Abel à Paris

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Abel à Paris

Évènement reporté sine die

January 20th, 2022

The Institut Henri Poincaré is happy to host the "Abel à Paris" Conference.
L’Institut Henri Poincaré est ravi d'accueillir la conférence “Abel à Paris ”.

Main scientific event - Mandatory registration

Abel à Paris, January 20th, 2022.

Program

10:00-11:00 Yves Meyer
Crystalline measures and applications
11:30-12:30 Sylvia Serfaty
Des supraconducteurs aux gaz de Coulomb: questions de cristallisation
Déjeuner
14:30-15:30 Alexander Lubotzky
Stability, non-approximated groups and high-dimensional
expanders
16:00-17:00 Jean-Pierre Serre
Souvenirs mathématiques
18:00 Réception à l’ambassade de Norvège (sur inscription)

Speakers - Abstracts

Alexander Lubotzky

Title: Stability, non-approximated groups and high-dimensional expanders

Abstract: Several well-known open questions, such as: "are all groups sofic / hyperlinear?", have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, some of these versions, showing that there exist finitely presented groups that are not approximated by U(n) with respect to the Frobenius (=L_2) norm and many other norms. The strategy is via the notion of "stability": Some higher dimensional cohomology vanishing phenomenon is proven to imply stability. Using the Garland method ( a.k.a. high dimensional expanders as quotients of Bruhat-Tits buildings), it is shown that some non-residually-finite groups are stable and hence cannot be approximated. These groups are central extensions of some lattices in p-adic Lie groups (constructed via a p-adic version of a result of Deligne). We will also discuss the connection between "stability" and "testability" from computer science. All notions will be explained with some suggestions for further research.

Based on joint works with M. De Chiffre, L. Glebsky , A. Thom, I. Oppenheim, O. becker and J. Mosheiff.

Yves Meyer

Title: Crystalline measures and applications

Abstract: An atomic measure μ on Rn is a crystalline measure if (a) μ is supported by a locally finite set and if (b) its distributional Fourier transform ̂μ is also an atomic measure supported by a locally finite set. Crystalline measures were introduced in 1959 by Andrew Guinand, Jean-Pierre Kahane, and Szolem Mandelbrojt. To any crystalline measure μ a generalized zeta function ζμ(s) can be associated as it is proved by Guinand et al. If μ is a Dirac comb ζμ(s) is the Riemann zeta function. The discovery of quasicrystals by D. Shechtman et al. renewed the in-terest in Guinand’s work. This led to the following time frequency interpolation problem: We are given two locally finite sets E ⊂Rn and F ⊂Rn. We say that E and F complement each other if a Schwartz function f is uniquely defined by its restriction to E together with the restriction of its Fourier transform to F. A spectacular example is given by Maryna Viazovska. Finally it applies to the solution by Viazovska of the packing Kepler problem in dimensions 8 and 24. This line of investigation is related to the multi-layered decompositions of signals and images.

Yves F. Meyer. Measures with locally finite support and spectrum, PNAS (2016) 113 3152-3158.
D. Radchenko and M. Viazovska, interpolation on the real line, Publ. Math.
IHES 129, (2019) 5181.
J. Tropp On the linear independence of spikes and sines. Journal of Fourier analysis and applications. 14, (2008) 838-858.

Sylvia Serfaty

Title: Des supraconducteurs aux gaz de Coulomb: questions de cristallisation

Abstract: Le physicien Abrikosov avait prédit l'apparition de réseaux de vortex triangulaires dans certains matériaux superconducteurs. Dans l'étude des gaz de Coulomb, motivée par des questions d'approximation, de matrices aléatoires et de physique statistique, on s'attend à voir la même forme de réseaux pour les états d'énergie minimale. Ce phénomène de cristallisation se trouve être directement relié à la conjecture de Cohn et Kumar sur l'universalité optimale de certains réseaux particuliers en dimension 2, 8 et 24, récemment résolue en dimension 8 et 24. On discutera aussi de la relation de la cristallisation et de la température dans ces systèmes.

Jean-Pierre Serre

Title: Souvenirs mathématiques

Organizers

The Abel committee

Contact

Claire Voisin, CNRS, IMJ-PRG

January 2022