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TBA
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TBA
In this talk we shall discuss the notion of hyperlattices, that is preorders somewhat similar to lattices, but where instead of binary supremums and infimums we consider sets of minimal elements of upper bounds and maximal elements of lower bounds. We will utilize this notion to study superpowersets, that is families of functions defined on nonempty sets with values in hyperlattices, and superpowergroups, that is algebras defined on superpowersets with group-like properties. Numerous examples of such objects appear in topology, logic and algebra, and, in particular, in the algebraic theory of quadratic forms. Finally, we shall distinguish one class of superpowergroups that we call relational superpowergroups, and will see how the category of relational superpowergroups is almost a topos — that is, it satifsies all the axioms of a topos except the subobject classifier axiom. This is joint work with Hugo Mariano and Kaique Matias de Andrade Roberto.
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.