Séminaires

Let $G/K$ be a noncompact Riemannian symmetric space, where $G$ is a noncompact connected semisimple Lie group with finite center. Via the Iwasawa decomposition $G=ANK$, we may also view $G/K$ as the solvable, non-unimodular Lie group $S=AN$. The wave equation on symmetric spaces associated with two Laplace-like operators—the Laplace–Beltrami operator of $G/K$ and the distinguished Laplacian of $S$—has been extensively studied. Numerous results on the boundedness of its solutions are available in the literature. In this talk, I will briefly review some developments in this area and then present joint work with Yulia Kuznetsova on the $L^p-L^q$ boundedness of solutions to the shifted wave equation.

En 1999, Wen Chao Lu a donné une démonstration par l’analyse réelle du théorème des nombres premiers avec terme d’erreur, “à epsilon près” celui obtenu un siècle auparavant par La Vallée Poussin au moyen de l’analyse complexe. En 2024, Gozé a, dans sa thèse, donné une version quantitative de ce résultat.

Dans un travail en cours avec Gozé et Bruno Martin, nous reprenons les démonstrations de Lu et Gozé, et tentons d’en dégager les idées essentielles. L’exposé présentera sous une forme simple certaines d’entre elles.

Let $X$ be a compact hyperbolic surface with finite order singularities and $X_1$ its unit tangent bundle. We consider the twisted Selberg zeta function $Z(s; \rho)$ associated with a representation $\rho: \pi_1(X_1) → GL(V_\rho)$. In this talk, we will present recent results concerning a relation between the twisted Selberg zeta function $Z(s; \rho)$ and the regularized determinant of the twisted Laplacian. The main tool we use is the Selberg trace formula.  If $X$ has no finite order singularities, we obtain as a corollary a corresponding relation. These results can be viewed as an extension to the non-unitary twists case of the results by Sarnak and Naud. This is joint work with Jay Jorgenson and Lejla Smajlovic.

The discrepancy of a distribution of $N$ points in the torus $T^d$ with respect to a given family of test sets measures how far the points are from being uniformly distributed over that family. When the family consists of all translates of a fixed set, one can consider the $L^2$-average of the discrepancy over translations and use Fourier analytical methods to understand its size. Sharp lower bounds for such $L^2$ discrepancy in terms of $N$ are known for wide classes of sets in $T^2$, but much less is known in higher dimensions. In this talk, I will report on recent progress in this direction, focusing on a family of test
sets with “cylindrical” symmetry that can be defined in any dimension. In three dimensions, these sets have the shape of a barrel. They are particularly
interesting because they exhibit geometric features known to play a key role in discrepancy theory: flat regions, curved regions, and corners. Joint work with
Luca Brandolini and Alessandro Monguzzi.

En mécanique quantique, le principe d’incertitude d’Heisenberg stipule qu’on ne peut connaître simultanément avec précision la position et la vitesse d’une particule. Cette célèbre inégalité relie en réalité une fonction et sa transformée de Fourier.
En 1989, motivés par des applications en traitement du signal, Donoho et Stark donnent un nouveau principe d’incertitude, non plus pour des fonctions définies sur $\mathbb{R}$ mais sur un groupe abélien fini. Ce dernier a ensuite été significativement amélioré : en 2006, Tao prouve ce qu’on appelle un principe d’incertitude fort pour des fonctions définies sur $\mathbb{Z}/p\mathbb{Z}$, où $p$ est premier. Plus récemment, en 2021, Garcia, Karaali et Katz généralisent ce principe aux corps finis, pour des fonctions vérifiant une certaine condition de symétrie qu’on détaillera.
Dans cet exposé, on présentera une généralisation du principe d’incertitude fort pour des groupes abéliens finis quelconques. Nous verrons à quel point ce dernier est restrictif, et nous décrirons les cas pour lesquels il est vérifié. Enfin, nous terminerons avec une application en combinatoire additive, plus précisément un théorème de type Cauchy-Davenport sur les corps finis.
Cet exposé est issu d’un travail en collaboration avec Angelot Behajaina.

We study the large-time $\ell^p$ behavior of transition densities of an isotropic random walk in homogeneous trees, which are infinite, connected, acyclic graphs in which every vertex has the same degree, and can be thought as discrete counterparts of hyperbolic space. Caloric functions of interest are then convolutions of these transition densities with a finitely supported initial condition, and we are interested in their large time behavior in $\ell^p$ norm.

For each $p \in [1, \infty]$, we introduce a notion of a $p$-mass function and prove that caloric functions with compactly supported initial data, asymptotically decouple as the product of this mass function the transition density. Using tools of Fourier analysis available on such graphs, we show that this function even boils down to a constant, still depending on $p$, if the initial condition is radial, that is, depends only on the distance to the origin. Determining the spatial concentration of the densities in $p$-norm plays an important role, in turn clarifying the interplay between the exponential volume growth of the graph and heat diffusion. The results extend to affine buildings, even exotic ones beyond the Bruhat–Tits framework.

Joint work with B. Trojan.