Institut de recherche mathématique avancée

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.