**December 20 - 22, 2021**

Bielefeld University

Faculty of Mathematics

* online* (ZOOM link available after registration)

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Monday 20 | Tuesday 21 | Wednesday 22 | ||

9.00 - 10.00 | Zhan Shi I | Maximilian Engel II | Nils Berglund III | |

10.15 - 11.15 | Nils Berglund I | (Geom. Analysis Seminar) | Zhan Shi II | |

11.30-12.30 | Nils Berglund - YouTube Channel | |||

Lunch time | ||||

14.00-15.00 | Nils Berglund II | |||

15.00-16.00 | Maximilian Engel I | 15.15-16.15 | Maximilian Engel III | |

16.15-17.15 | Richard Sowers I | 16.30-17.00 | Richard Sowers II |

**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.

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.

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

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.