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Opérateurs sur des espaces de fonctions analytiques
02 dec 2024 - 06 dec 2024 • CIRM (Marseille Luminy), France
Organisateur:
CIRM – Centre International de Rencontres Mathématiques
Identifiant de l'évènement:
1574935
Sujets apparentés:
2
RESEARCH SCHOOL — Arithmetic, Algebraic and Analytics Dynamics
20 jan 2025 - 24 jan 2025 • CIRM (Marseille Luminy), France
Organisateur:
CIRM – Centre International de Rencontres Mathématiques
Résumé:
The main goal of these Etats de la recherche is to review some of the most recent spectacular developments in algebraic, analytic and arithmetic dynamics. Covered themes will include a variety of problems related to the iteration of rational maps on algebraic varieties as well as the study of their dynamical moduli spaces. Depending on the structure of the ambient field over which these maps are defined, methods may vary drastically. Over the field of complex numbers, one speaks of holomorphic dynamics in which quasi-conformal deformations play a key role. The case of maps defined over general metrized fields have led to the developments of dynamics over Berkovich spaces. One can also consider maps over number fields whose study lies at the core of arithmetic dynamics. Interactions between these various fields have deepened in the recent years. Complex pluripotential theory combined with tools from arithmetic geometry has been used to prove equidistribution results with applications to problems of unlikely intersection, and to explore dynamical moduli spaces and their special varieties. Methods in p-adic analysis and from the minimal model program have revolutionized our understanding of groups of birational transformations. Non-archimedean dynamics has been turned into a very efficient tool to analyze degeneration problems in holomorphic dynamics.
Identifiant de l'évènement:
1633900
3
RESEARCH IN RESIDENCE — Dispersion correction for the convected Helmholtz equation
20 jan 2025 - 24 jan 2025 • CIRM (Marseille Luminy), France
Organisateur:
CIRM – Centre International de Rencontres Mathématiques
Résumé:
Solving time-harmonic wave equations with iterative methods is very challenging because they combine multiple difficulties like e.g. highly oscillatory solutions and complex non-Hermitian and indefinite discrete operators. In addition, the wave number associated to the numerical solution is different from the one of the exact solution thus leading to so-called numerical dispersion which requires very fine mesh to be controlled. There exists numerical schemes designed to reduce the dispersion error where the most used technique to build them relies on stencils involving free parameters that are next obtained by numerically minimizing the dispersion error. In the past few years, we develop an alternative approach called asymptotic dispersion correction. The latter is based on the introduction of a perturbation of the wavenumber (the shift) in the numerical scheme. We then explicitly compute the shift that minimizes the dispersion error for small enough meshsize. In addition, we show that our method can be applied to any finite difference scheme and reduces the relative error. We emphasize that most of works dealing with dispersion reducing methods consider the Helmholtz equation. The goal of the Research in Residence is to extend the asymptotic dispersion correction to the convected Helmholtz equation that can be used to model time-harmonic wave propagation in a moving flow.
Identifiant de l'évènement:
1633943
4
RESEARCH IN RESIDENCE — Unsolvable problems in dynamical systems
24 fév 2025 - 08 mar 2025 • CIRM (Marseille Luminy), France
Organisateur:
CIRM – Centre International de Rencontres Mathématiques
Résumé:
Starting in the 2011 Annals of Mathematics paper of Foreman, Rudolph and Weiss there have many results showing that classical problems in dynamical systems are unsolvable using inherently countable resources. The initial results were for abstract measure preserving systems, but more recent work has focussed on smooth systems on compact manifolds. Gerber and Kunde have made very significant contributions for smooth transformations that are weakly mixing and for Kautomorphisms. They also showed that the Kakutani equivalence relation is not Borel. These results raise more questions than they solve! The proposed Research inResidence is to bring the four researchers together to share knowledge and techniques and hopefully make progress on the open problems.
Identifiant de l'évènement:
1634039
Sujets apparentés:
5
RESEARCH IN RESIDENCE — Keller-Segel and Chemotaxis Models in Lipschitz-Domains
05 mai 2025 - 09 mai 2025 • CIRM (Marseille Luminy), France
Organisateur:
CIRM – Centre International de Rencontres Mathématiques
Identifiant de l'évènement:
1634096
6
RESEARCH IN RESIDENCE — Explicit local solubility and applications
09 mai 2025 - 20 mai 2025 • CIRM (Marseille Luminy), France
Organisateur:
CIRM – Centre International de Rencontres Mathématiques
Résumé:
The question of whether or not a collection of equations has a solution in the integers is notoriously challenging, with the answer depending heavily on the geometry of the constituent equations. An important reduction is localization – looking at the equations modulo a prime or over the real numbers. Having solutions locally is a key necessary condition for the equations to have an integral solution and much more tractable to determine explicitly. The proposed research involves determining explicit and exact expressions for how often certain families of equations have local solutions, which can sometimes yield explicit results for how often an equation has an integral solution. In particular, we propose an approach to determine how often a degree 3 polynomial in n+1 variables has an integral zero, by reducing to the local probabilities (via recent work of Browning, Le Boudec, and Sawin) and determining them by a careful recursive argument.
Identifiant de l'évènement:
1634156
7
RESEARCH IN RESIDENCE — Elliptic and parabolic equations originating from real world problems
29 sep 2025 - 10 oct 2025 • CIRM (Marseille Luminy), France
Organisateur:
CIRM – Centre International de Rencontres Mathématiques
Résumé:
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.
Identifiant de l'évènement:
1634175


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