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.