In this seminar, I will focus on the characterisation of bijective digitized rotations and reflections. Although the characterisation of bijective digitized rotations in 2D is well known, the extension to 3D is still an open problem. A certification algorithm exists that allows to verify that a digitized 3D rotation defined by a quaternion is bijective. In this seminar, we show how we use geometric algebra to represent a bijective digitized rotation as a pair of bijective digitized reflections. Visualization of bijective digitized reflections in 3D using geometric algebra leads to a conjectured characterization of 3D bijective digitized reflections and, thus, rotations. So far, any known quaternion that defines a bijective digitized rotation verifies the conjecture. An approximation method of any 3D digitized reflection by a conjectured bijective one is also proposed. Some experimental results will be shown with DGtal.
We introduce open parity games, which is a compositional approach to parity games. This is achieved by adding open ends to the usual notion of parity games. We introduce the category of open parity games, which is defined using standard definitions for graph games. We also define a graphical language for open parity games as a prop, which have recently been used in many applications as graphical languages. We introduce a suitable semantic category inspired by the work by Grellois and Melliès on the semantics of higher-order model checking. Computing the set of winning positions in open parity games yields a functor to the semantic category. Finally, by interpreting the graphical language in the semantic category, we show that this computation can be carried out compositionally. We also discuss current work on an efficient implementation of a compositional solver of graph games.
Monadic interpreters have been used for a long time as a mean to embed arbitrary computations in purely functional contexts. At its core, the idea is elementary: the object language of interest is implemented as an executable interpreter in the host language, and monads are simply the abstraction used to embed features such as side effects, failure, non-determinism. By building these interpreters on top of the free monad, the approach has offered a comfortable design point notably enabling an extensible syntax, reusable modular components, structural compositional definitions, as well as algebraic reasoning. The approach has percolated beyond its programming roots: it is also used as a way to formalize the semantics of computational systems, programming languages notably, in proof assistants based on dependently typed theories. In such assistants, the host language is even more restricted: programs are all pure, but also provably terminating. Divergent programs can nonetheless be embedded using for instance the Capretta monad: intuitively, a lazy, infinite (coinductive) tree models the dynamic of the computation. Interaction trees are a specific implementation, in the Coq proof assistant, of a set of tools realizing this tradition. They provide a coinductive implementation of the iterative free monad, equipped with a set of combinators, allowing notably for general recursion. Each iterator comes with its equational theory established with respect to a notion of weak bisimulation --- i.e. termination sensitive, but ignoring the amount of fuel consumed --- and practical support for equational reasoning. Further effects are implemented into richer monads via a general notion of interpretation, allowing one to introduce the missing algebras required for proper semantic reasoning. Beyond program equivalence, support for arbitrary heterogeneous relational reasoning is provided, typically allowing one to prove a compilation pass correct. Introduced in 2020, the project has spawned realistic applications --- they are used to model LLVM IR's semantics notably ---, as well as extensions to reduce the necessary boilerplate, or to offer proper support for non-determinism. In this talk, I will attempt to paint an illustrative overview of the core ideas and contributions constitutive of this line of work.
Environmental changes threaten many species and ecosystems. To assess their impacts, we use a mathematical approach based on reaction dispersion models. I will investigate evolutionary adaptation of population structured by a phenotypic trait under a changing environment. I will derive PDE model from stochastic model and using Hamilton-Jacobi approach and large deviation technics, I will present some approximations of these models. Then I will present a new approach to track ancestral lineages in quantitative genetic model.
This talk proposes full convexity as an alternative definition of digital convexity, which is valid in arbitrary dimension. It solves many problems related to its usual definitions, like possible non connectedness or non simple connectedness, while encompassing its desirable features. Fully convex sets are digitally convex, but are also connected and simply connected. They have a morphological characterisation, which induces a simple convexity test algorithm. Arithmetic planes are fully convex too. Full convexity implies local full convexity, hence it enables local shape analysis, with an unambiguous definition of convex, concave and planar points. We propose also a natural definition of tangent subsets to a digital set. It gives rise to the tangential cover in 2D, and to consistent extensions in arbitrary dimension. We present two applications of tangency: the first one is a simple algorithm for building a polygonal mesh from a set of digital points, with reversibility property, the second one is the definition and computation of shortest paths within digital sets. In a second part of the talk, we study the problem of building a fully convex hull. We propose an iterative operator for this purpose, which computes a fully convex enveloppe in finite time. We can even build a fully convex enveloppe within another fully convex set (a kind of relative convex hull). We show how it induces several natural digital polyhedral models, whose cells of different dimensions are all fully convex sets. As perspective to this work, we study the problem of fully convex set intersection, which is the last step toward a full digital analogue of continuous convexity.
Nous ferons 'quelques observations sur le schéma des arcs'.
L'allostérie est un phénomène d'importance fondamentale en biologie qui permet la régulation de la fonction et l'adaptabilité dynamique des enzymes et protéines. Malgré sa découverte il y a plus d'un siècle, l'allostérie reste une énigme biophysique, parfois appelée « second secret de la vie ». La difficulté est principalement associée à la nature complexe des mécanismes allostériques qui se manifestent comme l'altération de la fonction biologique d'une protéine/enzyme (c-à-d. la liaison d'un substrat/ligand au site active) par la liaison d'un « autre objet » (``allos stereos'' en grec) à un site distant (plus d'un nanomètre) du site actif, le site effecteur. Ainsi, au cœur de l'allostérie, il y a une propagation d'un signal du site effecteur au site actif à travers une matrice protéique dense, où l'un des enjeux principal est représenté par l'élucidation des interactions physico-chimiques entre résidus d'acides aminés qui permettent la communication entre les deux sites : les chemins allostériques. Ici, nous proposons une approche multidisciplinaire basée sur la combinaison de méthodes de chimie théorique, impliquant des simulations de dynamique moléculaire de mouvements de protéines, des analyses (bio)physiques des systèmes allostériques, incluant des alignements multiples de séquences de systèmes allostériques connus, et des outils mathématiques basés sur la théorie des graphes et d'apprentissage automatique qui peuvent grandement aider à la compréhension de de la complexité des interactions dynamiques impliquées dans les différents systèmes allostériques. Le projet vise à développer des outils rapides et robustes pour identifier des chemins allostériques inconnus. La caractérisation et les prédictions de points allostériques peuvent élucider et exploiter pleinement la modulation allostérique dans les enzymes et dans les complexes ADN-protéine, avec de potentielles grandes applications dans l'ingénierie des enzymes et dans la découverte de médicaments.
J'introduirai une notion de décomposition en anses pincées pour les complexes simpliciaux finis et en montrerai l'existence après subdivisions stellaires en des facettes. Ces décompositions étendent les effeuillages classiques. Toute fonction de Morse discrète induit une telle décomposition sur la deuxième subdivision barycentrique.
Dans cet exposé, on considérera les ensembles algébriques réels munis d'une structure polynomiale de groupe et on mettra en avant quelques différences avec les groupes algébriques complexes. Nous nous intéresserons ensuite au cas des groupes algébriques réels compacts, aux propriétés plus proches de celles des groupes algébriques complexes. Enfin, on étudiera quelques propriétés géométriques des actions polynomiales des groupes algébriques réels compacts sur les ensembles algébriques réels, notamment en termes d'orbites et de quotients.
In a previous paper we have introduced the notion of geometric directional bundle of a singular space, in order to introduce global bi-Lipschitz invariants. Then we have posed the question of whether or not the geometric directional bundle is stabilised as an operation acting on singular spaces. In this talk we give a positive answer in the case where the singular spaces are subanalytic sets, thus providing a new invariant associated with the subanalytic sets.
Reconstructing digital humans is a key problem in 3D vision with many applications for autonomous driving, robotics, Virtual and Augmented Reality and has attracted a lot of research for decades. In this talk I will discuss about non-invasive hardware-based solutions to jointly capture human body shape and motion. We will see that efficient modelisation of human body deformation is key to enable real-time tracking. I will also present recent works about AI-based solutions for both human shape reconstruction from a single color images and full body animation with minimum driving signal such as a skeleton. We will see that deep learning opens new perspectives and possibilities to create real digital humans and animate them in the digital spaces.
This work deals with the interaction of waves with a floating structure immersed in a 2D fluid in coastal area. The horizontal plane is decomposed into two regions: the exterior region where the surface of the fluid is in contact with the air, and the interior region where it is in contact with the bottom of the object. In the exterior region, we have the standard equations, where the surface is free but the pressure is constrained (equal to the atmospheric pressure). In the interior region, this is the reverse: the pressure is free but the surface is constrained, which changes the structure of the equations. Finally, coupling conditions between both regions are needed. We show how to implement this program in the case where the waves are described by the nonlinear dispersive Boussinesq equations. We also show a numerical method that exploits the added-mass effect and the dispersive boundary layer.
Tout polynôme multivarié P(X_1,...,X_n) peut être écrit comme une somme de monômes, i.e., une somme de produits de variables et de constantes du corps. La taille naturelle d'une telle expression est le nombre de monômes. Mais, que se passe-t-il si on rajoute un nouveau niveau de complexité en considérant les expressions de la forme : somme de produits de sommes (de variables et de constantes) ? Maintenant, il devient moins clair comment montrer qu'un polynôme donné n'a pas de petite expression. Dans cet exposé nous verrons que ce problème est lié à des conjectures célèbres de complexité algorithmique (comme P <> NP) et nous montrerons ensuite comment le résoudre. Plus précisément, nous pouvons montrer que certains polynômes explicites n'ont pas de représentations ``somme de produits de sommes'' (SPS) de taille polynomiale. Nous pouvons aussi obtenir des résultats similaires pour les SPSP, SPSPS, etc... pour toutes les expressions de profondeur constante.
I will present a generalization of Gabrielov's rank theorem for families of rings of power series which we call W-temperate. Examples include the family of complex analytic functions and of the Eisenstein series. I will provide the definition of Eisenstein series, and will discuss how the result allows us to give new proofs of the following two results of W. Pawlucki: I) The non regular locus of a complex or real analytic map is an analytic set. II) The set of semianalytic or Nash points of a subanalytic set X is a subanalytic set, whose complement has codimension two in X. This is a work in collaboration with Octave Curmi and Guillaume Rond.
Dans cet exposé, nous aborderons la problématique de l’observabilité de systèmes d'EDPs linéaires, notamment des systèmes hyperboliques du premier ordre. Nous examinerons en particulier le coût de la détection de paquets d’ondes hautes fréquences. Il s’agit d’un travail en collaboration avec Roberta Bianchini et Vincent Laheurte.
On considère un écoulement de fluide incompressible visqueux dans une boite B (dans R^n) autour d'un obstacle K (dans B), avec au bord de K une condition de Navier, et on s’intéresse à la minimisation de la traînée parmi tous les obstacles K de mesure fixée. Je présenterai des résultats théoriques d'existence en toute dimension et de régularité en dimension 2 de tels minimiseurs.
I will talk about a joint paper with Jacek Bochnak containing an appendix written by János Kollár. Let X be a real algebraic variety and let Y be a homogeneous space for some linear real algebraic group. We prove that a continuous map f: X -> Y can be approximated by regular maps in the compact-open topology if and only if it is homotopic to a regular map. Taking Y=S^p , the unit p-dimensional sphere, we obtain solutions of several problems that have been open since the 1980's and which concern approximation of maps with values in the unit spheres. This has several consequences for approximation of maps between unit spheres. For example, we prove that for every positive integer n every continuous map from S^n into S^n can be approximated by regular maps. Up to now such a result has only been known for five special values of n, namely, n=1, 2, 3, 4 or 7.
We consider the minimization/maximization problem of the product Tp(Ω)λp(Ω), where Ω varies among different classes of admissible sets. Here Tp and λp denote respectively the p-torsional rigidity and the p-principal frequency associated to the p-Laplace operator. We present some new results in the case p≠2 and we list some interesting open problems.