Collective Dynamics 1000-1M25WDK

The lecture aims to familiarize participants with foundational models from the field of collective dynamics. Special emphasis will be placed on theories related to artificial intelligence, such as transformers and sentence generation in large language models (LLMs). Participants will be exposed to several modern applications of differential equations. Some sessions will take the form of workshops.

Lecture Outline:

1. Introduction to collective dynamics. The linear consensus problem—its stability, asymptotic behavior, and connections with graph theory. The concept of algebraic connectivity.
2. Gradient flow and its connections to dynamics and optimization.
3. Examples of first-order models: the Hegselmann-Krause model of opinion dynamics and consensus, the Aw-Rascle model of traffic flow. The Kuramoto model and its wide range of applications.
4. An example of a second-order model: the Cucker-Smale model of flock formation.
5. A mathematical perspective on the transformer architecture. Sentence generation in LLMs as a collective dynamics problem and its relationship with the Kuramoto model.

Advanced Topics (Optional):
6. Collective dynamics in large populations. Partial differential equations and the mean-field limit.
7. Predicting phase transitions (i.e., significant structural changes in populations) based on the work of C. Kuehn.

Due to the open-ended nature of the problems discussed, the course naturally provides topics suitable for bachelor's or master's theses.

We welcome all interested students at the undergraduate, master’s, and doctoral levels.

Learning Outcomes:
The student is familiar with examples of collective behavior and can describe them mathematically. They understand the fundamental connections between collective dynamics, optimization, and artificial intelligence. They are aware of possible directions for further development of this knowledge.