250043 VU Kinetic Theory Applied to Biology (2024S)

Emergent phenomena are ubiquitous in nature: it corresponds to the appearance of large-scale structure from underlying microscopic dynamics. At the microscopic level particles or agents interact following some rules, but the macroscopic structures are not encoded directly in these rules and, therefore, it is a challenge to explain how the macroscopic or observable dynamics emerge from the microscopic ones. Examples of emergence are collective dynamics (flocks of birds, school of fish, pedestrians…), network formation (capillary formation, leaf venation, formation of gullies…), opinion dynamics, tumor growth, tissue development… Understanding emergence in science is key to explaining why observable phenomena take place. The mathematical tools to studying emergence come from kinetic theory, which originally was developed to study problems in Mathematical Physics in the field of gas dynamics. The application of these tools to explore questions coming from biology poses many new interesting challenges at the level of the modeling and mathematical analysis.
This course will be a short introduction to classical and modern techniques in kinetic theory to derive continuum equations from discrete equations. The course will be based on learning and applying techniques rather than on giving rigorous proofs.
The topics covered in this course include:
1. What is emergence and how does kinetic theory contribute to its study?
2. Agent-based models (deterministic and stochastic).
2. Mean-field limits: from agent-based models to transport equations.
3. Hydrodynamic limits: from transport equations to macroscopic models.
4. Applications to collective dynamics.

PREVIOUS KNOWLEDGE ASSUMED: knowledge from courses in mathematical analysis and introductory courses to ordinary differential equations and partial differential equations.