Dans cet exposé j'introduirai les surfaces K3 et leurs groupes d'automorphismes, en particulier je montrerai comme la théorie des réseaux joue un rôle clé dans cette étude. Je montrerai des progrès récents sur les automorphismes qui agissent non-symplectiquement et qui sont d'ordre un multiple de sept. Il s'agit ici d'un des cas qui est encore ouvert en vue d'une classification complète des groupes d'automorphismes finis qui agissent sur les surfaces K3. Si le temps le permet je donnerai des exemples qui utilisent les fibrations elliptiques. Ces résultats sont obtenus en collaboration avec R. Bell, P. Comparin, J. Li, A. Rinc'on-Hidalgo, A. Zanardini.
Je présenterai des travaux récents qui mettent en scène des hypersurfaces cubiques projectives complexes de dimension trois et les revêtements cycliques ramifiés au-dessus, pour étudier la riche et belle géométrie de la variété de Fano des droites qu'ils contiennent et le comportement de l'automorphisme du revêtement lors de la dégénérescence vers une cubique à singularités isolées.
La conjecture 3n+1 est sans doute le problème ouvert actuel le plus élémentaire de toutes les mathématiques. En itérant la très simple fonction arithmétique T(n)= n/2 ou (3n+1)/2 selon que n est pair ou impair, on ne maîtrise plus grand chose sur l’évolution des trajectoires. En effet, on conjecture depuis des décennies que quel que soit l’entier de départ n0, on finira toujours par tomber sur 1. Mais il semble qu’on soit encore très loin de pouvoir le prouver. Le but de l’exposé est de présenter deux ou trois résultats partiels, tant théoriques qu’expérimentaux, sur ce problème.
Dans cet exposé, on montrera que toute variété algébrique réelle de dimension n contient des hypersurfaces algébriques réelles de degré d dont les nombres de Betti croissent en O(d^n), lorsque le degré d tend vers l’infini. Ceci est l'ordre de croissance maximal autorisé par l'inégalité de Smith-Thom. L’existence de telles hypersurfaces est obtenue à l’aide de techniques probabilistes.
I will be talking about simplicial sets, which are combinatorial objects widely used in topology. But this time, I'll explain how some of them can characterize graphs, and by extension categories. After defining what simplicial sets are, I will show how to associate to each graph a simplicial set, called its nerve. I will then give a nerve for partial order and categories as well. Then, we will study which simplicial sets arise as nerves from a category, given by the so- called Segal condition. Finally, I will show how to recover graphs, partial orders and categories from their nerve.
Weinstein a montré que toute variété de Poisson holomorphe est localement le produit d'une variété symplectique et d'une variété de Poisson dont le rang est nul au point considéré. En particulier, toute variété de Poisson possède un feuilletage naturel dont les feuilles sont des variétés symplectiques. Dans un travail en collaboration avec Jorge Pereira, Brent Pym et Frédéric Touzet, nous montrons que si une variété de Poisson compacte kählerienne X a une feuille compacte L dont le groupe fondamental est fini alors, à un revêtement étale fini près, X est le produit du revêtement universel de L et d'une autre variété de Poisson.
En sémantique des langages, l'équivalence contextuelle de programmes est une notion clef, mais sa définition se prête mal à la manipulation, on cherche donc des modèles corrects et complets lui donnant une expression plus pratique. Les jeux et la sémantique interactive sont de tels modèles, applicables à beaucoup de langages et donnant une équivalence coïnductivement définie. Dans cette présentation je vais esquisser visuellement et de manière intuitive un formalisme de jeux général qui se prête bien à la formalisation dans un assistant de preuve. Je vais également décrire différents genres d'arbres inductifs et coïnductifs ainsi qu'une manière pratique de construire des jeux à partir de composants plus élémentaires.
11h Jacques Blum (Univ Côte d’Azur) : exposé de Mathematiques pour grand public (Amphi Nivolet) 14h Michel Pierre (ENS Rennes) (salle TLR) 15h Laurent Véron (Univ Tours) (salle TLR)
La journée en l'honneur de Pierre aura lieu le lundi 24 octobre de 9h45 à 16h Programme provisoire 10h - 10h45 : hommages de Ph. Galez, Ph. Briand, P. Orro, G. Angénieux, N. Kardos (amphi Nivolet) 11h - 11h45 : exposé de J. Blum (amphi Nivolet) 12h - 14h : buffet au bâtiment EVE 14h - 14h45 : exposé de Michel Pierre (salle TLR) 15h - 15h45 : exposé de Laurent Véron (salle TLR)
Most of you probably heard about ZFC, this formal theory of sets from the early 20th that is said to be the fundamental language for modern math. Much rarer are the mathematicians that actually make use of it! Since the introduction of ZFC, formal logic has evolved a lot and since several decades other formal systems, more general (some would say more useful!) have been brought up. I will try to introduce the motivations for constructive systems, the links with computation, and demystify the law of excluded middle.
Cet exposé présentera l’aspect feuilleté des géométries (connexions) de Cartan qui sont des structures géométriques infinitésimalement modelées sur des espaces homogènes. Après une introduction du cadre classique, nous allons montrer des résultats de classification pour les feuilletages holomorphes avec des géométries de Cartan transverses sur les variétés de Calabi-Yau et sur les variétés rationnellement connexes. L’exposé s’attachera à introduire le cadre classique et les motivations de manière géométrique et accessible.
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'.