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Christopher Deninger
Primes, knots and periodic orbits
17 janvier 2025 - 16:00Salle de conférences IRMA
In the 1960s Manin, Mazur and Mumford noted that from there was an intriguing analogy between prime numbers embedded into the spectrum of the integers and knots in 3-space. Later Kapranov, Reznikov, Morishita and other authors discovered further analogies between number rings and the topology of 3-manifolds. For example, the Iwasawa zeta function corresponds to the Alexander polynomial of a knot. The search for a cohomology theory related to the Riemann zeta function led to the discovery of analogies between number rings and a class of 3-dimensional dynamical systems, where the primes would correspond to the periodic orbits. For example, Riemann’s explicit formulas in analytic number theory correspond to a transversal index theorem in the dynamical context, proved by Álvarez-López, Kordyukov and Leichtnam. The dynamical systems analogy refines the previous analogy because forgetting its parametrization, a periodic orbit gives a knot. Recently, we have constructed foliated dynamical systems for number rings and even for all arithmetic schemes that have some but not yet all the expected properties. -
Patrick Massot
TP informatique : découverte de l’assistant de preuve Lean
28 février 2025 - 10:00Salle de conférences IRMA
Le logiciel Lean permet de parler de maths de tout niveau à son ordinateur. Il peut aussi servir à enseigner le raisonnement mathématique rigoureux, par exemple en L1. Ce TP sera une introduction à l’utilisation de Lean en pratique. Il n’y a aucun pré-requis si ce n’est de venir avec un ordinateur portable -
Patrick Massot
Pourquoi raconter des maths à un ordinateur ?
28 février 2025 - 16:00Salle de conférences IRMA
Dans cet exposé j’expliquerai ce que signifie « expliquer des mathématiques à un ordinateur » et pourquoi je trouve cela intéressant et utile. Je montrerai à quoi ressemble concrètement l’utilisation d’un logiciel permettant d’encoder informatiquement des définitions, énoncés et démonstrations. Je présenterai les applications de ces techniques pour vérifier, expliquer, enseigner ou créer des mathématiques. Je mentionnerai des exemples de projets non-triviaux dans ce domaine et j’évoquerai brièvement les liens avec l’IA. Il n’y a aucun pré-requis. -
Katharina Schratz
Resonances as a computational tool
7 mars 2025 - 16:00Salle de conférences IRMA
A large toolbox of numerical schemes for dispersive equations has been established, such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). These classical schemes are based on linearised time dynamics and in many situations allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene such as for problems at low regularity and high oscillations. Classical schemes fail to capture the oscillatory nature of the solution, and this may lead to severe instabilities and loss of convergence. In this talk I present a new class of resonance based schemes. The key idea in the construction of the new schemes is to tackle and deeply embed the underlying nonlinear structure of resonances into the numerical discretization. As in the continuous case, these terms are central to structure preservation and offer the new schemes strong geometric properties at low regularity.