• Zhan Shi

AMSS, Chinese Academy of Sciences

Title: The Derrida-Retaux model

Abstract: The Derrida-Retaux model is a toy version of hierarchical renormalisation systems in statistical physics. It brings in a number of very simple mathematical questions, most of which remain unanswered so far. I am going to make elementary discussions on some of these questions.

• Nils Berglund

University of Orleans

Title: Metastable dynamics of Markov processes

Abstract: A Markov process is called metastable if it spends long time spans in states different from its equilibrium state. In these lectures, we will mainly be interested in diffusion processes defined by stochastic differential equations (SDEs), but we will use Markov chains in order to introduce some of the concepts in a simpler setting, and discuss some extensions to stochastic PDEs.
The topics include:

• The trace process, and its applications to eigenvalue estimation.
• Random Poincaré maps for SDEs with periodic orbits.
• Quasistationary distributions.
• The potential-theoretic approach to reversible Markov processes.
• Applications to metastable SPDEs.

Notes part 1 - Notes part 2 - Notes part 3

and

Title: Science outreach with a YouTube channel: a personal experience

Abstract: The channel https://www.youtube.com/c/NilsBerglund/ contains simulations of mathematical and physical systems intended for a general audience of science enthusiasts. It was relatively little known until April 2021, when it suddenly encountered unexpected success. In this presentation, I will try to give some reasons for this success, and discuss some aspects of science outreach I believe are important.

Slides of this talk

• Maximilian Engel

Free University Berlin

Title: Lyapunov exponents in random dynamical systems: synchronization, chaos and bifurcations

Abstract: I will introduce the framework of Random Dynamical Systems (RDS), bringing together notions from dynamical systems theory and stochastic analysis and calculus, with a focus on stability and bifurcation theory. In my first lecture, I will discuss basic elements of RDS theory and introduce important examples. The second lecture will present essential results on the stability and geometry of RDS, yielding Lyapunov exponents and associated invariant subspaces as core objects in Multiplicative Ergodic Theory. The third lecture will show how Lyapunov exponents help to detect chaotic or synchonizing structures (and transitions between them), touching upon parts of my research in recent years.

Lecture notes - Lecture on bifurcations

Lecture Intro - Lecture 1 - Lecture 2

• Richard Sowers

University of Illinois at Urbana-Champaign

Title: Big data and mobility: a mathematician looks at traffic

Abstract: We discuss some recent problems and models related to urban mobility problems and related data. We consider some problems of unsupervised learning (i.e., nonnegative matrix factorization) in datasets of traffic. We in particular want to identify and understand some natural behaviors and understand how these techniques can lead to an understanding of complexity in traffic. We secondly look at some routing and safety and try to answer the question “how much time would you give up to be safer?”. We thirdly look at some questions of connectedness of congestion, using persistent homology.

Our discussion will be focussed on mathematical analyses of real datasets (rather than theorem-proof).

and

Title: Small-noise asymptotics of car-following models

Abstract: A number of car-following models have been developed to model the behavior of platoons of vehicles, often with the goal of informing the development of connected and autonomous vehicles. A number of these models have singularities which are intended to prevent collision. We look at some simple questions of how small noise can influence these equations and develop a boundary-layer analysis showing that small noise can lead to collisions, but (unsurprisingly) in large time interals.

Please register for the winter school. After the registration you will receive the ZOOM link to be able to participate via email.