V5F3 – Advanced Topics in Stochastic Analysis

Modelling across Scales

WS 2020

Lectures: Mondays and Wednesdays, 12.00-14.00 (online via zoom)

Course Instructor: Prof. Dr. Franca Hoffmann

Course Description
This course gives an overview of different mathematical tools to analyze partial differential equations encountered across mathematics, biology, physics, engineering and social science. We will look at the main properties of different classes of linear and nonlinear PDEs and the behavior of their solutions using tools from functional analysis and calculus of variations with an emphasis on applications.
We will focus on representative models from different areas which may include: the Fokker-Planck equation, the Boltzmann equation, the Fisher-KPP equation, Burger’s equation, swarming models, models for opinion dynamics, bio-mathematics models for cell movement and bacterial chemotaxis (Patlack-Keller-Segel model), SIR models from epidemiology, Lotka-Volterra equations, predator-prey systems, chemical reactions, enzymatic reactions.
The above list is flexible and depends on the audience. If you are interested in this course, feel free to contact me at franca.hoffmann@hcm.uni-bonn.de with (types of) models you would like to study.

Content outline

We will study the theories involved in understanding the following key concepts:

  • Connections between different scales: Micro-Meso-Macro
  • Agent-based models, kinetic equations and corresponding macroscopic descriptions
  • Mean Field Limits and Hydrodynamic Scalings
  • Weak Derivatives and Theory of Distributions
  • Transport Equations

Other core concepts may include:

  • Gradient Flows and Entropy Methods
  • Epidemiology Models
  • Conservation Laws
  • Instabilities and Pattern Formation
  • Dimensional Scaling Analysis
  • Self-Similar Scalings
  • Introduction to Numerical Methods for ODEs and PDEs
  • Introduction to Inverse Problems