We discuss the method of picking a convenient coordinate system adapted to vector fields. Let $X_1, …, X_q$ be either real or complex $C^1$ vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g. $C^m$ or $C^\infty$, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as « sub-Hermitian ».

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.

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.

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.

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.

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.

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.)

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.

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].

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 introduit une version quantitative de la cohomologie polynomiale des groupes et on montre qu’elle coïncide avec la cohomologie de groupes classique sous des hypothèses de remplissage polynomial. Dans cet exposé on présentera une ou deux applications de ce résultat, en fonction du temps.
1) On montre que les nombres de Betti des groupes nilpotents sont invariants par équivalence mesurable L^p, à partir d’un p assez grand explicite.
2) On obtient des nouvelles annulations de cohomologie unitaire et banachique pour des réseaux non uniformes dans des groupes de Lie simples de rang 1.

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).

In 1996 Gilles Carron introduced the notion of nonparabolic operators at infinity and showed that such operators are Fredholm in the usual sense. The question that we could ask ourselves is: can we extend the notion of Carron to $\Gamma$-operators on Galois coverings and still have a notion of Fredholmness upstairs ?

In this talk we will first introduce the notion of an $\mathcal{N}\Gamma$-Hilbert space as introduced by Wolfgang Lück in 1997. Then, we will introduce the notion of $\Gamma$-Fredholm operators defined between $\mathcal{N}\Gamma$-Hilbert spaces. And finally, the goal of the presentation is to define nonparabolic $\Gamma$-near infinity operators and show how they induce $\Gamma$-Fredholmness on admissible domains.

La construction des nombres surréels de Conway donne également une nouvelle approche aux nombres réels, que je qualifierais de « quantique », opposée à  l’approche « usuelle », que je qualifierais de « classique ». J’essaierai d’expliquer cette opposition « quantique-classique » en la mettant dans le contexte de la théorie des ensembles (je résumerai en partie mon cours « Ensembles et nombres » que je donne actuellement dans le cadre de notre école doctorale). Cette séance se veut plus un forum de discussion qu’une présentation formelle de résultats : le sujet du « quantique » est pour ainsi dire la musique de fond du travail pour beaucoup d’entre nous, et il me semble intéressant de poser et de discuter des questions sur ce fond. 

[Exposé en ligne diffusé dans la salle de séminaire]

A common question for geometric structures is that of rigidity. Infinitesimally rigidity is often controlled by the vanishing of a cohomology group. A common question then becomes when infinitesimal rigidity actually implies rigitidy. In this talk we discuss the case of regular foliations.

In the first part we define the terms involved and give an overview of the current state of the art. Moreover, we highlight a relation with the rigidity of group actions.

One encounters two issues in the literature. Many results require the foliation to have compact leaves and there is a general lack of examples.

In the second part of the talk, we take a step towards addressing those issues, by outlining a construction of infinitesimally rigid foliations with dense leaves. This is based on joint work with Stephane Geudens.

In this talk I will introduce Quantum Field Theory, its Path Integral formulation, and Segal’s Axioms. Then I will indicate some recent progress on rigorous construction of some concrete models using probability. After that I slightly extend Segal’s framework and show how it can be related to a useful physical quantity called “entanglement entropy”, where I suggest a geometric way of rigorously deriving relevant formulae based on recent works of the speaker and B. Estienne (2501.19014).