Various mathematical problems are challenged by the fact they involve functions of a very large number of variables. Such problems arise naturally in learning theory, partial differential equations or numerical models depending on parametric or stochastic variables. They typically result in numerical difficulties due to the so-called "curse of dimensionality". We shall first discuss how these difficulties may be theoretically handled in the context of stochastic-parametric PDE’s through the concept of sparse polynomial approximation. We shall then focus on a class of concrete algorithms based on least-squares fitting that provably achieve the convergence properties of these approximations.

We discuss Approximation in High Dimensions, focusing on entropy widths, optimal recovery, model classes, etc. We will round out the discussion with polynomial approximation in high dimensions and neural networks, which is only peripherally high dimensional.

Dr. Kutyniok’s main research topics include applied harmonic analysis, compressed sensing, data science, deep learning, frame theory, high dimensional data analysis, imaging science, inverse problems, machine learning, numerical analysis of partial differential equations, sparse approximation.

Collective dynamics is driven by alignment that tend to self-organize the crowd, and by different external forces that keep the crowd together. Prototype models based on environmental averaging are found in opinion dynamics of human networks, self-organization of biological organisms, and rendezvous of mobile systems. Different emerging equilibria are self-organized into parties, flocks, tissues, etc. I will overview recent results of collective dynamics driven by different "rules of engagement". I begin with a survey of several classical models of agent-based systems, and follow with two fundamental questions that arise in the context of such systems, namely --- their large time behavior and their large crowd dynamics. In particular, I address the question how short-range interactions lead, over time, to the emergence of long-range patterns, comparing geometric vs. topological interactions. I conclude with a general framework which describes the competition between pairwise alignment with external forcing.

We will discuss kinetic models with singular hydrodynamic limits. The asymptotic preserving (AP) schemes aim to provide a universal solver for both the full system and the limit system, in the sense that the stability does not depend on the parameter. For systems that have mono-kinetic singular limits, standard AP schemes lose accuracy when the parameter is close to the limit. To overcome such difficulty, we introduce a velocity scaling method that transforms the singular limit to a non-singular one, and build AP schemes on the transformed systems.