Si X est une variété algébrique sur un corps non archimédien complet, son analytifié à la Berkovich $X^{an}$ contient de nombreuses parties, les squelettes, ayant une structure naturelle d’espace linéaire par morceaux. Si X est intègre, si S est un squelette de $X^{an}$ et si f est une fonction rationnelle non nulle sur X, log |f| est bien définie sur S et sa restriction à S est linéaire par morceaux. Que dire de l’ensemble E des fonctions PL sur S obtenues de cette façon ? Je présenterai dans cet exposé un résultat issu d’un travail en commun avec E. Hrushovski et F. Loeser, qui assure que E est un groupe stable sous min et max, et est de type fini modulo les constantes pour les opérations (+,-, min, max).
Type theory is a family of formal systems ranging from programming language semantics to the foundations of mathematics. In practice, type theories are defined by means of “inference rules”. Everyone in the community understands them to some extent, but they do not have any commonly accepted rigorous interpretation. Or, rather, they have several interpretations, none of which is entirely satisfactory.
In this work, after a brief overview of the literature, we introduce a rigorous, semantic notion of inference rule, our thesis being that most syntactic inference rules written in practice may be directly interpreted in this framework. If time permits, we will sketch how this covers quantitative type theories.
This is joint work in progress with André Hirschowitz and Ambroise Lafont.
I will introduce the field, the kind of objects we manipulate, the sub fields and the kind of problems we want to solve... I will also focus on discrete diferential geometry and our current work on trying to adapt existing tools to geometry defined by surfels (surfaces of voxels).
In the first part of this talk we will give an overview of our past and present research activity, highlighting the different fields of applied mathematics that have been considered so far. In the second part of the talk, I present a novel Finite Volume (FV) scheme on unstructured polygonal meshes that is provably compliant with the Second Law of Thermodynamics and the Geometric Conservation Law (GCL) at the same time. The governing equations are provided by a subset of the class of symmetric and hyperbolic thermodynamically compatible (SHTC) models introduced by Godunov in 1961. Specifically, our numerical method discretizes the equations for the conser- vation of momentum, total energy, distortion tensor and thermal impulse vector, hence accounting in one single unified mathematical formalism for a wide range of physical phenomena in continuum mechanics, spanning from ideal and viscous fluids to hyperelastic solids. By means of two conservative corrections directly embedded in the definition of the numerical fluxes, the new schemes are proven to satisfy two extra conservation laws, namely an entropy balance law and a geometric equation that links the distortion tensor to the density evolution. As such, the classical mass conservation equation can be discarded. Firstly, the GCL is derived at the continuous level, and subsequently it is satisfied by introducing the new concepts of general potential and generalized Gibbs relation. The new potential is nothing but the determinant of the distortion tensor, and the associated Gibbs relation is derived by introducing a set of dual or thermodynamic variables such that the GCL is retrieved by dot multiplying the original system with the new dual variables. Once compatibility of the GCL is ensured, thermodynamic compatibility is tackled in the same manner, thus achieving the satisfaction of a local cell entropy inequality. The two corrections are orthogonal, meaning that they can coexist simultaneously without interfering with each other. The compatibility of the new FV schemes holds true at the semi-discrete level, and time integration of the governing PDE is carried out relying on Runge-Kutta schemes. A large suite of test cases demonstrates the structure preserving properties of the schemes at the discrete level as well.
À venir
En 1927 Artin a résolu le 17ème problème de Hilbert en montrant qu'un polynôme positif sur $\mathbb{R}^n$ est somme de carrés de fonctions rationnelles. Ce résultat marque le début du développement de l'algèbre réelle. Dans cet exposé on s'intéresse à la réciproque du 17ème problème de Hilbert dans un cadre général. Soit $A$ un anneau intègre de corps des fractions $K$, on va décrire les lieux où la positivité des éléments de $A$ est équivalente à être une somme de carrés dans $K$. Lorsque $A$ est l'anneau de coordonnées d'une variété algébrique réelle irréductible affine $V$, ces lieux sont fortement liés aux singularités de $V$. Il s'agit d'un travail en commun avec Goulwen Fichou et Ronan Quarez.
In this talk, I will introduce you to the main electrical concepts to model the electrical response of a solar module. We will answer some questions like 'Why do we need an electrical model?', 'What is the electrical data we can extract from the real world?', and 'How do we treat such data?'. For this purpose, the main concepts will be conceptualized with a simple example.
Wave breaking is a challenging subject that is not encompassed in the usual mathematical description of water waves. This is the consequence of the impossibility to represent the water-air interface as the graph of a function. In the first part of this presentation, we shall exhibit the strong non-linear nature of the breaking phenomena through the mathematical study of two water waves models: (1) KdV, whose solutions do not break and (2) Camassa-Holm, whose non-global solutions do break at some point. Next we shall discuss the way to incorporate multi-valued interfaces in the usual water waves problem before discussing whether or not this model efficiently describes the breaking phenomenon. In a third, ultimate, part we will present new numerical results that have been obtained using a finite-element code to solve the free-surface Navier-Stokes equations for an initial condition leading to a plunging jet. We compare these results with those obtained solving the Euler equation in the exact same configuration and conclude about the convergence of the two methods whenever the Reynolds number increases. We also discuss the influence of viscous dissipation on the overall shape of the wave.
I describe a P-time heuristic to reconstruct a subclass of degree sequences of 3-uniform hypergraphs. The heuristic bases on some geometrical properties of the involved hypergraphs and also produces a small set of ambiguous hyperedges that has to be individually considered.
We consider the problem of detecting and reconstructing degree sequences of 3-uniform hypergraphs. The problem is known to be NP-hard, so some subclasses are inspected in order to detect instances that admit a P-time reconstruction algorithm. Here, I consider instances having specific numerical patterns and I show how to easily reconstruct them.
Logarithmic-analytic functions are iterated compositions (from either side) of globally subanalytic functions (i.e. functions definable in the o-minimal structure $\R_{an}$ of restricted analytic functions) and the global logarithm. Their definition is kind of hybrid. From the viewpoint of logic, log-analytic functions are definable in the o-minimal expansion $\R_{an,exp}$ of $\R_{an}$ by the global exponential function; in fact they generate the whole structure $\R_{an,exp}$. But from the point of analysis their definition avoids the exponential function and should therefore also not exhibit properties of the function $\exp(−1/x)$ as flatness or infinite differentiability but not real analyticity. This seems to be obvious. But the problem is that a composition of globally subanaytic functions and the logarithm allows a representation by ’nice’ terms only piecewise. Moreover, the ’pieces’ are in general not definable in $\R_{an}$ but only in $\R_{an,exp}$. And the existing preparation results for log-analytic functions involve functions which are not log-analytic. But by elaborating on the preparation theorems one can identify situations where the preparation can be carried out inside the log-analytic category. And these situations are sufficient to obtain the following results: We show that the derivative of a log-analytic function is log-analytic. We prove that log-analytic functions exhibit strong quasianalytic properties. We establish the parametric version of Tamm’s theorem for log-analytic functions. It seems also to be obvious that log-analytic functions are polynomially bounded. This is indeed true in the univariate case. But, surprisingly, multivariate log-analytic functions can exhibit exponential growth. We give examples and present structural results on the growth.
We initiate the study of the algorithmic complexity of Maker-Breaker games played on edge sets of graphs for general graphs. We mainly consider three of the big four such games: the connectivity game, perfect matching game, and H-game. Maker wins if she claims the edges of a spanning tree in the first, a perfect matching in the second, and a copy of a fixed graph H in the third. We prove that deciding who wins the perfect matching game and the H-game is PSPACE-complete, even for the latter in graphs of small diameter if H is a tree. Seeking to find the smallest graph H such that the H-game is PSPACE-complete, we also prove that there exists such an H of order 51.
On the positive side, we show that the connectivity game and arboricity-k game are polynomial-time solvable. We then give several positive results for the H-game, first giving a structural characterization for Breaker to win the P4-game, which gives a linear-time algorithm for the P4-game. We provide a structural characterization for Maker to win the K_{1,l}-game in trees, which implies a linear-time algorithm for the K_{1,l}-game in trees. Lastly, we prove that the K_{1,l}-game in any graph, and the H-game in trees are both FPT parameterized by the length of the game. We leave the complexity of the last of the big four games, the Hamiltonicity game, as an open question.
Motivic integration is a powerful tool in algebraic geometry for studying singularities. The theory was conceived by Kontsevich in 1989 to provide a shorter proof of Batyrev's theorem. In 2009, a more "modern" form of this theory has emerged, spearheaded by Cluckers and Loeser. First, I'll talk about p-adic integration and motivations. Then I'll try to introduce the theory's basic objects, such as model theory and Grothendieck groups. Finally, if time permits, I'll set out some axioms of the motivic integral.
L'imagerie haute résolution de la Terre, et en particulier de la croûte, est fondamentale pour la transition énergétique: pour la massiffication du stokage de CO2, une technologie mise en avant par le GIEC pour lutter contre le réchauffement climatique, mais aussi pour l'exploitation des ressources nécessaires pour la construction des infrastructures énergétiques éoliennes et salaires, et les batteries électriques. L'état de l'art pour l'imagerie haute résolution de la croûte repose sur une méthode appelée "inversion des formes d'ondes complètes". D'un point de vue mathématique, ceci revient à un problème d'estimation de paramètres d'une équation aux dérivées partielles (EDP) modélisant la propagation d'ondes dans le sous-sol à partir de données collectées ponctuellement en surface. Dans cette présentation, on introduit les bases géophysiques et mathématiques autour de cette méthode, avant de passer en revue des travaux de recherche menés au sein du projet SEISCOPE. Ces travaux recoupent les thèmes suivants: méthode d'optimisation de second-ordre basées sur des méthodes adjointes d'ordre deux, utilisation de distances transport optimal pour lutter contre le caractère mal posé du problème inverse, modélisation numérique 3D de la propagation d'onde dans l'approximation élastique, théorie des milieux équivalents (homogénisation) pour la propagation d'ondes en milieux élastiques, estimation des incertitudes pour les problèmes inverse de grande taille en se basant sur une méthode de filtre de Kalman d'ensemble.
Après un rappel sur les équations d'Euler 2D, je parlerai des tourbillons concentrés. J’exposerai les arguments principaux pour montrer la persistance de la concentration vers des points vérifiant le système des points vortex. Dans la seconde partie, je présenterai les équations des lacs qui peuvent se voir comme une généralisation d’Euler 3D axisymétrique sans swirl. Je montrerai que les points vortex se déplacent selon une loi de type « courbure binormale ». Ce travail est en collaboration avec Lars Eric Hientzsch et Evelyne Miot.
Je vais montrer les caractères topologiques d'une variété complexe projective qui déterminent le degré de la variété duale.
Ce sont des caractéristiques d’Euler-Poincaré associées à la stratification de Whitney minimale de la variété.
Tous les termes utilisés seront expliqués.
ML modules offer large-scale notions of composition and modularity. Provided as an additional layer on top of the core language, they have proven both vital to the working OCaml and SML programmers, and inspiring to other use-cases and languages. Unfortunately, their meta-theory remains difficult to comprehend, and more recent extensions (abstract signatures, module aliases) lack a complete formalization. Building on a previous translation from ML modules to Fω, we propose a new comprehensive description of a significant subset of OCaml modules, including both applicative and generative functors and transparent ascription -- a useful new feature. By exploring the translations both to and from Fω, we provide a complete description of the signature avoidance issue, as well as insights on the limitations and benefits of the path-based approach of OCaml type-sharing.
Geophysical phenomenon such as magnetic field reversal are a challenge to observe numerically. They are quite demanding in terms of numerical resources, and with the upcoming generation of exascale computers, it becomes necessary to ensure an efficient and full usage of such clusters. In geophysical and astrophysical flows, the classical method for parallelism is data parallelism. The physical domain is distributed across a large number of cores. The scaling of such a distribution can quickly saturate when the number of cores increases. Introducing an additional pipeline parallelism, through a distribution of a time interval across a number of cores, is a potential solution to use a larger number of cores, and perform numerical simulations of magnetic field reversals. When used by parallel in time schemes, time-steppers need to validate a few of criteria. We extracted 18 time-steppers from literature, from second to eighth order of accuracy. We will compare the accuracy and efficiency of those time steppers in the context of liquid planetary cores, in order to identify potential candidates to build parallel in time schemes.
We start from categorical semantics of dependent types and propose structural rules involving a new notion of subtyping. We begin by recalling a few classical models, which traditionally have been heavily set-based: this is the case for categories with families, categories with attributes, and natural models. In particular, all of them can be traced back to certain discrete fibrations. We extend this intuition to the case of general, non necessarily discrete, fibrations, and use the newly found structure on the fibers to interpret a form of subtyping. Interestingly, the emerging notion turns out to be closely related to that of coercive subtyping. This is joint work with J. Emmenegger.