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CIRM – Centre International de Rencontres Mathématiques

We would like to tackle the numerical resolution of some isoperimetric problems involving non-local competing energies of different nature. The competing energy we consider would either be inspired by Gamow’s liquid drop model for the atomic nucleus (e.g., given by the self-interaction of a set with respect to a general G, or by a fractional perimeter) or would be of Optimal-Transport type (e.g., involving the Wasserstein distances between measures). Although these problems have already been studied theoretically in some regimes, there are many interesting regimes for which little is known ; only some conjectures were formulated. We want to investigate numerically these problems to test these conjectures or formulate new ones.

CIRM – Centre International de Rencontres Mathématiques

This project focuses on the development of unconditionally stable conforming space—time methods for the linear time‑dependent Schrödinger equation. The main challenge is to overcome the lack of inf‑sup stability in the standard Petrov‑Galerkin formulation for this equation. In fact, the conforming space—time finite element discretization requires a restrictive CFL condition. To address this issue, we propose two approaches. ‑The first one is to augment the sesquilinear form of the standard scheme by adding a carefully chosen inconsistent penalty term to improve the discrete properties of the scheme. By introducing this term, we aim to improve stability without compromising convergence order. ‑The second approach involves the composition of the continuous sesquilinear form with an appropriate transform of the test functions. This modification aims to endow the new sesquilinear form with higher stability properties than its unmodified counterpart.

Mathematical Physics,    Computational Physics and Numerical Simulation

CIRM – Centre International de Rencontres Mathématiques

Our project focuses on understanding the interplay between elasticity and the classical Hele-Shaw model, with the long-term of simplifying the model and improving its suitability for numerical applications and algorithmic development. Our first aim is to show existence of solutions via the convergence of an implicit time-discretization using the Wasserstain distance. We expect the limit to satisfy the equation in a distributional sense as well as an optimal energy-dissipation relation. Similarly, we construct weak solutions to the Cahn-Hilliard equation with degenerate mobility and elastic misfits. Finally, we aim to prove that the sharp interface limit is a distributional solution to the elastically modified Hele–Shaw flow with optimal energy-dissipation rate.

CIRM – Centre International de Rencontres Mathématiques

The conference, in advance of a thematic program planned for IHP in 2026, addresses the key role played by completely integrable systems as models for wave phenomena in myriad physical settings such as surface water waves and internal waves, quantum excitations in Bose-Einstein condensates, plasma oscillations, optical fiber telecommunication systems, and solid-state physics. The principal objectives of the CIRM conference are to focus specifically on soliton gases, explicit formulas, and asymptotic behavior in integrable models, and to highlight corresponding novel results and new progress on several previously impenetrable fronts.

Calculus, Differential Equations and Integration,    Mathematical Physics

CIRM – Centre International de Rencontres Mathématiques

The school will introduce participants to several hot research topics at the frontier between geophysical fluid dynamics, fundamental physics and mathematics. The main themes will be spontaneous stochasticity, waves in geophysical and astrophysical fluids and fundamental challenges in hydrodynamic and wave turbulence. The school will consist of several lecture series, plus a number of research seminars, covering mathematical, theoretical and experimental aspects. The school is organized in the framework of an international collaboration which started in 2019 and which is financially supported by the Simons Foundation (https://cims.nyu.edu/wave-turbulence/people/). This “Simons Collaboration on Wave Turbulence” brings together physicists and mathematicians with the aim of addressing fundamental questions about the Wave Turbulence theory and its applicability to real systems and especially to the Earth climate system.

Courses and Events for Math Students and Early Career Researchers,    Computational Physics and Numerical Simulation

CIRM – Centre International de Rencontres Mathématiques

The abstract foundation of the book concerning functional analysis, special issues on Sobolev type function scales, and elliptic and parabolic regularity in case of non-smooth data, is already solid. However, the culminating last part of the book is still under discussion and editing. There, it is intended, using the theory established before, to give a condensed and unified treatment of wellposedness for several real-world problems, namely the thermistor problem, the Keller-Segel chemotaxis model, and the macroscopic semiconductor equations (Van Roosbroeck system) in case of Avalanche generation, all for the case of non-smooth spatial domains. These problems were open for decades and solved only recently by the authors and collaborators in journal articles.

Analysis,    Calculus, Differential Equations and Integration