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Résumé : Almost toric geometry is a generalization of toric geometry, in which certain four-dimensional symplectic manifolds are encoded by two-dimensional base diagrams. The goal of this talk is to give an introduction to this topic, discuss many examples and then move to applications. If time permits we will discuss so-called Lagrangian needle's eyes. These are Lagrangian subspaces which exhibit intersection rigidity with the trace of certain Hamiltonian isotopies of Lagrangian tori. Almost toric fibrations are crucially used in their construction.

Résumé : I will explain how to define a new class of "C^0 Lagrangians" in a cotangent bundle, that we call "bare Lagrangian branes", and that arise as Hausdorff limits of exact Lagrangian submanifolds with control on their Liouville primitives. I will address some rigidity and flexibility questions for such objects: if a bare Lagrangian brane is smooth, is it an exact Lagrangian brane? When does a bare Lagrangian brane admit a unique primitive (up to an additive constant)? The answer to this last question provides us with a lower regularity version of Arnaud's multidimensional Birkhoff theorem: if the support of a bare Lagrangian brane admitting a unique primitive is fixed by the time-1 flow of a Tonelli Hamiltonian, then it is a Lipschitz graph over the base. This is based on joint work with Skander Charfi. The talk is complementary to Charfi's talk on November 2, but no previous knowledge will be assumed.

Résumé : The $H^{2|2n}$ model generalizes the $H^{2|2}$ model originally introduced by Zirnbauer to study disordered systems. Surprisingly these models are closely related to probabilistic models such as vertex-reinforced jump processes and the arboreal gas. These connections provide further motivation for investigating the decay of correlation functions. In this talk we establish exponential decay of correlations in one dimension in some simple cases and compare the resulting rates of decay with those obtained in previous work in the high temperature regime. To determine the optimal rate we introduce the necessary tools from Grassmann calculus and define a transfer operator acting on a Grassmann algebra. Exploiting the hyperbolic symmetry of the model we can diagonalize the operator and explicitly compute the eigenvalues to deduce the desired decay. Based on ongoing joint work with Luca Fresta.

Résumé : Abstract: Goodwillie calculus gives a structured way of approximating a functor by simpler ones, namely by functors that satisfy a generalized form of the excision property familiar from homology. These approximations are built from the "derivatives" of the functor, which turn out to be strongly related to the theory of operads. In this talk, I will introduce Goodwillie calculus and describe how the relation to operads allows one to state versions of the chain rule and the product rule. Perhaps somewhat surprisingly, Goodwillie calculus turns out to be a powerful tool for studying operads in spectra. I will end the talk with describing such applications. This talk is based on joint work with Max Blans.

Résumé : Abstract: Diffusion processes with sticky boundaries describe particles that move inside a domain but may also spend a positive amount of time on its boundary, where they undergo a tangential diffusion. At the PDE level, this leads to a coupled system of bulk and boundary diffusion equations, with mass exchange between the two. In this talk, I will discuss the optimal transport geometry associated with these processes. Starting from their short-time large deviations, I will explain how the relative strength of bulk and boundary diffusion determines the effective transport cost. A sharp transition occurs when diffusion along the boundary becomes faster than in the interior: travelling along the boundary can then provide a shortcut, leading to a transport geometry that differs from the usual Euclidean one. I will then discuss the connection with entropy gradient flows and the variational formulation of the corresponding bulk–boundary diffusion equations, highlighting the difficulties caused by the coexistence of interior and boundary mass. The talk is based on joint work with Jean-Baptiste Casteras and Léonard Monsaingeon.

Résumé : The medial axis represents a 3D shape through its maximal inscribed spheres and their skeletal structure, combining implicit geometric primitives with explicit connectivity. This hybrid nature makes it a promising bridge between explicit and implicit representations. However, its sensitivity to boundary perturbations makes accurate and compact approximations difficult to construct.

In this talk, we will present two complementary approaches that exploit the medial axis in geometry processing.

First, we introduce a variational formulation that directly fits a set of medial spheres to a target shape. By combining local and global geometric information, our approach achieves a favorable trade-off between approximation accuracy, compactness, and computational efficiency.

We then show how surface reconstruction from unsigned distance fields (UDFs) can be reformulated as a medial axis extraction problem. By extracting the medial axis of an α-offset volume around the zero level set of a UDF, we obtain a unified, grid-free framework for reconstructing accurate and compact meshes of closed and open, manifold and non-manifold, as well as mixed-dimensional geometry. When the input field itself encodes a medial axis, the same approach can also recover that medial axis directly.

Taken together, these methods highlight the potential of the medial axis to bridge explicit and implicit geometric representations through shape approximation and surface reconstruction.


About the speaker : Qijia Huang is a PhD candidate in computer graphics and geometry processing at the ICube laboratory, Université de Strasbourg, supervised by Prof. Dominique Bechmann and Dr. Pierre Kraemer. His research focuses on compact, editable, and structure-aware representations of 3D shapes, particularly medial axis approximation, implicit and hybrid shape representations, and neural distance fields. He develops methods for recovering accurate and interpretable geometric structures, with applications in shape approximation and surface reconstruction.

Résumé : I will review the random phase approximation for Fermi gases, which was initiated by Bohm--Pines and Gell-Mann--Brueckner in the 1950s. In particular, I will discuss recent joint work with Emanuela L. Giacomelli, Christian Hainzl, and Robert Seiringer concerning the mathematical justification of the Huang--Yang formula on the ground state energy of dilute gases. Our approach is based on an adaptation of the bosonic Bogoliubov theory to the Fermi system, where the correlation structure of fermionic particles is incorporated by a modified zero-energy scattering equation that takes into account the presence of the Fermi sea, in the spirit of the Bethe-Goldstone equation.

Résumé : The aim of geometric representation theory is to study representations using geometric tools, including sheaves. This inherently requires the choice of a sheaf theory, i.e., of a cohomology theory. Many phenomena in geometric representation theory turn out to be independent of the choice of cohomology theory; more precisely, they are of motivic origin. This is not a priori clear for the Casselman-Shalika formula, which requires the existence of an exponential local system. In this talk, I will explain how to use Gaitsgory's Kirillov model to bypass the choice of an exponential local system, and give an alternative, purely geometric, formulation of the geometric Casselman-Shalike formula. This can be used to construct a motivic Casselman-Shalika equivalence. This is joint work with Robert Cass and Jakob Scholbach.

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Résumé : AI has made remarkable progress in recent years across many domains, and more recently in mathematics. Hard to imagine just a few years ago: general-purpose LLMs can now find proofs of statements that long resisted human efforts. This talk will retrace how AI for mathematics got to where it is today over the past eight years, from two angles: we will first discuss AI systems dedicated to automated theorem proving, and then see how models specifically trained for mathematics can provide valuable insights and intuitions, and even predict solutions to problems from a variety of fields. We will then try to discuss the next steps and the future of mathematics that may be ahead.

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Résumé : Cet exposé sera un panorama sur l'utilisation de méthodes combinatoires et graphiques à travers les toiles (certains types de graphes) et les mousses (certains cobordimes naturels entre eux) dans le contexte de la théorie de représentations et aussi des homologies d'entrelacs. On fera le lien (si le temps le permet) avec la théorie des bimodules de Soergel qui donne une catégorification de l'algèbre de Hecke et on parlera de catégorification d'autres algèbres centralisatrices. Cet exposé suit plusieurs travaux en collaboration avec L-H. Robert, M. Khovanov, J. Grlj et Z. Lei.

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Résumé : Let (W, lambda) be a Liouville manifold and consider an exact Lagrangian isotopy (Lt) in W for t in [0,1]. We seek a lower bound for the intersection number of L0 and L1 when L0 and L1 intersect transversely. In the particular case where W is a cotangent bundle, analogous results are proved via the theory of generating families by Laudenbach and Sikorav in 1985 and by Chekanov in 1996. Laudenbach also proved such a result for a general Liouville manifold W in 1995, with the additional hypothesis pi1(W, L) = 0, pi2(W, L) = 0.

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Résumé : On dit qu'un système dynamique est surjonctif si tout endomorphisme injectif du système est surjectif. Je présenterai une caractérisation géométrique de la surjonctivité pour les sous-décalages de type fini sur $\mathbb{Z}$. Cette caractérisation fait intervenir un ordre naturel sur l'ensemble des composantes irréductibles du sous-décalage. Il s'agit d'un travail en collaboration avec Tullio Ceccherini-Silberstein et Ville Salo.

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Résumé : Nous présentons dans ce travail un cadre unifié pour les inéquations variationnelles elliptiques de première espèce. Ce cadre regroupe plusieurs problèmes issus de la mécanique du contact. Il repose sur une formulation mixte faisant intervenir un multiplicateur de Lagrange. Nous proposons une discrétisation basée sur une méthode de Galerkine. Des discrétisations conformes et non conformes peuvent être considérées. Le résultat principal de ce travail établit la convergence de la méthode de discrétisation mixte sous certaines hypothèses. Ces hypothèses doivent ensuite être vérifiées pour chaque méthode de discrétisation que nous souhaitons étudier. Nous proposons d'appliquer ce cadre général à un problème de contact elliptique, discrétisé à l’aide d’une méthode d’éléments finis et d’une méthode hybride d’ordre élevé (HHO). Dans les deux cas, une discrétisation mixte P2 − P0 est considérée. Des estimations d’erreur a priori sur la solution numérique sont fournies pour les deux discrétisations. Ces méthodes de discrétisation constituent une variante des travaux précédents [1] et [2]. Dans [1], des fonctions « bulle » additionnelles sont introduites afin d’assurer la stabilité inf-sup du problème, tandis que nous démontrons ici que ces inconnues supplémentaires peuvent être supprimées sous certaines hypothèses sur le maillage. Dans [2], l’analyse porte uniquement sur la variable primale. Le multiplicateur de Lagrange est introduit à des fins de simulation. Dans notre travail, l’analyse numérique prend en compte le multiplicateur de Lagrange, qui est choisi dans un espace de dimension plus petite. Ceci est un travail en collaboration avec Jad Dabaghi. [1] T. Gustafsson, R. Stenberg, J. Videman. Mixed and stabilized finite element methods for the obstacle problem, SIAM J. Numer. Anal. 55 (2017), no. 6, 2718–2744. [2] M. Cicuttin, A. Ern, T. Gudi. Hybrid high-order methods for the elliptic obstacle problem, J. Sci. Comput. 83 (2020), paper no. 8, 18

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Résumé : Jointly with Stefan Nemrovski in our GAFA, GT and GP paper we showed Legendrian linking of the spheres of light rays through the two points does (nearly always) tell if the two event points in the universe are causally related, that is if they know of each other existence. We review these results and related Bott-Samelson type results of Nemirovski and myself and of Fraunenfelder-Labrousse-Shlenk. After that we introduce generalized spacetime that are singular Lorentz manifolds whose critical levels of a singular timelike Morse function cut them into globally hyperbolic pieces. Strong Cosmic censorship conjecture of Penrose suggests that if you delete black holes along horizons then what is left is globally hyperbolic. We introduce virtual Legendrian linking coming from the unit cotangent bundles of the spacetlike sections of such pieces and jointly with Sadykov we formulate the conjectures relating causality in generalized spacetimes to virtual Legendrian linking. In the remaining time we explore the theory results on virtual Legendrian links of Sadykov and myself.