# Séminaire PAS

Organisateurs : Erwan Saint-Loubert Bié et Christoph Kriegler
Les exposés ont lieu le mardi à 14h45 en salle 218 du bâtiment de mathématiques (consulter le plan d'accès au laboratoire).
Agenda global au format

### Mars 2023

• Mardi 28 mars 2023 - Athanasios Kouroupis (NTNU Trondheim, Norvège)

Beurling primes and Hardy spaces of Dirichlet series

Given an arbitrary increasing sequence q = {qn},
1 < qn → ∞, such that {log qn}
is linearly independent over the rationals, we will denote by Nq = {νn} the set of numbers that
can be written (uniquely) as finite products with factors from q, ordered in an increasing
manner. The numbers qn are known as Beurling primes, and the numbers νn are Beurling
integers. The corresponding generalized Dirichlet series are of the form
f (s) = ∑ an vn^(-s).

The talk comprises a first part on Bohr's theorem
(joint work with Frederik Broucke and Karl–Mikael Perfekt)
and a second part on composition operators on
the Dirichlet Hardy space H2.

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• Mardi 07 mars 2023 - Carlos Gomez Cabello, Universidad de Sevilla

Semigroups of composition operators on Banach spaces of Dirichlet series

In this talk, we shall consider continuous semigroups of analytic functions $\{\Phi_t\}_{t\geq0}$ in the so-called \emph{Gordon-Hedenmalm class} $\mathcal{G}$, that is, the family of analytic functions $\Phi:\mathbb{C}_+\to\mathbb{C}_+$ giving rise to bounded composition operators in the Hardy space of Dirichlet series $\mathcal{H}^2$. We will show the existence of a one-to-one correspondence between continuous semigroups $\{\Phi_{t}\}_{t\geq0}$ in the class $\mathcal G$ and strongly continuous semigroups of composition operators $\{T_t\}_{t\geq0}$, where $T_t(f)=f\circ\Phi_t$, $f\in\mathcal{H}^2$. Then, we will characterise the infinitesimal generators of continuous semigroups in the class $\mathcal G$ as the Dirichlet series sending $\mathbb C_{+}$ into its closure. We will extend these results to the range $p\in[1,\infty)$ and for the case $p=\infty$, we shall prove that there are no non-trivial strongly continuous semigroups of composition operators in $\mathcal{H}^\infty$. If time permits, a brief comment will be made regarding the symbols of the bounded composition operators in the algebra $\mathcal{A}(\mathbb C_+)$ of Dirichlet series as well as the interplay between the corresponding semigroups of these symbols and the associated semigroups of composition operators.

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### Novembre 2022

• Mardi 15 novembre 2022 - Rémi BOUTIN, Université Paris Cité

Embedded Topics in the Stochastic Block Model

Communication networks such as emails or social networks are now ubiquitous and their analysis has become a strategic field. In many applications, the goal is to automatically extract relevant information by looking at the nodes and their connections. Unfortunately, most of the existing methods focus on analysing the presence or absence of edges and textual data is often discarded. However, all communication networks actually come with textual data on the edges. In order to take into account this specificity, we consider in this paper networks for which two nodes are linked if and only if they share textual data. We introduce a deep latent variable model allowing embedded topics to be handled called ETSBM to simultaneously perform clustering on the nodes while modelling the topics used between the different clusters. ETSBM extends both the stochastic block model (SBM) and the embedded topic model (ETM) which are core models for studying networks and corpora, respectively. The inference is done using a variational-Bayes expectation-maximisation algorithm combined with a stochastic gradient descent. The methodology is evaluated on synthetic data and on a real world dataset.

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• Mardi 08 novembre 2022 - Claire BRECHETEAU, Université de Rennes 2, 14h00

Approcher des données par une union de boules ou d'ellipsoïdes et partitionnement

Dans cet exposé, il sera question de construire un proxy de le fonction
distance à un compact, à partir d'un nuage de points générés sur ce
compact, avec du bruit.
Ce proxy sera construit à partir d'un critère de type k-means, avec une
divergence de Bregman. Ses sous-niveaux seront des unions de boules. Je
présenterai l'utilisation de ce proxy à des fins de partitionnement de
données.

Il s'agit de travaux publiés dans :
- Claire Brécheteau and Clément Levrard, A k-points-based distance for
robust geometric inference. Bernoulli 2020, Vol. 26, No. 4, 3017-3050
- Claire Brécheteau and Aurélie Fischer and Clément Levrard, Robust
Bregman Clustering. Annals of Statistics 2021, Vol. 49, No. 3, 1679-1701

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### Octobre 2022

• Vendredi 14 octobre 2022 - GUNNAR TARALDSEN

Mathematics of improper priors and posteriors

Taraldsen et al (2022) have recently proved a generalization of Bayes theorem that includes both
improper priors and improper posteriors. This theory links the Renyi (1955) conditional probability
spaces with the theory of disintegration presented by Bourbaki (1959). In this talk, the Basu
theorem, the factorization theorem, and optimal frequentist inference based on Bayesian or fiducial
arguments are reconsidered in this more general context

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