The main question that we will investigate in this talk is: what does the spectrum of a quantum channel typically looks like? We will see that a wide class of random quantum channels generically exhibit a large spectral gap between their first and second largest eigenvalues. This is in close analogy with what is observed classically, i.e. for the spectral gap of transition matrices associated to random graphs. In both the classical and quantum settings, results of this kind are interesting because they provide examples of so-called expanders, i.e. dynamics that are converging fast to equilibrium despite their low connectivity. We will also present implications in terms of typical decay of correlations in 1D many-body quantum systems. If time allows, we will say a few words about ongoing investigations of the full spectral distribution of random quantum channels. This talk will be based on: arXiv:1906.11682 (with D. Perez-Garcia), arXiv:2302.07772 (with P. Youssef) and arXiv:2311.12368 (with P. Oliveira Santos and P. Youssef).[-]

In this talk, we discuss functional inequalities and gradient bounds for the heat kernel on the Vicsek set. The Vicsek set has both fractal and tree structure, whereas neither analogue of curvature nor obvious differential structure exists. We introduce Sobolev spaces in that setting and prove several characterizations based on a metric, a discretization or a weak gradient approach. We also obtain $L^{p}$ Poincaré inequalities and pointwise gradient bounds for the heat kernel. These properties have important applications in harmonic analysis like Sobolev inequalities and the Riesz transform. Moreover, several of our techniques and results apply to more general fractals and trees.[-]

About fifteen years ago, Patrick Gérard and I introduced the cubic Szegö equation$$\begin{aligned}i \partial_{t} u & =\Pi\left(|u|^{2} u\right), \quad u=u(x, t), \quad x \in \mathbb{T}, t \in \mathbb{R} \\u(x, 0) & =u_{0}(x) .\end{aligned}$$Here $\Pi$ denotes the Szegö projector which maps $L^{2}(\mathbb{T})$-functions into the Hardy space of $L^{2}(\mathbb{T})$-traces of holomorphic functions in the unit disc. It turned out that the dynamics of this equation were unexpected. This motivated us to try to understand whether the cubic Szegö equation is an isolated phenomenon or not. This talk is part of this project.
We consider a family of perturbations of the cubic Szegö equation and look for their traveling waves. Let us recall that traveling waves are particular solutions of the form$$u(x, t)=\mathrm{e}^{-i \omega t} u_{0}\left(\mathrm{e}^{-i c t} x\right), \quad \omega, c \in \mathbb{R}$$We will explain how the spectral analysis of some operators allows to characterize them.
From joint works with Patrick Gérard.[-]

For an open set $\Omega \subset \mathbb{R}^{d}$ with an Ahlfors regular boundary, solvability of the Dirichlet problem for Laplaces equation, with boundary data in $L^{p}$ for some $p<\infty$, is equivalent to quantitative, scale invariant absolute continuity (more precisely, the weak- $A_{\infty}$ property) of harmonic measure with respect to surface measure on $\partial \Omega$. A similar statement is true in the caloric setting. Thus, it is of interest to find geometric criteria which characterize the open sets for which such absolute continuity (hence also solvability) holds. Recently, this has been done in the harmonic case. In this talk, we shall discuss recent progress in the caloric setting, in which we show that quantitative absolute continuity of caloric measure, with respect to surface measure on the parabolic Ahlfors regular (lateral) boundary $\Sigma$, implies parabolic uniform rectifiability of $\Sigma$. We observe that this result may be viewed as the solution of a certain 1-phase free boundary problem. This is joint work with S. Bortz, J. M. Martell and K. Nyström.[-]

The aim of this project is a deeper investigation of off-diagonal estimates. In the ISem lectures, in Theorem 11.16, it has already been shown that off-diagonal estimates in combination with Sobolev embeddings lead to $L^{p}$-extrapolation for the resolvents of an elliptic operator $L$ in divergence form on $\mathbb{R}^{n}$. More precisely, if $\left|\frac{1}{p}-\frac{1}{2}\right|<\frac{1}{n}$, then there exists $C>0$ such that $\left\|(1+t L)^{-1} u\right\|_{p} \leqslant C\|u\|_{p}$ for all $t>0, u \in L^{p} \cap L^{2}\left(\mathbb{R}^{n}\right)$.
A related (more difficult!) question is for what range of $p \in(1, \infty)$ the norm equivalence $\|\sqrt{L} u\|_{2} \simeq\|\nabla u\|_{2}$ from Theorem 12.1 (the Kato square root property for $L$ !) extrapolates to $L^{p}\left(\mathbb{R}^{n}\right)$. It turns out that there are different ranges of $p$ for the two estimates $\|\sqrt{L} u\|_{p} \lesssim\|\nabla u\|_{p}$ and $\|\nabla u\|_{p} \lesssim\|\sqrt{L} u\|_{p}$. The latter estimate is generally known as $L^{p}$-boundedness of the Riesz transform, and this is what shall be the core of the project.
Starting point of the project is the AMS memoir [1], which starts with an excellent introduction into the topic; you can find a preprint version of the memoir on the arXiv (with different numbering of theorems than in the published version, unfortunately). An important abstract $L^{p}$-extrapolation result is Theorem 1.1 in [1], the application Riesz transforms on $L^{p}$ can be found in Section 4.1. This approach is due to Blunck and Kunstmann [2, 3]. If time permits, we can also study the approach of Shen [4] to Riesz transforms. The precise selection of topics will be decided among the participants of the project.[-]

We investigate the mean-field limit of large networks of interacting biological neurons. The neurons are represented by the so-called integrate and fire models that follow the membrane potential of each neuron and captures individual spikes. However we do not assume any structure on the graph of interactions but consider instead any connection weights between neurons that obey a generic mean-field scaling. We are able to extend the concept of extended graphons, introduced in Jabin-Poyato-Soler, by introducing a novel notion of discrete observables in the system. This is a joint work with D. Zhou.[-]

Let $\mathrm{X}$ be a topological space of holomorphic functions on the open unit disc $D$. The study of the geometry of a space $X$ is centered on the identification of the linear isometries on $\mathrm{X}$, and there is an obvious connection between weighted composition operators and isometries. This connection can be traced back to Banach himself and emphasized by Forelli, El-Gebeily, Wolfe, Kolaski, Cima, Wogen, Colonna and many others. A characterisation is given of all the linear isometries of Hol($\Omega$), the Fr´ echet space of all holomorphic functions on $\Omega$ when $\Omega$ is the unit disc or an annulus, endowed with one of the standard metrics. Further, the larger class of operators isometric when restricted to one of the defining seminorms is identified. This is a joint work with Lucas Oger and Jonathan Partington.[-]

The classical von Neumann inequality provides a fundamental link between complex analysis and operator theory. It shows that for any contraction $T$ on a Hilbert space and any polynomial $p$, the operator norm of $p(T)$ satisfies
$\|p(T)\| \leq \sup _{|z| \leq 1}|p(z)|$
Whereas Andô extended this inequality to pairs of commuting contractions, the corresponding statement for triples of commuting contractions is false. However, it is still not known whether von Neumann's inequality for triples of commuting contractions holds up to a constant. I will talk about this question and about function theoretic upper bounds for $\|p(T)\|$.[-]

Given a separable Banach space $X$ of infinite dimension, we can consider on the algebra $\mathcal{B}(X)$ of continuous linear operators on $X$ several natural topologies, which turn its closed unit ball $B_1(X)=\{T \in \mathcal{B}(X) ;\|T\| \leq 1\}$ into a Polish space - that is to say, a separable and completely metrizable space.
In this talk, I will present some results concerning the "typical" properties, in the Baire category sense, of operators of $B_1(X)$ for these topologies when $X$ is an $\ell_p$-space, with $1 \leq p<+\infty$. One motivation for this study is the Invariant Subspace Problem, which asks for the existence of non-trivial invariant closed subspaces for operators on Banach spaces. It is thus interesting to try to determine if a "typical" contraction on a space $\ell_p$ has a non-trivial invariant subspace (or not). I will present some recent results related to this question.

This talk will be based on joint work with Étienne Matheron (Université d'Artois, France) and Quentin Menet (UMONS, Belgium).[-]