Dunkl operators are commuting differential-reflection operators on $\mathbb R^n$ which generalize the usual partial derivatives by additional reflection terms. They are associated with some underlying finite reflection group and its root system. Dunkl operators allow a rich harmonic analysis, including a generalization of the Fourier transform with many similar properties. In this talk, we give a basic introduction to Dunkl theory and explain some uncertainty principles related to them.

Large random matrices tend to exhibit universal fluctuations. Beyond the well-known Wigner-Dyson and Tracy-Widom eigenvalue distributions, we overview other universality results for Hermitian and non-Hermitian matrices. We discuss the emergence of normal distribution involving eigenvectors, especially the random matrix version of quantum unique ergodicity. We also explain why results on non-Hermitian random matrices are much harder than their Hermitian counterparts and highlight our new methods to tackle them.

The new wave of artificial intelligence is impacting industry, public life, and the sciences in an unprecedented manner. However, one current major drawback is the lack of reliability as well as the enormous energy consumption of AI systems.
In this talk we will take a mathematical viewpoint towards this problem, showing the power of such approaches in the field of AI. We will first provide an introduction into this vibrant research area. We will then discuss key mathematical research directions toward reliability of AI and survey some results on, in particular, performance guarantees as well as explainability. This is followed by a discussion of fundamental limitations also in terms of sustainability. Our mathematical viewpoint will lead us naturally to the necessity of novel (analog) hardware such as neuromorphic computing and the related model of spiking neural networks. We will finish with some very recent mathematical results for spiking neural networks.

Metric measure spaces with synthetic Ricci bounds have attracted great interest in recent years, accompanied by spectacular breakthroughs and deep new insights. In this survey, I will provide a brief introduction to the concept of lower Ricci bounds as introduced by Lott-Villani and myself, and illustrate some of its geometric, analytic, and probabilistic consequences, among them Li-Yau estimates, coupling properties for Brownian motions, sharp functional and isoperimetric inequalities, rigidity results, and structural properties like rectifiability. In particular, I will explain its crucial interplay with the heat flow and its link to the curvature-dimension condition formulated in functional-analytic terms by Bakry-Émery. This equivalence between the Lagrangian and the Eulerian approach then will be further explored in various recent research directions.

In recent years, data science, machine learning, and artificial intelligence have emerged both as entirely new majors and as electives within many mathematics, physics, or computer science curricula. While sometimes perceived as competitors to traditional areas of mathematics, the core of many data science or machine learning methods is of course classical linear algebra, analysis, probability theory, and, especially in the context of neural networks, functional analysis. In my talk I will outline the latter connection. In particular, I will show that Cybenko's 1990s theorem on the so-called expressivity of neural networks fits perfectly into a classical functional analysis course---with the advantage that all the prerequisites for a full proof will be available.

We discuss the algebraic geometry of low rank symmetric matrices that have zero blocks along the main diagonal. In theoretical physics, these arise as Gram matrices for kinematic variables in quantum field theories.

On the full, flat space, Egidi-Veselic and Wang-Wang-Zhang-Zhang established that given a measurable set $\omega$, the following three properties are equivalent: (i) thickness: $\omega$ has equidistributed measure in every ball at some scale, (ii) a spectral inequality bounding the $L^2$ norm of functions with compact spectral support by their $L^2$ norm on $\omega$, and (iii) observability of the heat semigroup from $\omega$. The aim of the talk will be to discuss how these properties interact in the manifold setting and how to generalize the equivalence to non-compact manifolds, under relevant bounds on curvature. I will focus on ongoing work with Alix Deleporte and Jean Lagacé, in which we aim to prove that (i) implies (ii) on manifolds with bounded curvature, with a constant that depends only on curvature bounds rather than the metric itself. The proof crucially relies on elliptic estimates by Logunov and Malinnikova, combined with tools from geometric analysis.

We consider a class of dynamical systems that are described by time and space dependent partial differential equations. This class fits perfectly the port-Hamiltonian framework. We cover the wave equation, Maxwell's equations, the Kirchhoff-Love plate model, piezo-electromagnetic systems and many more. Our goal is to characterize boundary conditions that make the systems passive (the energy of solutions decays). This is done by constructing a boundary triple for the underlying differential operator. As a by-product we develop the theory of quasi Gelfand triples, which enables us to regard L2 boundary conditions even though the ``natural`` boundary spaces are neither included nor covering L2.