| 14h - 15h |
Sonae Hadama (Kyoto University) |
Global well-posedness of the Hartree equation for infinitely many particles with singular potential Abstract: We consider the Hartree equations describing the time evolution of wave functions of infinitely many interacting fermions in three-dimensional space. These equations can be formulated using density operators and have infinitely many stationary solutions. We give a global well-posedness result for initial data with finite relative entropy without requiring any smoothness of the interaction potential. This work is a joint work with Younghun Hong at Chung-Ang University. |
| 15h15 - 16h15 | Pascal Millet (Sorbonne Paris-Nord) |
Spectral aspects of wave propagation on black hole
spacetimes
Abstract: The study of wave propagation on black hole spacetimes has been an active field of research in recent decades. This interest has been driven by the stability problem for black holes and by questions related to scattering theory. I will focus on the spectral approach, which relates the asymptotic properties of the solution to the properties of the resolvent operator. These properties are constrained by the geometric features of the black hole background and by the lower-order structure of the equation under consideration. I will illustrate this interplay, mainly in the case of the Teukolsky equation which governs the evolution of various massless fields around spinning black holes. In this context, resolvent analysis can be used to precisely determine the late-time asymptotics of initially localized solutions. |
| 14h - 15h |
Antoine Mouzard (Paris Nanterre) |
Schrödinger operator with distributional potential and Strichartz inequalities Abstract: In this talk, I will present an approach to study Schrödinger operators with distributional potentials. The motivation comes from stochastics PDEs with the Anderson operator which is the Schrödinger operator with white noise potential. I will explain how paracontrolled calculus allows the construction and study of the operator and associated evolution PDEs with the example of the nonlinear Schrödinger equation. On the circle, we are able to consider generic rough potentials of Hölder regularity greater than one and prove invariance of the associated Gibbs measure, following works from Lebowitz, Rose and Speer and Bourgain. On compact surface, even the construction of the Anderson Hamiltonian requires a probabilistic renormalization procedure. Based on joint works with A. Debussche and I. Zachhuber. |
| 15h15 - 16h15 | Benoit Dagallier (Paris Dauphine) |
Stochastic dynamics and the Polchinski equation
Abstract: I will discuss a general framework to obtain large-scale properties of statistical mechanics and field theory models. A well known idea is to introduce a dynamics that samples from the model and controls its long time behaviour. The Langevin dynamics is a popular and successful choice, but is hard to use in a number of cases as I will explain. In this talk I will introduce another object, the Polchinski dynamics, based on renormalisation group ideas. This dynamics has appeared in very different contexts in connection with convex analysis (it is the same as Eldan's stochastic localisation) and optimal transport. I will motivate the construction of the dynamics from a physics perspective and explain how it can be used to prove functional inequalities (Poincaré, log-Sobolev) via a generalisation of Bakry and Emery's convexity argument. The talk is based on joint work with Roland Bauerschmidt and Thierry Bodineau, and the review paper https://arxiv.org/abs/2307.07619 |
| 14h - 15h |
Hakim Boumaza (Sorbonne Paris-Nord) |
Localization for a quasi-one-dimensional random Dirac model. Abstract: In this talk, we focus on the phenomena of Anderson localization and dynamical localization in the context of quasi-one-dimensional random models. For these models, the question of localization is reduced to the study of an algebraic object, the Furstenberg group. The introduction of this group will be linked to that of objects typical of dimension one: transfer matrices, Lyapunov exponents and Kotani theory. In the context of Dirac-like quasi-one-dimensional operators, we will present a localization criterion involving only properties of the Furstenberg group. Finally, I will study the Furstenberg group for a particular example of a Dirac-type quasi-one-dimensional model. This is a joint work with S. Zalczer. |
| 15h15 - 16h15 | Cyril Letrouit (CNRS & Paris-Saclay) |
Spectral methods for quantitative stability of optimal transport maps
Abstract: Optimal transport consists in sending a given source probability measure to a given target probability measure, in a way which is optimal with respect to some cost. On bounded subsets of Rd, if the cost is given by the squared Euclidean distance and the source measure is absolutely continuous, a unique optimal transport map exists. The question we will discuss is the following: how does this optimal transport map change if we perturb the target measure? This question, motivated by numerical aspects of optimal transport, has started to receive partial answers only recently, under quite restrictive assumptions on the source measure. We will show how methods coming from spectral theory (Poincaré and Cheeger inequalities for instance) allow to handle much more general cases. This is a joint work with Quentin Mérigot. |
| 14h - 15h |
Denis Grebenkov (CNRS & Ecole Polytechnique). |
The Steklov spectral problem: asymptotic results and probabilistic applications Abstract: In this overview talk, I will present the encounter-based approach to diffusive processes in Euclidean domains and highlight its fundamental relation to the Steklov spectral problem. So, the Steklov eigenfunctions turn out to be particularly useful for representing heat kernels with Robin boundary condition and disentangling diffusive dynamics from reaction events on the boundary. In the second part of the talk, I will discuss the asymptotic behavior of the Steklov eigenvalues for the exterior Steklov problem. Some open questions related to spectral, probabilistic and asymptotic aspects of this problem will be outlined. |
| 15h15 - 16h15 | Valentina Ros (CNRS & Université Paris-Saclay) |
The geometry of high-dimensional Gaussian landscape, and what can it tell us about dynamics
Abstract: High-dimensional random landscapes (i.e., random fields on high-dimensional manifolds) have been extensively studied as models of the potential energy landscapes of complex systems, such as glasses. Understanding their structure has been—and continues to be—crucial for determining the evolution of local dynamics in associated glassy systems. In recent years, these landscapes have also emerged as generalized cost functions in a variety of high-dimensional optimization, inference, and control problems. In this talk, I will present results on a simple yet paradigmatic model: an isotropic Gaussian landscape on a high-dimensional hypersphere. I will describe techniques, relying on the Kac-Rice formula and on Random Matrix Theory, for characterizing the “landscape geometry”—namely, the distribution and connectivity of stationary points in configuration space. Finally, I will comment on how this information could be useful for understanding the exploration of the landscape through local dynamics, beyond the asymptotic, "mean-field" limit of infinite dimensionality. |