Some duality properties for induced representations of enveloping algebras involve the character $Trad_{\goth g}$. We extend them to deformation Hopf algebras $A_{h}$ of a noetherian Hopf $k$-algebra $A_{0}$ satistying $Ext^{i}_{A_{0}}(k, A_{0})=\{0\}$ except for $i=d$ where it is isomorphic to $k$. These duality properties involve the character of $A_{h}$ defined by right multiplication on the one dimensional free $k[[h]]$-module $Ext^{d}_{A_{h}} (k[[h]], A_{h})$. In the case of quantized enveloping algebras, this character lifts the character $Trad_{\goth g}$. We also prove Poincar{\'e} duality for such deformation Hopf algebras in the case where $A_{0}$ is of finite homological dimension. We explain the relation of our construction with quantum duality.

Sophie Chemla

The notion of double Lie algebroid was defined by M. Van den Bergh and was illustrated by the double quasi Poisson case. We give new examples of double Lie algebroids and develop a differential calculus in that context. We recover the non commutative de Rham complex and the double Poisson-Lichnerowicz cohomology (Pichereau-vanWeyer) as particular cases of our construction.

Sophie Chemla

We study integral theory for left (or right) Hopf left bialgebroids. Contrary to Hopf algebroids, the latter ones don't necessary have an antipode $S$ but, for any element $u$, the elements $u_{(1)} \otimes S(u_{(2)})$ (or $u_{(2)}\otimes S^{-1}(u_{(1)})$ ) does exist. Our results extend those of G. B\"ohm who studied integral theory for Hopf algebroids. We make use of recent results about left Hopf left bialgebroids. We apply our results to the restricted enveloping algebra of a restricted Lie Rinehart algebra.

Sophie Chemla

Let $k$ be a field of positive characteristic $p>2$. We prove a duality property concerning the kernel of coinduced representations of Lie superalgebras. This property was already proved by M. Duflo for Lie algebras in any characteristic under more restrictive finiteness conditions. It was then generalized to Lie superalgebras in characteristic 0 in previous works of the author. In a second part of the article, we study the links between coinduced representations and induced representations in the case of restricted Lie superalgebras.

Sophie Chemla

We present large deviations principles for the moments of the empirical spectral measure of Wigner matrices and empirical measure of $\beta$-ensembles in three cases : the case of Wigner matrices without Gaussian tails, that is Wigner matrices whose entries have tail distributions decreasing as $e^{-ct^{\alpha}}$, for some constant $c>0$ and with $\alpha \in (0,2)$, the case of Gaussian Wigner matrices, and the case of $\beta$-ensembles associated with a convex potential with polynomial growth.

Fanny Augeri

We generalize probabilistic amplitude shaping (PAS) with binary codes to the case of non-binary codes defined over prime finite fields. Firstly, we introduce probabilistic shaping via time sharing where shaping applies to information symbols only. Then, we design circular quadrature amplitude modulations (CQAM) that allow to directly generalize PAS to prime finite fields with full shaping.

Joseph J. Boutros Fanny Jardel Cyril Méasson

We propose a general framework to study language emergence through signaling games with neural agents. Using a continuous latent space, we are able to (i) train using backpropagation, (ii) show that discrete messages nonetheless naturally emerge. We explore whether categorical perception effects follow and show that the messages are not compositional.

Nur Geffen Lan Emmanuel Chemla Shane Steinert-Threlkeld

In this article, $X$ will denote a ${\cal C}^{\infty}$ manifold. In a very famous article, Kontsevich showed that the differential graded Lie algebra (DGLA) of polydifferential operators on $X$ is formal. Calaque extended this theorem to any Lie algebroid. More precisely, given any Lie algebroid $E$ over $X$, he defined the DGLA of $E$-polydifferential operators, $\Gamma (X, ^{E}D^{*}_{poly})$, and showed that it is formal. Denote by $\Gamma (X, ^{E}T^{*}_{poly})$ the DGLA of $E$-polyvector fields. Considering $M$, a module over $E$, we define $\Gamma (X, ^{E}T_{poly}^{*}(M))$ the $\Gamma (X, ^{E}T^{*}_{poly})$-module of $E$-polyvector fields with values in $M$. Similarly, we define the $\Gamma (X, ^{E}D^{*}_{poly})$-module of $E$-polydifferential operators with values in $M$, $\Gamma (X, ^{E}D^{*}_{poly}(M))$. We show that there is a quasi-isomorphism of $L_{\infty}$-modules over $\Gamma (X, ^{E}T^{*}_{poly})$ from $\Gamma (X, ^{E}T^{*}_{poly}(M))$ to $\Gamma (X, ^{E}D^{*}_{poly}(M))$. Our result extends Calaque 's (and Kontsevich's) result.

Sophie Chemla

In this essay, I argue for the following theses. First, Kummer's concept of ''ideal prime factors of a complex number'' was inspired by Poncelet's introduction of ideal elements in geometry as well as by the reconceptualization that Michel Chasles put forward for them in 1837. In other words, the idea of ideal divisors in Kummer's ''theory of complex numbers'' derives from the introduction of ideal elements in the new geometry. This is where the term ''ideal'' comes from. Second, the introduction of ideal elements into geometry and the subsequent reconceptualization of what was in play with these elements were linked to philosophical reflections on generality that practitioners of geometry in France developed in the first half of the 19${}^{\rm th}$ century in order to devise a new approach to geometry, which would eventually become projective geometry. These philosophical reflections circulated as such and played a key part in the advance of other domains, including in Kummer's major innovation in the context of number theory.

Karine Chemla

We present two approaches for next step linear prediction of long memory time series. The first is based on the truncation of the Wiener-Kolmogorov predictor by restricting the observations to the last $k$ terms, which are the only available values in practice. Part of the mean squared prediction error comes from the truncation, and another part comes from the parametric estimation of the parameters of the predictor. By contrast, the second approach is non-parametric. An AR($k$) model is fitted to the long memory time series and we study the error made with this misspecified model.

Fanny Godet