We give explicit upper and lower bounds on the size of the coefficients of the modular polynomials $\Phi_N$ for the $j$-invariant. These bounds make explicit the best previously known asymptotic bounds. This is joint work with Fabien Pazuki and Florian Breuer.

Exposé du rencontre OAK2 (Observables – Actions – Quantification)

We discuss factorization problems for representations of a real Lie group G. First we discuss a factorization theorem of Dixmier-Malliavin type for the space of analytic vectors $E^{\omega}$ for representations of G on, for example, a Banach space E: There exists a natural algebra of superexponentially decreasing analytic functions A(G), such that $E^{\omega} = A(G) * E^{\omega}$. Such theorems can be deduced from simple properties of the wave equation on G, which provides detailed information about functions of the Laplacian. We then formulate a factorization theorem for real reductive groups which implies the Helgason conjecture, and we outline a new and elementary proof. (joint work with Krötz and Lienau, resp. Krötz, Kuit and Schlichtkrull)

This talk, based on joint work with Francesco Cattafi, presents an extension of the classical correspondence between vector bundles and principal bundles to the Lie groupoid setting. To achieve this, we introduced the notion of a PB-groupoid with a structural Lie 2-groupoid, establishing a correspondence between VB-groupoids and PB-groupoids. This framework enhances our understanding of VB-groupoids, which play a fundamental role in Poisson geometry, and provides a natural perspective on the tangent space of smooth symmetries.

La variété de Rankin-Selberg est la variété homogène sphérique X=(GL_n x GL_{n+1}) / GL_n, définie sur un corps K. Dans cet exposé, on s’intéressera à différentes notions de spectres associées à X. Pour K un corps local de caractéristique zéro, il s’agira de déterminer l’ensemble des sous-représentations irréductibles de l’espace des fonctions lisses sur X. Je donnerai alors la réponse pour celles qui sont unitaires, complétant des travaux de K.Y. Chan, C. Chen et R. Chen. Pour K un corps global, j’expliquerai comment le spectre de X s’interprète en termes d’une décomposition spectrale automorphe de séries thêta, dont je donnerai ensuite une description explicite. Dans les deux cas, la méthode repose sur l’observation que les résidus des intégrales Zeta de Rankin-Selberg produisent des périodes sur le spectre non-générique.

A Harish-Chandra module is the algebraic “skeleton” of an irreducible continuous representation of a real reductive group. For a given Harish-Chandra module there are typically many continuous representations that correspond to it. In this talk we will explore to what extend continuous representations (in particular on Banach spaces) with the same Harish-Chandra module may differ from each other, and discuss some relations to automorphic forms. (This is joint work with Joseph Bernstein, Pritam Ganguly, Bernhard Krötz and Eitan Sayag.)

Montgomery (1973) suggested an approach to study the pair correlation of nontrivial zeros of the Riemann zeta-function, and proved the corresponding asymptotic formula within a limited range assuming the Riemann Hypothesis (RH). The extended behavior remains a conjecture which implies the famous Pair Correlation Conjecture (PCC) for these zeros. In my previous work with Siegfred Alan C. Baluyot, Daniel Alan Goldston, and Caroline L. Turnage-Butterbaugh, we have showed how to remove RH in Montgomery’s pair correlation method and recover known results on the proportion of simple zeros under hypotheses weaker than RH. We have in addition obtained the proportion of zeros lying on the critical line, which we simply call critical zeros for brevity. Getting results on critical zeros is only achieved since we do not assume RH. We also recently noticed that these proportions can be further improved if we take further advantage of the feature that we “may” have zeros off the critical line.
In follow-up work with Daniel Goldston, Junghun Lee and Jordan Schettler, we showed that PCC without RH implies that asymptotically 100% of the zeros are simple and critical, thus RH is asymptotically true. We remark that our method also works with other pair correlation conjectures. In this talk, I would like to briefly introduce these results and our key ideas.

Factorization homology can be used to assign functorial invariants to surfaces, starting from a braided monoidal category. If this category is obtained by deforming a symmetric monoidal category, this deformation can be tracked through factorization homology. Applying this to the cases of the Drinfeld-Jimbo quantum groups and Drinfeld’s quasi-triangular quasi-Hopf algebras, we obtain the  Fock-Rosly and the Alekseev-Malkin-Meinrenken bivectors as well as their quantizations, in terms of fusion and skein theory. Based on joint work with Eilind Karlsson, Corina Keller and Lukas Mueller [arxiv:2410.12516].

Dans cet exposé, on étudie les courses de nombres premiers entre résidus et non-résidus quadratiques modulo q. Les nombres de Skewes généralisés désignent les premières valeurs pour lesquelles les résidus quadratiques devancent les non-résidus. Je présenterai un travail en collaboration avec A. Bailleul et T. Untrau dans lequel nous réfutons une conjecture de Fiorilli portant sur la taille de ces nombres. De plus, sous l’hypothèse de Riemann généralisée, ainsi qu’une hypothèse d’indépendance linéaire effective, nous établissons des bornes supérieures pour ces nombres en fonction de q.

Let G be a Lie group whose Lie algebra carries an Ad-invariant, nondegenerate symmetric pairing. Then its classifying space has a 2-shifted symplectic structure. In the first part of the talk, I will present concrete models for this structure coming from differential geometry and mathematical physics. In the second part, I will study the analogue of Lagrangian submanifolds and show their relation to Poisson and Dirac structures. This talk is based on joint work with Chenchang Zhu, and with Daniel Álvarez and Henrique Bursztyn.

On présentera dans un premier temps le cadre mathématique développé pour la conception du chiffrement symétrique moderne, cadre basé sur la notion de transformée de Fourier booléenne. Dans une seconde partie, on présentera quelques résultats qu’on a obtenus ces dernières années sur les attaques par canaux cachés sur ces chiffrement et basés aussi sur une approche spectrale (Travaux en collaboration avec H. Belbachir, S. Guilley, Ph. Guillot, M. Mahar et M. Ouladj).