Les résultats obtenus par Rolin et Servi sur les classes quasianalytiques fournissent une méthode très générale pour la construction de structures o-minimales polynomialement bornées. Néanmoins, les structures o-minimales obtenues par le biais de cette méthode possèdent toutes la décomposition cellulaire lisse. En particulier, toute fonction définissable à variable réelle est lisse sauf en un nombre fini de points. Pour autant, Le Gal et Rolin ont construit une structure o-minimale sans la décomposition cellulaire lisse en 2009. Nous présenterons une généralisation des résultats de Rolin et Servi qui admet la structure de Le Gal et Rolin comme cas particulier. Nous montrerons ensuite qu'il existe une structure o-minimale dans laquelle on peut définir une fonction nulle part lisse.
In 1992, Andrew Blass published "A game semantics for linear logic", the first attempt to give meaning to proofs in linear logic in terms of winning strategies for a game. However, the composition of his construction was not associative, the two possible ways to compose three strategies were not necessary equal. Most of the subsequent works saw the non-associativity as an issue that needed to be corrected. In this talk, after explaining the Blass games in details, I will embrace their non-associativity, reveal their actual structure by using Munch-Maccagnoni's duploids, a non-associative model of effects, and show what it tells us about games.
Cet exposé porte sur l'analyse mathématique d'un modèle de couplage océan-atmosphère, décrit par un couplage de deux équations paraboliques avec des conditions d'interface non linéaires. Nous introduisons un modèle vertical 1D correspondant à un problème de couche limite d'Ekman couplée avec des viscosités turbulentes, l'intérêt de ce modèle réside dans sa proximité avec des modèles réalistes en considérant les stratégies numériques employées pour prendre en compte les échelle turbulentes. Les conditions d'interfaces issues de ces modèles réalistes induisent une dépendance entre les profils de viscosité et la trace de la solution et donne au problème un caractère non local. Nous étudions tout d'abord le caractère bien posé du modèle dans son état stationnaire et non stationnaire en considérant des viscosités turbulentes paramétrées généralisées. Nous établissons des critères suffisants sur les profils de viscosité pour l'unicité de la solution et constatons qu'ils ne sont pas satisfaits pour des paramètres de l'ordre de grandeur utilisé dans les modèles océaniques et atmosphériques. Afin d'identifier précisément les conditions qui permettent l'existence et l'unicité de solution cohérent avec la physique, nous établissons un critère nécessaire et suffisant pour l'état stationnaire. Nous montrons que la solution n'est pas unique lorsque l'on considère les profils de viscosité typiques des modèles océaniques et atmosphériques. Finalement, nous illustrons que la non-unicité est produite par une incohérence entre le profil de viscosité et la paramétrisation de la couche limite.
What do the usual set operations (intersection, union, complement,…) and the structure of first-order formulas have in common? In this talk, we will see how both can be described in the context of category theory through the notion of “Doctrines”. After a brief review of first-order logic, we will compare two worlds: we will observe that the set of all subsets of a given set and the set of all first-order formulas that depend on some fixed free variables can be endowed with the structure of a Boolean algebra. This will allow us to show that the powerset functor and indexed families of first-order formulas are both examples of first-order Boolean doctrines. These are categorical structures where logical connectives and quantifiers are interpreted as algebraic operations and adjunctions. Finally, I will give a hint about my current research topic, where I am exploring the concept of “quantifier-free formula” in the doctrinal setting.
We generalize the seminal polynomial partitioning theorems of Guth and Katz [1, 2] to a set of semi-Pfaffian sets. Specifically, given a set $\Gamma \subseteq \mathbb{R}^n$ of $k$-dimensional semi-Pfaffian sets, where each $\gamma \in \Gamma$ is defined by a fixed number of Pfaffian functions, and each Pfaffian function is in turn defined with respect to a Pfaffian chain $\vec{q}$ of length $r$, for any $D \ge 1$, we prove the existence of a polynomial $P \in \mathbb{R}[X_1, \ldots, X_n]$ of degree at most $D$ such that each connected component of $\mathbb{R}^n \setminus Z(P)$ intersects at most $\sim \frac{|\Gamma|}{D^{n - k - r}}$ elements of $\Gamma$. Also, under some mild conditions on $\vec{q}$, for any $D \ge 1$, we prove the existence of a Pfaffian function $P'$ of degree at most $D$ defined with respect to $\vec{q}$, such that each connected component of $\mathbb{R}^n \setminus Z(P')$ intersects at most $\sim \frac{|\Gamma|}{D^{n-k}}$ elements of $\Gamma$. To do so, given a $k$-dimensional semi-Pfaffian set $\gamma \subseteq \mathbb{R}^n$, and a polynomial $P \in \mathbb{R}[X_1, \ldots, X_n]$ of degree at most $D$, we establish a uniform bound on the number of connected components of $\mathbb{R}^n \setminus Z(P)$ that $\gamma$ intersects; that is, we prove that the number of connected components of $(\mathbb{R}^n \setminus Z(P)) \cap \gamma$ is at most $\sim D^{k+r}$. Finally, as applications, we derive Pfaffian versions of Szemer\'edi-Trotter-type theorems and also prove bounds on the number of joints between Pfaffian curves. This is joint work with Martin Lotz and Nicolai Vorobjov.
These results, together with some of my other recent work with Adam Sheffer on bounding the number of distinct distances on plane Pfaffian curves, are steps in a larger program - pushing discrete geometry into settings where the underlying sets need not be algebraic. I will also discuss this broader viewpoint in the talk.
[1] Larry Guth, Polynomial partitioning for a set of varieties, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 159, Cambridge University Press, 2015, pp. 459–469.
[2] Larry Guth and Nets Hawk Katz, On the Erdős distinct distances problem in the plane, Annals of mathematics (2015), 155–190.
On présente dans cet exposé une classe de systèmes décrivant la propagation d'ondes de surface en insistant sur les nouveaux problèmes mathématiques qu'offrent ces systèmes par rapport à leur version scalaire.
Many recent works have studied how analogue models work, compared to classical digital ones. By “analogue” models of computation, we mean computing over continuous quantities, while “digital” models work on discrete structures. It led to a broader use of Ordinary Differential Equation (ODE) in computability theory. From this point of view, the field of implicit complexity has also been widely studied and developed. We show here, using arguments from computable analysis, that we can algebraically and implicitly characterise PTIME and PSPACE for functions over the reals using ODEs.
Within the study of the semantics of programming languages, computational effects may be modelled with monads, and weak distributive laws between monads are then a tool to combine two such effects. The first part of the talk will be dedicated to introducing (monotone) (weak) distributive laws.
In both the category of sets and the category of compact Hausdorff spaces, there is a monotone weak distributive law that combines two layers of non-determinism. Noticing the similarity between these two laws, we study in the second part of the talk whether the latter can be obtained automatically as some sort of lifting of the former.
More specifically, we show how a framework for constructing monotone weak distributive laws in regular categories lifts to categories of algebras, giving a full characterization for the existence of monotone weak distributive laws therein. We then exhibit such a law, combining probabilities and non-determinism, in compact Hausdorff spaces; but we also show how such laws do not exist in a lot of other cases.
Cassandre Lebot "Introduction to the low-Mach-number limit for compressible two-phase flows"
Abstract : This presentation provides an introduction to the low-Mach-number limit for compressible two-phase flows. Such models describe the coupled evolution of a liquid and a gas phase and are widely used in geophysical and industrial contexts. When the characteristic flow velocity is small compared with the speed of sound in the mixture, the governing equations are expected to simplify and approach an incompressible or quasi-incompressible regime.
The core of the talk focuses on the formal derivation of this asymptotic limit. Starting from a general compressible two-phase system with one single velocity, we introduce the nondimensional formulation and analyse how the pressure and velocity terms behave as the Mach number tends to zero. This formal process highlights the constraints imposed by the incompressibility condition, the evolution of phase fractions, and the modifications to the momentum and mass equations.
Charlotte Tonnelier "Modeling problems around the Poisson-Nernst-Planck equations"
Abstract: The movements of charged particles can be modeled by the Poisson-Nernst-Planck equations, completed with a Stokes equation for hydrodynamic behavior. These dynamics are necessary for understanding the methods of energy recovery from saline gradients. We are interested here in modeling these phenomena in a nanofluidic exchanger. We then wish to understand how to model the different scales present in the geometry of the system, as well as the influence of the boundary conditions.
The rational base number system, introduced by Akiyama, Frougny and Sakarovitch in 2008, is a generalization of the classical integer base number system. Within this framework two interesting families of infinite words emerge, called minimal and maximal words. We conjecture that every minimal and maximal word is normal over an appropriate subalphabet. To support this conjecture, I will present the results of several numerical experiments that examine the richness threshold and the deviation from uniformity of these words. I will also discuss the implications that the validity of this conjecture would have for several long-standing open problem, such as the non-existence of the so-called Z-numbers and the "Collatz-inspired" 4/3-problem.
Soit C une courbe algébrique réelle. Un morphisme $\C \to P^1$ est séparant si la préimage des points réels de P^1 est exactement la partie réelle de C. Le degré d'un tel morphisme est nécessairement supérieur au nombre de composantes de la partie réelle de C. Mais existe-t-il des morphismes séparants de degré égal au nombre de composantes ? Dans cet exposé on présentera une obstruction à l'existence de morphismes séparants de petit degré. Il s'agit d'un travail en cours avec A. Demory et A. Toussaint, basé sur des idées de M. Manzaroli.
J'expliquerai comment la géométrie de Hilbert des convexes peut être généralisée aux corps ordonnés non archimédiens et utilisée pour étudier les propriétés à grande échelle des géométries de Hilbert réelles et leurs dégénérescences. En étudiant le cas des polytopes, nous obtenons ainsi une description explicite des cones asymptotiques des géométries de Hilbert des polytopes réels.
Travail en commun avec Xenia Flamm
Lake tsunamis represent an underrecognized but potentially devastating natural hazard for growing lakeside populations, infrastructure, and cultural heritage. In lacustrine environments, the combination of steep slopes, human-driven sediment input, and active tectonics makes lakes especially vulnerable to mass movements or delta collapses triggered by earthquakes or occurring spontaneously. Mass-wasting deposits (MWDs) in lakes have long been used as palaeoseismic archives, but their associated tsunami hazards remain poorly understood. This thesis advances the study of lake tsunamis by integrating numerical modelling with sedimentological, palaeoseismological, historical, and archaeological records to reconstruct past events and improve hazard assessments.
In this talk, four study sites (Lake Aiguebelette, Lake Bourget, Lake Iseo, and Lake Iznik) were prioritized based on geophysical and geological observations, enabling responses to different types of geodynamic contexts. A novel numerical model was developed that couples landslide dynamics, co-seismic deformation, and seismic wave propagation with both hydrostatic and dispersive tsunami models to capture the complete cycle of earthquake-landslide-induced tsunamis. This approach not only takes into account the combined effects of earthquakes and landslides in generating tsunami waves but also simulates basin-wide seiches excited by seismic shaking. We further demonstrate how seiche waves can interact with turbidites to form homogeneous layers and provide a framework to distinguish seismic from non-seismic triggers of megaturbidites. We also quantify dispersive effects in subaqueous landslideinduced tsunamis in closed lakes.
This is an expository talk introducing the first concepts of combinatorial category theory. The goal is to present the so-called co-Yoneda lemma, stating that any presheaf is canonically a colimit of representables. We will explain these terms and try to convey some intuition as to why and how the result holds. From my last talk, only categories and functors are expected to be remembered.
In this talk, we will look at some efficient algorithms to recover a message from a corrupted encoding of it. This is a phenomenon that happens all around us, from a phone call to scanning QR codes. We will consider encodings based on low-degree polynomials. We will start with some basic definitions in error-correcting codes, and then we will see few technical lemmas that is useful in designing our algorithms. If time permits, we will also see a cool application of these algorithms for hardness amplification in complexity theory. The talk is aimed at general math/CS audience.
Dans le contexte des champs de vecteurs analytiques réels en un point singulier, Cano, Moussu et Sanz ont introduit et étudié la notion de pinceau intégral de trajectoires en ce point afin d'obtenir des informations sur les possibles comportements dynamiques. Nous prolongeons cette approche du côté formel, en utilisant la calculabilité explicite des transséries (réticulées i.e. grid-based au sens d'Ecalle - van der Hoeven) pour la résolution d'équations différentielles. Plus précisément, étant donné un champ de vecteur formel du plan, nous introduisons la notion de trajectoire transsérielle, et fournissons une description explicite des différents pinceaux transseriels possibles. Il s'agit d'une première étape dans l'étude en cours de la même question en dim 3. Travail en commun avec Olivier Le Gal, Daniel Pananzzolo et Fernando Sanz.
Entropy and Fisher information, where the latter is also known as the entropy production, play essential roles in fields such as physics, biology, and information theory, and they are also powerful tools in mathematics, for instance, in analyzing large time behavior, regularity, and asymptotic properties of solutions. In this talk, we investigate the time evolution of Fisher information for nonlinear diffusion equations on bounded domains with Neumann boundary conditions, extending classical results for the linear heat equation and the porous medium equation on the whole space. In particular, we introduce an alternative formulation of one-dimensional nonlinear Fisher information that reveals its time monotonicity. As an application, the global existence of solutions to the one-dimensional critical quasilinear fully parabolic Keller–Segel system with nonlinear diffusion and nonlinear sensitivity is studied. This is based on joint work with Tomasz Cieślak (IMPAN, Poland) and Kentaro Fujie (Tohoku University, Japan).
Tout polynôme multivarié P(x1,…,xn) peut s’écrire comme une somme de monômes, c’est-à-dire une somme de produits de variables et de constantes. En général, la taille d’une telle expression correspond au nombre de monômes ayant un coefficient non nul. Que se passe-t-il si l’on ajoute une autre couche de complexité et que l’on considère des expressions sous la forme de sommes de produits de sommes (de variables et de constantes) ? Dans ce cas, il devient difficile de démontrer qu’un polynôme donné P(x1,…,xn) ne possède pas de petites expressions de ce type. Nous présenterons le contexte de cette question, ses liens avec la complexité booléenne classique ainsi que quelques résultats de base dans ce domaine. Enfin, nous exposerons certains résultats récents sur ces objets.
Les circuits algébriques sont un modèle de calcul de polynômes multivariés. Les roABP (read-once oblivious algebraic branching programs) forment également un modèle de calcul pour de tels polynômes. On s'intéresse à une classe particulière : la classe des polynômes calculables facilement par des roABP. Nous pourrions nous demander si cette classe est close par factorisation, c'est-à-dire si tout facteur irréductible d'un polynôme calculable par un petit roABP est également calculable par un petit roABP. Nous étudions également les propriétés de clôture de cette classe spécifique pour d'autres opérations, en comparaison avec d'autres classes de circuits, tels que les formules ou les circuits de profondeur constante.
Many natural phenomena, such as volcanic eruptions, involve complex multi-phase flows. These flows often feature a mix of compressible and incompressible behaviors, making their modeling particularly challenging. A widely adopted framework is provided by the Baer–Nunziato equations for compressible two-phase flow. We present a semi-implicit solver for this system, featuring a novel linearly implicit discretization for both the pressure fluxes and the relaxation source terms, while the nonlinear convective terms are treated explicitly. This formulation leads to a CFL-type stability condition on the maximum admissible time step only based on the mean flow velocity, rather than on the sound speed of each phase, so that the novel scheme works uniformly for all Mach numbers. Implicit terms are discretized with central finite differences on Cartesian grids, avoiding artificial numerical diffusion in the low-Mach regime, whereas shock-capturing finite volume schemes are employed for the convective fluxes to guarantee robustness at high Mach numbers. The discretization of nonconservative terms preserves moving equilibrium solutions, making the method well-balanced, while the asymptotic-preserving property ensure consistency in the low-Mach limit of the mixture model. Second-order accuracy in space and time is achieved through the IMEX time-stepping scheme combined with TVD reconstruction. The proposed method is validated through a series of benchmark problems spanning a wide range of Mach numbers, demonstrating both its accuracy and robustness.