## Les événements de décembre 2017

• ### Vendredi 15 décembre 2017 15:15-16:15 - Sushma Kumari - Kyoto University

Séminaires doctorant : Universal consistency of $k$-NN rule in $\sigma$-finite dimensional metric spaces

Résumé :
The $k$-nearest neighbour (NN) rule is one of the important learning rules in machine learning. We start with the definition of the $k$-NN rule and universal consistency. Charles Stone proved the universal consistency of $k$-NN rule. The essential ingredient for Stone’s theorem was the so-called geometric Stone’s lemma but the proof of the geometric Stone’s lemma is restricted to finite dimensional normed spaces only.
In this work, we try to extend the geometric Stone’s lemma to the $\sigma$-finite dimensional metric spaces and reprove the universal consistency using the generalized Stone’s theorem. Further questions related to this work are also discussed.

Lieu : 316B

• ### Mardi 12 décembre 2017 13:45-15:00 - Monika Malczak - Greifswald

Lévy processes on braided *-bialgebras

Résumé : In order to use Majid’s Bosonization/Symmetrization of braided bialgebras for quantum
probability theory, we give an extension to braided *-bialgebras, which is based on a
joint work by Franz, Schott and Schürmann [1]. Furthermore we investigate a connection between quantum Lévy processes on braided *-bialgebras and their symmetrization. We present a realization as solution of quantum stochastic differential equations for Brownian motions on such braided structures, which can be considered as multi-dimensonal analogues of the Azéma martingales.
References
[1] Franz, Uwe ; Schott, René ; Schürmann, Michael : Lévy Processes and Brownian Motion on Braided Spaces. In : Franz, Uwe : The Theory of Quantum Lévy Processes, Chapter 5 of habilitation thesis.
2003. Available at https://arxiv.org/abs/math/0407488.

• ### Mardi 12 décembre 2017 15:00-16:00 - Philipp Varso - Greifswald

Central Limit Theorem for General Universal Products

Résumé : To model independence in quantum probability theory one uses so-called universal products, which are in general described by tensor categories of algebraic quantum probability spaces $(A,\varphi)$. In [1] Muraki has shown how to classify such products and in particular obtained that only -five universal products exist. But there are examples which do not -fit into Muraki’s framework, for instance if one wants to deal with a tuple of linear functionals $\varphi^(i)$ on the algebra $A$ like it has been done in the case of $c$-freeness by Bo\.zejko and Speicher in [2].
The case of bi-freeness by Voiculescu [3], where in particular the underlying algebra $A$ is isomorphic to the free product of two algebras $A^(1)$ and $A^(2)$, is also not covered by Muraki’s classi-cation. In [4] Schürmann and Manzel present a uni-fied approach to cumulants, which includes the above concepts of independence. This is achieved by considering a certain category of algebraic non-commutative probability spaces, denoted by $\rm algP_d,m$, which consist of an $m$-tuple of subalgebras and a $d$-tuple of linear functionals and therefore allows to investigate $(d,m)$-independence induced by a so-called u.a.u.-product.
The -first noncommutative version of a central limit theorem dates back to von Waldenfels [5]. In this talk we want to present a noncommutative version of a central limit theorem for a u.a.u.-product in $\rm algP_d,m$, where we make use of the so-called Lachs functor [6], which operates between certain tensor categories.
References
[1] Muraki, Naofumi : The -five independences as natural products. In-n. Dimens. Anal. Quantum Probab. Relat. Top. 6.3 (2003), 337-371.
[2] Bo\.zejko, Marek ; Speicher, Roland : $\psi$ -independent and symmetrized white noises. In : Quantum probability \& related topics. QP-PQ, VI, 219-236. World Sci. Publ., River Edge, NJ, 1991.
[3] Voiculescu, Dan-Virgil : Free probability for pairs of faces I. Comm. Math. Phys. 332.3 (2014), 955-980.
[4] Manzel, Sarah ; Schürmann, Michael : Non-Commutative Stochastic Independence and Cumulants. Preprint arXiv:1601.06779 (2016), 42 pages. To appear in IDAQP 20.2 (2017).
[5] von Waldenfels, W. : An algebraic central limit theorem in the anti-commuting case. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42.2 (1978), 135-140.
[6] Lachs, Stephanie : A New Family of Universal Products and Aspects of a Non-Positive Quantum Probability Theory. PhD thesis. Ernst-Moritz-Arndt-Universität Greifswald, 2015.

• ### Mardi 19 décembre 2017 13:45-15:00 - Ignacio Vergara - ENS Lyon

Résumé : Les multiplicateurs de Schur sont des fonctions à deux variables sur un ensemble $X$ qui définissent des opérateurs bornés sur $B(\ell_2(X))$ par multiplication des coefficients matriciels. Lorsque l’ensemble $X$ est un graphe, on peut étudier le cas particulier des multiplicateurs radiaux, c’est-à-dire, des fonctions que ne dépendent que de la distance entre chaque paire de sommets.
Les multiplicateurs radiaux sur un arbre homogène ont été caractérisés par Haagerup, Steenstrup et Szwarc en termes de certaines matrices de Hankel. Dans cet exposé, je présenterai des extensions de ce résultat à des produits d’arbres, des produits de graphes hyperboliques et aux complexes cubiques CAT(0) de dimension finie.

• ### Du 11 au 14 décembre 2017 -

Conférence "Equations aux dérivées partielles et semi-groupes"

Lieu : Centre International de Séjour (CIS), Besançon

• ### Jeudi 7 décembre 2017 20:30-22:00 -

Concert "Musique et mathématiques" : Anamorphose sonore (Besançon)

Lieu : FRAC, Cité des Arts, 2 passage des arts, Besançon

• ### Jeudi 7 décembre 2017 16:30-18:00 -

Conseil de Laboratoire

Lieu : Salles 316B et 316Bbis

• ### Jeudi 14 décembre 2017 14:00-16:00 -

Soutenance de thèse de Adrien FAIVRE

Lieu : Amphi B (UFR ST)