Pour tout renseignement complémentaire, veuillez contacter les
organisateurs, Hakim
Boumaza, Mathieu
Lewin
ou Stéphane
Nonnenmacher.
| 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. |