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Résumé : Cette séance sera l'occasion de dresser un court bilan du séminaire ITI et de discuter des perspectives pour l'année 2025.
Suite à une perte de vitesse du séminaire, nous avons besoin de vos idées pour le faire revivre et le transformer.

Résumé : The bounded derived category of coherent sheaves D(X) is an important invariant of an algebraic variety X. While the structure of derived categories is generally quite intricate, in certain cases when D(X) admits a so-called full exceptional collection, D(X) can be described explicitly. Some of the earliest examples of full exceptional collections were constructed by Kapranov in 1983 for classical Grassmannians. Since then, a folklore conjecture says that full exceptional collections exist in the derived categories of all rational homogeneous varieties. In my talk I will outline the proof of this conjecture for all rational homogeneous varieties associated with symplectic groups. This is joint work with Sasha Novikov.

Résumé : Nambu structures on a smooth manifold are a special case of Poisson structures. This notion can be seen as a generalization of symplectic structures on odd-dimensional manifolds. We will present what a Nambu structure is and try to give some elements of comparison between Nambu mechanics and Hamiltonian mechanics.

References :
Y. Nambu, Generalized Hamiltonian mechanics, Phys. Rev. D7 (1973), 2405-2412
L. Takhtajan, On foundation of the generalized Nambu mechanics, Comm. Math. Phys. 160 (1994), 295-315
J.-P. Dufour, N. T. Zung, Poisson structures and their normal forms, coll. Progress in Mathematics, Birkhauser, 2005

Résumé : Le modèle Euler compressible bifluide ne présente pas de difficultés théoriques supplémentaires comparé au cas monofluide. Mais sa résolution numérique est notoirement plus difficile à cause du phénomène d'oscillations de pression à l'interface entre fluides. Nous présentons une approche basée sur un échantillonnage aléatoire "à la Glimm" à l'interface, qui permet de s'affranchir de ce défaut. Le schéma obtenu est applicable à des maillages non structurés, il a d'excellentes propriétés de robustesse et de convergence. Nous l'appliquons à des cas de déferlement.

Résumé : Given a closed compact surface S and a reductive Lie group G, Goldman introduced a symplectic structure on the character variety Hom(pi_1(S),G)//G (the space of representations modulo conjugation). For Teichmüller space, the form coincides with the Weil-Petersson form. The symplectic structure gives rise to Hamiltonian flows associated to functions on character varieties, and therefore gives ways of deforming representations. I will talk about joint work with Anna Wienhard and James Farre, where we focus on a particular type of function on the character variety, whose input includes families of curves on S. The result I will present states that the Hamiltonian flows of such functions are what we call subsurface deformations, which roughly means that the flow is concentrated on a subsurface of S that depends on the curves defining the function. If time permits, I will discuss some applications to Hamiltonian flows of length functions associated to some self intersecting curves.

Résumé : La fonction zeta locale la plus élémentaire est la fonction $s\longmapsto \frac{1}{1-p^{-s}}$ où $s\in\C$ et où $p$ est un nombre premier. On reconnaît évidemment là les facteurs qui apparaissent dans l'expresssion de la fonction $\zeta $ de Riemann en produit infini. Dans sa thèse (1950) John Tate a généralisé cette fonction en une ``fonction zeta locale" qui est une distribution sur le corps des nombres p-adiques $\Q_{p}$ (ou $\R$) d\'ependant d'un paramètre complexe $s$ et d'un caractère de $\Q_{p}^*$ ($\R^*$). Cette fonction zeta vérifie une équation fonctionnelle qui traduit une relation surprenante entre transformée de Fourier additive (sur $\Q_{p}$) et multiplicative (sur $\Q_{p}^*$). Par la suite, dans un exposé au séminaire Bourbaki en 1966, André Weil a re-interprété l'équation fonctionnelle de Tate en terme d'unicité de distributions homogènes et suggéra une généralisation de la situation ($\Q_{p}^*=GL_{1}(\Q_{p}), \Q_{p}$) (cas de Tate) à ($GL_{n}(\Q_{p}),M_{n}(\Q_{p})$) Cette généralisation a été obtenue, dans un cadre plus large que ne le suggérait Weil, par Godement et Jacquet, en 1972. En faisant intervenir la théorie des espaces préhomogènes, Pascale Harinck et moi-même avons étendu (partiellement) les résultats de Godement et Jacquet à certains espaces symétriques (i.e $\Omega=G/H$ où $G$ est un groupe réductif et $H$ les points fixes d'une involution). Note: Aucun pré-requis concernant les corps p-adiques n'est nécessaire.

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Résumé : I will explain Cieliebak-Mohnke's conjectures about extremal Lagrangian tori and our progress so far. If time permits, I will also discuss some applications to symplectic embedding problems.

Résumé : Multiple zeta values (and their motivic version) is the gadget lying in the heart of many subjects, such as mixed Tate motives over Z. The geometric relations between them are, therefore, crucial for these subjects. The associator relations are supposed to be the strongest among all relations. Regularized double shuffle relations form another set of relations. The interaction between these two sets seems to be an important question. Deligne and Terasoma initiated a geometric approach to interpreting regularized double shuffle relations. This approach explains the form of these relations: group-likeness of a certain element of a Hopf algebra. The tensor category standing behind this Hopf algebra is a certain category built of perverse sheaves, the tensor product being given by convolution. I will present my version of this approach, which (in my opinion) clarifies and simplifies some points.

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Résumé : In this talk some recent developments on matrix distributions and their connection to absorption times of inhomogeneous Markov processes will be discussed, in particular how and why such constructions are natural tools for modelling in applied probability, with a particular emphasis on non-life and life insurance applications. We also illustrate how certain extensions to the non-Markovian case involve fractional calculus and lead to matrix Mittag-Leffler distributions, which turn out to be a flexible and parsimonious class for the modelling of large but rare insurance loss events.

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Résumé : Abstract - Thurston defined a mapping class group-equivariant spine for Teichm\"uller space; the ``Thurston spine''. This spine is a CW complex, consisting of the points in Teichm\"uller space at which the set of shortest geodesics - the systoles - cut the surface into polygons. The systole function is a map from Teichm\"uller space to $\mathbb{R}_{+}$ whose value at any point is given by the length of the systoles. It is known that the systole function is a topological Morse function on Teichm\"uller space, whose critical points are contained in the Thurston spine. This talk surveys what the systole function tells us about the Thurston spine.