-intersection $\mu = T \wedge T$ du courant de Green $T$ de $f$. Les exposants de Lyapunov sont égaux à $(\log d)/2$ si et seulement si $\mu \lll \text{Leb}$ (Ledrappier). C'est équivalent à dire que $T$ est lisse $>0$ sur un ouvert, et c’est aussi équivalent à dire que $f$ est un exemple de Lattès (Berteloot-Loeb, Berteloot-Dupont).

Il est naturel de se demander si l’on peut caractériser de façon similaire la minimalité d’un seul exposant. R. Dujardin a démontré (2012) que l’absolue continuité de $\mu$par rapport à la trace de $T$ implique qu’un exposant est égal à $(\log d)/2$. J’ai ensuite démontré que la réciproque est vraie, donnant une première caractérisation. Les exemples connus d'applications possédant un seul exposant minimal sont donnés par des applications préservant un pinceau de droites avec une dynamique de type Lattès sur le pinceau. À partir d’une relation d'absolue continuité entre $\mu$ et $T$ (en un sens fort), on peut démontrer l’existence d’un tel pinceau avec une dynamique Lattès.

Les démonstrations sont basées sur l'utilisation de formes normales pour la dynamique, ainsi que sur la théorie du pluripotentiel. Pendant l'exposé, je reviendrai sur ces différentes notions.

Venez coopérer ou trahir vos amis dans la bonne humeur autour de vos jeux préférés. Si vous avez des jeux de société chez vous, n'hésitez pas à les apporter pour la soirée !

Les fractions rationnelles postcritiquement finies jouent un rôle important en dynamique complexe à une variable. Elles sont liées à des phénomènes de bifurcations maximales et forment un ensemble dense pour la topologie de Zariski dans l’espace de modules des fractions rationnelles de degré d. En dimensions supérieures, nous montrons, avec Thomas Gauthier et Gabriel Vigny, que leurs analogues ne sont pas Zariski denses dans l’espace de modules des endomorphismes de degré d de l’espace projectif Pk, dès que d et k sont supérieurs ou égaux à 2. La preuve combine des arguments issus de l’analyse complexe, de la géométrie arithmétique et de la dynamique réelle. Deux ingrédients essentiels sont l’utilisation d’ensembles hyperboliques spéciaux appelés mélangeurs, ainsi que l’indépendance des multiplicateurs des points périodiques. Ce dernier point a été récemment généralisé dans un travail en collaboration avec Igors Gorbovickis.

Hub Labeling (HL) is a state-of-the-art method for answering shortest-distance queries between node pairs in weighted graphs. It provides very fast query times but also requires considerable additional space to store the label information. Recently, a generalization of HL, called Landmark Hub Labeling (LHL), has been proposed, that conceptionally allows a storage of fewer label information without compromising the optimality of the query result. However, query answering with LHL was shown to be slower than with HL, both in theory and practice. Furthermore, it was not clear whether there are graphs with a substantial space reduction when using LHL instead of HL. 

In this talk, we describe a new way of storing label information of an LHL such that query times are significantly reduced and then asymptotically match those of HL. We establish novel bounds between different labeling variants and provide a comparative experimental study between approximation algorithms for HL and LHL. We demonstrate that label sizes in an LHL are consistently smaller than those of HL across diverse benchmark graphs, including road networks.

Les solutions des systèmes différentiels ou aux différences singuliers réguliers ont de bonnes propriétés analytiques. Par exemple, les matrices fondamentales de solutions des systèmes différentiels singuliers réguliers ont des coefficients qui ont une croissance modérée en 0. Des algorithmes pour reconnaître les systèmes singuliers réguliers ont été donnés pour de nombreux cas, par exemple pour les systèmes différentiels et ceux aux q-différences. Mais, ils ne s'appliquent pas aux systèmes dits de Mahler. Ces derniers ont des liens avec de nombreux domaines, par exemple la théorie des automates finis. Nous expliquerons comment reconnaître les systèmes de Mahler singuliers réguliers. C'est un travail en commun avec Colin Faverjon.

Let $d \equiv 5 \bmod 8$ be a square-free positive integer and consider the fundamental unit $u_d$ of the real quadratic field $K = \mathbb{Q}(\sqrt{d})$. Since 2 is inert in $K$, there are three possible residue classes of $u_d$ modulo (the prime above) 2. All other things being equal, one expects each of the three residue classes to occur equally often. In particular, one expects $u_d \equiv 1$ one third of the time; we call such $d$’s Eisenstein discriminants. Stevenhagen showed in 1990s that Eisenstein discriminants $d$ are related to cubic number fields of discriminant $4d$.

In this talk, I will explore this relationship and in particular compare the counting functions of Eisenstein discriminants and of cubic fields of discriminant $4d$. Some results can be proved, but tantalising mysteries remain.

In this talk, we will present the study of a model for particle suspensions in a non-Newtonian Ostwald-DeWaele fluid with a potentially degenerate viscosity coefficient.

The analysis of problems associated with such systems is a very active research topic, the source of many recent results, particularly in the case of particles that sediment in a Newtonian fluid (see, for example, the works of D. Cobb, R. Höfer, A. Mecherbet, R. Schubert, F. Sueur). We will consider the case where particles are suspended in a non-Newtonian fluid. From a mathematical point of view, this is characterized by a Stokes-Transport equation with the particularity that the Stokes equation is nonlinear, the viscosity term being expressed as a p-Laplacian for the symmetrized gradient. After briefly contextualizing the problem, we will present our result: the existence of global weak energy solutions.

In order to show that we do define an active scalar equation, i.e. one in which the relative density of the particles suspended in the fluid gives meaning to a solution of the system through an inverse mapping, it is then necessary to use monotonicity methods in conjunction with techniques derived from DiPerna-Lions theory to establish the existence of suitable weak solutions. We will therefore present the main ideas for establishing the existence of such ones.

Les technologies numériques ont ouvert un nouvel espace permettant d'accélérer les échanges d'informations entre les personnes. La délégation d'une part de plus en plus importante de nos processus intellectuels à des automatismes transforme nos modes de pensée, soumis à des demandes d'interaction de plus en plus rapides. Les libertés nouvelles offertes par ces technologies s'accompagnent cependant d'une pression plus forte sur les individus, dont les traces numériques peuvent être exploitées à grande échelle. Quelle peut être la place de la personne à l'ère numérique ?

Many problems, especially in machine learning, can be formulated as optimization problems. Using optimization algorithms, such as stochastic gradient descent or ADAM, has become a cornerstone to solve these optimization problems. However for many practical cases, theoretical proofs of their efficiency are lacking. In particular, it has been empirically observed that adding a momentum mechanism to the stochastic gradient descent often allows solving these optimization problems more efficiently. In this talk, we introduce a condition linked to a measure of the gradient correlation that allows to theoretically characterize the possibility to observe this acceleration.

We prove Landis-type uniqueness results for both the semidiscrete heat and the stationary discrete Schrödinger equations. To establish a nomenclature, we refer to Landis-type results when we are interested in the maximum vanishing rate of solutions to equations with potentials. The results are obtained through quantitative estimates within a spatial lattice which manifest an interpolation phenomenon between continuum and discrete scales. In the case of the elliptic equation, these quantitative estimates exhibit a rate decay which, in the range close to continuum, coincides with the same exponent as in the classical results of the Landis conjecture in the Euclidean setting. 

Joint work with Aingeru Fern\'andez-Bertolin and Diana Stan.

In this talk, we propose a novel collocation-based Model Order Reduction (cMOR) strategy for solving parametric advection-diffusion PDEs on moving Chimera grids. Unlike traditional projection-based MOR, cMOR solves the High-Dimensional Model on a small subset of collocation points and extends the solution to the entire domain using a global reduced basis. By leveraging the ADER method on unsteady Chimera meshes, cMOR addresses the computational challenges posed by convective dominated problems, particularly the Kolmogorov N-width barrier. Our results demonstrate the efficiency of cMOR in reducing the computational cost while maintaining accuracy.

La systole d'une surface hyperbolique est la longueur de la géodésique fermée la plus courte sur la surface. Déterminer la systole maximale possible d'une surface hyperbolique d'une topologie donnée est une question classique en géométrie hyperbolique. Je vais parler d'un travail commun avec Mingkun Liu sur la question de ce que les constructions aléatoires peuvent apporter à ce problème d'optimisation.

Soit X une variété sur un corps fini. Etant donné un ordre R dans une algèbre semi-simple sur les rationnels et un faisceau constructible de R-modules F sur X, on peut considérer une fonction L non commutative naturellement associée à F. Dans cet exposé, je présenterai une formule de valeurs spéciales aux entiers négatifs pour cette fonction L, exprimée en terme de cohomologie Weil-étale ; le résultat est conditionnel au "bon comportement" de celle-ci. Cette formule est un analogue géométrique de, et implique, la conjecture équivariante des nombres de Tamagawa de Burns-Flach dans le cas d'un motif de Tate et de ses twists négatifs sur un corps global de caractéristique p. Elle généralise aussi les résultats de Lichtenbaum et Geisser sur les valeurs spéciales de fonctions zeta, et le travail de Burns-Kakde dans le cas des fonctions L non-commutatives provenant d'un recouvrement Galoisien de variétés.

After explaining the notions of symmetry and differential equations, we review possibilities of symmetry methods and advantages of their usage in the theory of differential equations and mathematical physics.

As a specific example, we discuss the history of the (real potential symmetric) dispersionless  Nizhnik equation and its applications and overview its extended symmetry analysis carried out in our papers. More specifically, we construct essential megaideals of the maximal Lie invariance algebra of this equation. Using the original version of the algebraic megaideals-based method, we compute the point- and contact-symmetry pseudogroups of this equation as well as the point-symmetry pseudogroups of its Lax representation and the original real symmetric dispersionless Nizhnik system. This is the first example in the literature, where there is no need to use the direct method for completing the computation.

In addition, we also find geometric properties of the dispersionless Nizhnik equation that completely define it. Lie reductions of this equation are classified, which results in wide families of its new closed-form invariant solutions. We also study hidden generalized symmetries, hidden cosymmetries and hidden conservation laws of this equation.

Exceptionnellement, l'accueil de la Cellule Informatique au bureau 225

• ne sera ouvert que de 10h à 12h et de 14h à 16h
• et sera fermé vendredi

en raison de la participation d'une partie de l'équipe informatique aux JRES (journées réseaux de l'enseignement et de la recherche).

• -> Pensez à anticiper les retraits de matériel de prêt.

In this talk, we will define the class group of a number field, which is an important object in number theory and in some areas of cryptography.

Given a number field K, there exists a classic algorithm to compute its class group Cl(K) : the Buchmann algorithm. However, the time complexity is exponential in the degree d of the number field K.

The goal in the second part will be to describe some algorithms that can compute the class group of a large degree number field K inductively, by reducing the problem to the same computation over others smaller degree number fields.

These methods make use of the properties of norm relations and generalised norm relations, which are relations in the G-module Q[G], where G is the Galois group of the Galois closure of K.

Real-world industrial applications frequently confront the task of decision-making under uncertainty. The classical paradigm for these complex problems is to use both machine learning (ML) and combinatorial optimization (CO) in a sequential and independent manner. ML learns uncertain quantities based on historical data, which are then used used as an input for a specific CO problem. Decision-focused learning is a recent paradigm, which directly trains the ML model to make predictions that lead to good decisions. This is achieved by integrating a CO layer in a ML pipeline, which raises several theoretical and practical challenges.

In this talk, we aim at providing a comprehensive introduction to decision-focused learning. We will first introduce the main challenges raised by hybrid ML/CO pipelines, the theory of Fenchel-Young losses for surrogate differentiation, and the main applications of decision-focused learning. As a second objective, we will present our ongoing works that aim at developing efficient algorithms based on the Bregman geometry to address the minimization of the empirical regret in complex stochastic optimization problems.

Statistical estimation in a geometric context has become an increasingly important issue in recent years, not least due to the development in ML of non-linear dimension reduction techniques, which involve projecting high-dimensional data onto a much lower-dimensional sub-manifold. The search for statistical guarantees justifying both the use and the effectiveness of these algorithms is now a much-studied area. In this talk, we will take a geometric view of the issue, and see how some usual curvature quantities are translated into algorithmic guarantees. First, we will see that upper bounds on sectional curvatures give good properties for barycenter estimation, and then we will see that a lower bound on Ricci curvature implies the existence of depth points, giving rise to robust statistical estimators. Those works are based on joint works with Victor-Emmanuel Brunel (ENSAE Paris) and Shin-ichi Ohta (Osaka University).

Given a functional inequality whose extremizers are known, the question of stability is as follows: If a function almost saturates the inequality, is it close to some extremizer? The most famous example is perhaps that of the isoperimetric inequality: The extremizers are the balls, and the question of stability comes down to showing that the isoperimetric deficit controls a certain distance from the ball. There are 4 methods for obtaining such results: the symmetrisation method, the transport mass approach, the selection principle and the ABP method. In this talk, I will present a recent work in which I introduce a fifth method, Stein's method. In particular, I will show how it proves a stability result for the first Steklov eigenvalue.

In addition, I will also present a stability result for the Brascamp-Lieb inequality, which is a functional inequality encoding certain weighted isoperimetric properties. This last result is based on joint work in progress with Bonnefont (IMB Bordeaux) and Joulin (IMT Toulouse).

On dit qu'une classe de groupes de type fini satisfait une alternative de Tits si chacun de ces groupes est soit "petit" (le sens peut dépendre du contexte), soit contient un groupe libre. L'alternative de Tits originelle concerne les groupes linéaires (et dans ce cas petit signifie virtuellement résoluble). Depuis, elle a été démontrée dans de nombreux contextes géométriques, souvent en courbure négative : groupes agissant sur des espaces hyperboliques, sous-groupes de groupes modulaires de surfaces ou de Out(F_N), groupes agissant sur des complexes simpliciaux avec des bonnes propriétés de courbure, etc.

Je présenterai une nouvelle preuve de l'alternative de Tits pour les groupes agissant sur des immeubles de type Ã_2 (objets que j'introduirai). La nouveauté de notre approche est qu'elle se base sur des marches aléatoires. On démontre également au passage un théorème "local-global" : un groupe dont tous les éléments fixent un point a un point fixe global. C'est un travail en commun avec Corentin Le Bars et Jeroen Schillewaert.

Dans cet exposé, nous étudierons la taille du groupe de Tate-Shafarevich de certaines surfaces abéliennes sur le corps de fonctions $\mathbb{F}_q(t)$. Hindry et Pacheco ont montré que, pour les variétés abéliennes sur des corps de fonctions, la taille du Sha (dès que finie) est majorée par la hauteur exponentielle. Nous montrerons qu’en dimension 2 leur borne est optimale. Pour cela, on construira une suite de Jacobiennes vérifiant la conjecture de BSD, puis nous calculerons explicitement leur fonction L à l’aide de sommes de caractères. Grâce à des méthodes analytiques, nous estimerons la taille de la valeur spéciale, pour retrouver finalement la borne souhaitée sur le cardinal de leur groupe de Sha.

We consider an inhomogeneous linear Boltzmann equation in a low temperature regime, in the presence of an external force deriving from a single-well potential and with a collision operator featuring multiple conservation laws. We start by giving a description of the purely imaginary spectrum of the associated operator. We then go further and provide a hypocoercive result on the spectrum with real part smaller than $h$. It enables us to obtain some information on the long time behavior of the solutions and in particular to show the existence of metastable states. This is a joint work with Frédéric Hérau and Dorian Le Peutrec.

We discuss a rather general algorithm for point counting on a (nice) curve over a finite field of characteristic p, assuming a lift to the valuation ring in a finite extension of Q_p has been given. Our method for computing the action of Frobenius is based on computing cup products in cohomology, not on explicitly rewriting 1-forms using exact 1-forms. These cup products can be computed locally, and the same holds for the local expansions of a lift of Frobenius that is determined uniquely by some global equations. This is joint work with Amnon Besser, Pengju Guan, and Muxi Li.

L'ordre du jour sera le suivant :

1. Approbation du compte rendu de la réunion du conseil scientifique du 10 septembre (vote):

2. Informations générales.

3. Renouvellement partiel des membres de la CC25 et de la CC26 (vote).

4. Présentation par Raphaël Loubère de la composition du comité de sélection MCF 26 Bordeaux INP/IMB en Calcul Scientifique et Modélisation.

5. Examen des demandes de contrats doctoraux fléchés de l'EDMI et première phase de sélection (vote).

6. Questions diverses.

The stationary problem for the free boundary incompressible Navier-Stokes equations lies at the confluence of two distinct lines of inquiry in fluid mechanics. The first views the dynamic problem as an initial value problem. In this context, the stationary problem arises naturally as a special type of global-in-time solution with stationary sources of force and stress. One then expects solutions to the stationary problem to play an essential role in the study of long-time asymptotics or attractors for the dynamic problem. The second line of inquiry, which dates back essentially to the beginning of mathematical fluid mechanics, concerns the search for traveling wave solutions. In this context, a huge literature exists for the corresponding inviscid problem, but progress on the viscous problem was initiated much more recently in the work of the speaker and co-authors. For technical reasons, these results were only able to produce traveling solutions with nontrivial wave speed. In this talk we will discuss the well-posedness theory for the stationary problem and show that the solutions thus obtained lie along a one-parameter family of slowly traveling wave solutions. This is joint work with Noah Stevenson.

Cette recherche est menée pour examiner une approche d'optimisation distributionnellement robuste appliquée au problème de dimensionnement de lots avec des retards de production et une incertitude de rendement sous des ensembles d'ambiguïté par événement. Les ensembles d'ambiguïté basés sur les moments, Wasserstein et le clustering K-Means sont utilisés pour représenter la distribution des rendements. Des stratégies de décision statiques et statiques-dynamiques sont également considérées pour le calcul d'une solution. Dans cette présentation, la performance de différents ensembles d'ambiguïté sera présentée afin de déterminer un plan de production qui soit satisfaisant et robuste face aux changements de l'environnement. Il sera montré, à travers une expérience numérique, que le modèle reste traitable pour tous les ensembles d'ambiguïté considérés et que les plans de production obtenus demeurent efficaces pour différentes stratégies et contextes décisionnels.