Among 3-dimensional sets of given Newtonian capacity, which shape minimizes the q-th moment (q>0) of electrostatic equilibrium measure? One readily shows it is the ball. But what if the set is confined to the plane? A centered disk is then the natural minimizer, yet the proof is quite different and involves a cylindrical variant of Baernstein’s star-function. The approach succeeds when 0 <q <= 2. Higher moments (q>2) remain a tantalizing open problem, as do the analogous questions for Riesz equilibrium measures.
Note: this talk does not assume any previous knowledge about capacities.
(Joint work with Carrie Clark, Univ. of Illinois Urbana–Champaign.)
In 2020, Parusinski and Rond proved that every algebraic set $V \subset \mathbb{R}^n$ is homeomorphic to a $\bar{\mathbb{Q}}^r$-algebraic set $V' \subset \mathbb{R}^n$, where $\bar{\mathbb{Q}}^r$ denotes the field of real algebraic numbers. Latter very general result motivates the following open problem: $\mathbb{Q}$-algebraicity problem: (Parusinski, 2021) Is every algebraic set $V \subset \mathbb{R}^n$ homeomorphic to some $\mathbb{Q}$-algebraic set $V' \subset \mathbb{R}^m$, with $m \ge n$? The aim of the talk is to introduce above open problem and to explain how our new approximation techniques over $\mathbb{Q}$ allowed us to provide some classes of real algebraic sets that positively answer the $\mathbb{Q}$-algebraicity problem.
Comment un résultat de catégories a inspiré une solution aux problèmes de performance de Darcs. Darcs est un système de contrôle de versions sorti en 2005, basé sur des patchs, ce qui le rendait extrêmement simple à utiliser et particulièrement rigoureux, en particulier dans sa gestion des conflits. Seul problème, il prenait parfois un temps exponentiel en la taille de l'histoire pour son opération la plus courante (appliquer des patchs). Je vous expliquerai comment nous avons résolu le problème, en utilisant des catégories et des structures de données purement fonctionnelles, pour concevoir un algo en temps logarithmique pour tous les cas (sauf un cas dégénéré en temps linéaire).
Le résultat est un système déterministe (ce qui le distingue de tous les autres outils de contrôle de versions), facile à apprendre et à utiliser, tout en passant à des échelles qu'aucun autre système de contrôle de versions distribué ne peut atteindre.
The topic of the talk is existence of weak solutions to the pressureless Navier-Stokes system with nonlocal attraction--repulsion forces. We construct the solutions on the whole three-dimensional space, assuming that the viscosity coefficients are density-dependent. For the nonlocal term it is further assumed that the interaction kernel has the quadratic growth at infinity and almost quadratic singularity at zero. The main point of the construction is the derivation of the analogs of the Bresch--Desjardins and Mellet--Vasseur estimates in the nonlocal setting.
Toute courbe complexe plane est munie d’une métrique riemannienne induite par la métrique ambiante de Fubini- Study du plan projectif complexe. Nous donnons des bornes inférieures probabilistes sur certaines quantités métriques et spectrales (telles que la systole ou le trou spectral) des courbes planes lorsque celles-ci sont choisies aléatoirement. Il s’agit d’un travail commun avec Damien Gayet.
I will be the speaker this time and I'll talk about Shape Optimization. I will explain what are the problems that we try to solve and give a few examples then I will present some of the main tools that we use (\gamma-convergence, Buttazzo- Dal Maso existence theorem and Lions Concentration Compactness principle) and finaly if we still have some time, I will try to introduce the work that I'm doing at the moment. All of this with maybe some proofs to give you an idea of how the field works.
Nous montrons comment la théorie de la classification locale des systèmes dynamiques analytiques discrets en une variable peut s'étendre au cadre formel des transséries et de certains germes transsériels. Ces résultats s'étendent également à certains corps de "transséries généralisées" contenus dans le corps des nombres surréels, en s'appuyant sur des considérations inspirées des travaux de Rosenlicht sur les corps de Hardy. Travail joint avec V. Mantova, D. Peran et T. Servi.
In a first part I'll will broadly cover the topic of what the Laplace- Beltrami is, its different characterization and its many uses in computer graphics.
Then I'll cover our works on the topic of building this operator (along with a wider digital calculus framework) on digital surfaces (boundary of voxels). These surfaces cannot benefit directly from the classical mesh calculus frameworks. In our recent work, we propose two new calculus frameworks dedicated to digital surfaces, which exploit a corrected normal field. First we build a corrected interpolated calculus by defining inner products with position and normal interpolation in the Grassmannian. Second we present a corrected finite element method which adapts the standard Finite Element Method with a corrected metric per element. Experiments show that these digital calculus frameworks seem to converge toward the continuous calculus, offer a valid alternative to classical mesh calculus, and induce effective tools for digital surface processing tasks.
Normal form bisimilarities are a natural form of program equivalence resting on open terms, first introduced by Sangiorgi in call-by-name. The literature contains a normal form bisimilarity for Plotkin’s call-by-value 𝜆-calculus, Lassen’s enf bisimilarity, which validates all of Moggi’s monadic laws and can be extended to validate 𝜂. It does not validate, however, other relevant principles, such as the identification of meaningless terms—validated instead by Sangiorgi’s bisimilarity—or the commutation of lets. These shortcomings are due to issues with open terms of Plotkin’s calculus. We introduce a new call-by-value normal form bisimilarity, deemed net bisimilarity, closer in spirit to Sangiorgi’s and satisfying the additional principles. We develop it on top of an existing formalism designed for dealing with open terms in call-by-value. It turns out that enf and net bisimilarities are incomparable, as net bisimilarity does not validate Moggi’s laws nor 𝜂. Moreover, there is no easy way to merge them. To better understand the situation, we provide an analysis of the rich range of possible call-by-value normal form bisimilarities, relating them to Ehrhard’s relational model.
The motion of a compressible viscous barotropic fluid is described by the Navier-Stokes system. It is a system of hyperbolic-parabolic mixed-type PDEs. In this talk, we will study the so-called density patch problem: If we are given a density that is initially discontinuous across a C^(1+\alpha) curve alpha and alpha- Hölder continuous on the two disjoint components delimited by gamma, is this structure preserved in time?
An important quantity in the mathematical analysis of this system is the so-called effective flux, which was discovered by Hoff and Smoller in 1985. More precisely, the mathematical properties of this quantity play a crucial role in the study of the propagation of oscillations in compressible fluids (Serre, 1991), in the construction of weak solutions (P-L Lions 1996) or the propagation of discontinuity surfaces (Hoff 2002), to cite just a few examples. In the case of density-dependent viscosities, the behavior of the effective flux degenerates, which renders the analysis more subtle.
We give a characterization of star-free languages (a well-known subclass of regular languages) in a λ-calculus with support for non-commutative affine types (in the sense of linear logic), via the algebraic characterization of the former using aperiodic monoids. This was the first result in a research program that Cécilia Pradic (Swansea University) and I have carried out since my PhD, on which I will say a few words.
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.