Liste des exposés
Mini-cours:
- Charlotte Hardouin (Toulouse) : Combinatoire des marches et théorie de Galois aux différences
Résumé : La classification algébrique des séries génératrices
comptant les marches dans le quart-plan à petit pas initiée par les travaux de Bousquet-Mélou et Mishna en 2009 est désormais quasiment complète. Ce résultat est le fruit de diverses approches mêlant combinatoire, calcul formel, problèmes aux limites, géométrie et théorie de Galois aux différences. Dans cette série d'exposés, j'essaierais de présenter certaines de ces approches et de montrer comment ces dernières s'articulent pour donner lieu à des stratégies systématiques et des méthodes effectives. Si le temps le permet, je discuterai de la généralisation de certaines de ces méthodes au cas des marches à grands pas.
- Ronnie Nagloo (Chicago) : Théorie des modèles et équations différentielles
Exposés:
- Jehanne Dousse (Genève) : Proving partition identities
A partition of a positive integer n is a non-increasing sequence of positive integers whose sum is n. A partition identity is a theorem stating that for all n, the number of partitions of n satisfying some conditions equals the number of partitions of n satisfying some other conditions. In this talk, we will discuss recent developments on the techniques which can be used to prove such identities. They include q-difference equations and recurrences, bijections, and representation theory.
- Evelyne Hubert (Inria Côte d'Azur) : Explicit semi-algebraic description of the orbit space of Weyl group actions
The action of the Weyl group on the compact or algebraic torus arise in the representation theory of compact Lie groups or reductive groups. Its invariants are connected with the characters and form a polynomial algebra. This action is also described on lattices associated to a crytallographic root system. As such Weyl groups arise in a number of other domains like Fourier analysis, codes or combinatorics.
For the infinite families of root systems, those of types An-1, Bn, Cn and Dn (also G2 actually), the Weyl group is a semi direct product of a group {1,-1}^k with the symmetric group Sn. Exploiting this decomposition we make explicit a polynomial matrix whose locus of positivity is the image of the compact torus by the fundamental invariants, the orbit space. Remarkably it is a unified formula for all the above types when written in terms of the generalized Chebyshev polynomials associated to the root system.
This explicit formula is an essential ingredient in our approach to optimize trigonometric polynomials with crytallographic symmetry. The problem can be reduced to polynomial optimization on a semi-algebraic set, a subject that has ripened in the last two decades after the seminal article of Lasserre. We applied our approach to compute the spectral bound for the chromatic number of some infinite set-avoiding graphs (a.k.a. Caley graphs).
This is joint work with Tobias Metzlaff, Philippe Moustrou and Cordian Riener.
https://hal.inria.fr/hal-03590007
https://hal.inria.fr/hal-03768067
- Oleg Lisovyy (Tours) : Problème de connexion pour les équations de Heun
Résumé : Les formules de connexion reliant les solutions de Frobenius d'EDOs linéaires associées aux différents points singuliers Fuchsiens peuvent être exprimées en termes de l'asymptotique de coefficients des séries correspondantes. Je montrerai que pour l'équation de Heun et certaines de ses versions confluentes, on peut calculer explicitement un développement perturbatif des amplitudes asymptotiques nécessaires. Cela
permet de vérifier une conjecture récente de Bonelli-Iossa-Panea-Tanzini qui relie la matrice de connexion de Heun aux blocs conformes semiclassiques de l'algèbre de Virasoro.
- Wodson Mendson (Rennes) : Arithmetic aspects of planar vector fields
Abstract: In this talk, I will discuss some topics around foliations on the projective plane in positive characteristic and how to relate it to problems in characteristic zero. I will discuss the notion of the p-divisor and I will show how to use this notion to study some algebraicity problems of foliations on the complex projective plane.
- Marina Poulet (Marseille) : Computing Galois groups of difference equations of order 3.
Abstract: One important application of the difference Galois theory is the study of the (differential) transcendence of solutions of difference equations. Roughly speaking, if the difference Galois group G of a difference equation is sufficiently big then the nonzero solutions of this equation are (differentially) transcendent. More generally, the larger G is, the fewer algebraic relations there are. However, the computation of difference Galois groups is in general a difficult task, we do not have a way to do it for general difference equations. For difference equations of order 1 or 2, many things are known and we can compute Galois groups of q-difference equations, Mahler equations and other well-known types of equations. The aim of this talk is to present the main ideas used to compute difference Galois groups. In particular, we will give an extension of these results for difference equations of order 3 and, in some cases, of order greater than 3. It is a joint work with Thomas Dreyfus.
- Titouan Sérandour (Lyon) : La monodromie des structures projectives méromorphes
Résumé : Une structure projective complexe compacte P est une courbe localement modelée sur la droite projective complexe. C'est notamment un objet géométrique naturellement associé à une EDO linéaire homogène du second ordre à coefficients holomorphes de la forme d²y/dx² + a(x)dy/dx + b(x)y = 0. À un tel objet géométrique, modulo isomorphisme, on associe un objet algébrique : une représentation de son groupe fondamental dans PGL(2,C), à conjugaison près. Celle-ci est définie comme la monodromie des prolongements analytiques d'une carte de P. L'application de monodromie, qui a une structure projective sur une surface compacte orientée fixée S associe sa monodromie, n'est ni injective ni surjective. Néanmoins, Hejhal a montré en 1975 qu'il s'agit d'un difféomorphisme local. Dans cet exposé, j'introduirai le sujet avant de présenter une généralisation du théorème de Hejhal aux structures projectives méromorphes obtenue pendant ma thèse sous la supervision de Frank Loray.
- Jacques-Arthur Weil (Limoges) : Darboux Transformations for orthogonal differential systems and differential Galois Theory
Abstract. This is joint work with Primitivo Acosta-Humanez, Moulay Barkatou and Raquel Sanchez-Cauce. Darboux developed an ingenious algebraic mechanism to construct infinite chains of ``integrable" second order differential equations as well as their solutions. After a surprisingly long time, Darboux's results were rediscovered and applied in many frameworks, for instance in quantum mechanics (where they provide useful tools for Supersymmetric Quantum Mechanics), in soliton theory, Lax pairs and many other fields involving hierarchies of equations. In this work, we propose a method which allows us to generalize the Darboux transformations algorithmically for tensor product constructions on linear differential equations or systems. We obtain explicit Darboux transformations for third order orthogonal systems ($\mathfrak{so}(3, C_K)$ systems) as well as a framework to extend Darboux transformations to any symmetric power of $\mathrm{SL}(2,\mathbb{C})$-systems.
- Changgui Zhang (Lille) : Some problems about a family of linear functional differential equations
Abstract. Given q between 0 and 1, one considers the following problems about the q-difference-differential equation
y′(x) = a y(qx) + b y(x) + f (x),
where a and b are two complex numbers and where f is a rational function:
(1) The pantograph equations following Kato and McLeod;
(2) The indexes of the associated operator d/dx − aσ_q − b;
(3) The connection formulas between zero and infinity;
(4) The asymptotic behavior of the solutions at infinity.
This talk is partially based on a joint work with H. Dai and G. Chen (HITSZ).
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