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Résumé : La notion de polytope est très ancienne, et même carrément antique pour la dimension 3. On discutera un peu d'histoire, puis on parlera des polytopes réflexifs et d'une construction récente.

Frédéric Chapoton est directeur de recherche en mathématiques au CNRS, dans l'équipe ART du laboratoire IRMA à Strasbourg. Il travaille en combinatoire algébrique et en théorie des représentations, notamment sur les interactions entre ordres partiels finis, représentations de carquois, opérades et polytopes.

Résumé : For an ergodic dynamical system, the cutoff describes an abrupt transition to equilibrium. Historically introduced in seminal work by Diaconis, Shahshahani and Aldous for card shuffling and other random walks on finite groups, there are now numerous examples of Markov chains and Markov processes where the cutoff has been established. Most of the current examples are on finite spaces. In this talk, we study cutoff for classical processes -- namely Brownian motion and geodesic paths -- on compact hyperbolic manifolds, and we develop a spectral strategy introduced by Lubetzky and Peres in 2016 for Ramanujan graphs and further developed in different geometric contexts. In particular, we extend results obtained by Golubev and Kamber in 2019 to any dimension and still are able to obtain cutoff under weaker hypothesis. Based on a joint work with C. Bordenave

Résumé : Since Mumford’s work in the 1960s, questions on the finite generation of Chow and Griffiths groups—such as the finiteness of dimension, the size of their torsion, or their divisibility—have been a central theme in the study of algebraic cycles. Motivic cohomology and morphic cohomology naturally generalize the Chow group (cycles modulo rational equivalence) and cycles modulo algebraic equivalence. In this talk, I will show how, over an algebraically closed field whose transcendence degree over its prime field is infinite (e.g., the complex number field), one can combine Schoen’s injectivity argument with Schreieder’s refined unramified cohomology to construct examples of motivic and morphic cohomology groups with infinitely many torsion elements. These appear to be the first known examples exhibiting infinite torsion in motivic or morphic cohomology. Joint work with Theodosis Alexandrou.

Résumé : Dans cet exposé on se propose d’étudier une algèbre naturellement associée à un graphe. Il s’agit d’une algèbre de Hall provenant d’un ensemble simplicial mais dont nous ne donnerons pas plus de détails que le nom. Étudier cette algèbre signifie, dans notre cas, essayer de donner une présentation par générateurs et relations. L’algèbre étant commutative on pourrait motiver l’intérêt d’une telle présentation pour obtenir cette algèbre comme l’anneau de cohomologie d’un espace topologique. Pour parvenir à notre fin, on se propose d’enrichir la structure d’algèbre avec une notion duale, celle de cogèbre. On s’intéressera alors aux éléments primitifs pour le coproduit qui sont souvent de bons candidats pour les générateurs de notre présentation. Presque arrivés au bout de cette histoire, c’est la compatibilité entre les deux structures duales s’associant dans la notion de bigèbre tordue qui nous donnera la présentation de l’algèbre. Cet exposé espère mettre en lumière à quel point l’apport de structure permet une meilleure compréhension des objets étudiés.

Résumé : We propose novel variable selection methods for sparse GLARMA (Generalised Linear Autoregressive Moving Average) models, which can be used for modelling discrete-valued time series. These models allow us to introduce some dependence in a Generalised Linear Model (GLM). The key idea behind our estimation procedure is first to estimate the coefficients of the ARMA part of the GLARMA model and then use a regularised approach, namely the Lasso, to estimate the regression coefficients of the GLM part of the model. Furthermore, we establish a sign-consistency result for the estimator of the regression coefficients in a sparse Poisson model without time dependence. The performance of our proposed methods was assessed on simulation studies in different frameworks and on several datasets in the field of molecular biology. Our approaches exhibit very good statistical performance, surpassing other methods in identifying non-null regression coefficients. Secondly, their low computational burden enables their application to relatively large datasets. Our proposed methods are implemented in R packages, which are publicly available on the Comprehensive R Archive Network (CRAN).

Résumé : D'habitude on utilise des groupes quantiques pour construire des invariants des nœuds. On va aller dans le sens inverse en définissant le groupe quantique et l'algèbre de Iwahori-Hecke (de type A_n) à partir de l'invariant HOMFLY. Cette technique permets de travailler avec les représentations de ces algèbres en utilisant les digrammes des enchevêtrements.

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Résumé : The space of (local) symmetries of a given geometric structure has the natural structure of a Lie (pseudo)group. Conversely, geometric structures admitting a local model can be described via the pseudogroup of symmetries of such local model. The main goal of this talk is to provide several examples and give an intuitive understanding of the slogan above, which can be made precise at various levels of generality (depending on the definition of "geometric structure") and using different tools/methods. Moreover, I will sketch a new framework, which include previous formalisms (e.g. G-structures or Cartan geometries) and allows us to prove integrability theorems. In particular, I will provide intuition on the relevant objects which make this approach work, namely Lie groupoids endowed with a multiplicative "PDE-structure" and their principal actions. Poisson geometry will give us the guiding principles to understand those objects, which are directly inspired from, respectively, symplectic groupoids and principal Hamiltonian bundles. This is based on a forthcoming book written jointly with Luca Accornero, Marius Crainic and María Amelia Salazar.

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Résumé : Résumé : J'expliquerai comment on peut approcher le type d'homotopie des espaces de plongements entre variétés différentiables en termes de modules sur l'opérade des petits disques. Cette méthode due à Goodwillie et Weiss devient particulièrement puissante quand on la combine avec des outils de théorie de l'homotopie rationnelle. En particulier, je formulerai un résultat récent obtenu avec Pedro Boavida de Brito donnant une description explicite du type d’homotopie rationnel d'un espace de plongements sous des hypothèses assez générales.

Résumé : In the derivation of the kinetic equation from the cubic NLS, a key feature is the invariance of the Schrödinger equation under the action of U(1), which allows the quasi-resonances of the equation to drive the effective dynamics of the correlations. In this talk, I will give an example of equation that does not enjoy such type of invariance and show that the exact resonances always take precedence over quasi-resonances. As a result, the effective dynamics is not of kinetic type but still nonlinear. I will present the problem, the ideas behind the derivation of the effective dynamics and some elements of the proof. This is based on a recent work in collaboration with de Suzzoni (Université Evry Paris-Saclay) and Touati (CNRS and Université de Bordeaux).

Résumé : Quantum analogues of real numbers are a generalization of q-deformed integers (also called Gaussian q-integers), which consist of replacing integers by polynomials in an indeterminate "q", in such a way that the specialization q=1 gives the initial number back. This idea can be found in generating series, and was already used by Euler to solve combinatorial problems. A good deformation must be compatible with the structural properties of the object that is being quantized. For instance, q-deformed binomial coefficients have a quantized Pascal rule. In 2020, Sophie Morier-Genoud and Valentin Ovsienko proposed a quantization of rational numbers, generalizing the q-deformed integers, with good combinatorial properties. In this talk, I will define these q-rational numbers by three different ways, first via continued fractions, then via the Farey graph, and finally by an action of the modular group. We will see how this last definition discloses a twin version of q-rational numbers, known as the left one. I will then extend the quantized action of the modular group to unify the left and the right versions. In the remaining time, I will introduce a combinatorial interpretation of q-rationals based on fence posets, bridging the gap to the cluster algebra theory.

Résumé : Résumé : Depuis leur introduction vers 2000, des liens entres les algèbres amassées et d’autres domaines on été établis, parmi eux la théorie de représentations de carquois et la géométrie des surfaces. Dans cet exposé, je vais montrer comment on peut définir des structures amassées en utilisant la combinatoire des surfaces. Les courbes correspondent aux variables amassées/aux objets indécomposable (rigides). Les mutations de variables amassées sont associées avec la relation de Ptolémée.

Résumé : Let $X = G/K$ be a Hermitian symmetric space of noncompact type (in rank one, $X$ is the unit ball in $\mathbb{C}^n$ and $G$ is the group $\mathrm{PU}(n,1)$), and let $\Gamma$ be a discrete and torsion-free subgroup of $G$. Can we find criteria on $\Gamma$ implying that the quotient of $X$ by $\Gamma$ is holomorphically convex, or that it contains no compact analytic subvariety of positive dimension? I will present criteria inspired by the work of Dey and Kapovich, which concern the critical exponent of the group (in rank one) or its entropy associated with some linear form (in higher rank). In both cases, the proofs involve Patterson–Sullivan measures, and the ultimate goal is to show that these quotients are Stein manifolds. The results in higher rank come from work in progress, in collaboration with Colin Davalo.

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