Description: Differential equations provide a unified language to quantitatively model phenomena in biology, chemistry, physics, engineering, social sciences, machine learning and many more disciplines. This course gives an overview over models in micro-, meso-, and macroscopic scales, modelling particle motion at different levels of detail. The behavior of their solution will be discussed and connections between different scales will be revealed based on sound mathematical theory and computational methods.
Mathematical tools may include
• functional analysis,
• calculus of variations,
• spectral analysis,
• long term behaviour,
• blow-up and extinction of solutions,
Depending on the audience, applications may include:
• Newtonian mechanics (ODEs),
• random walks (SDEs),
• neural ODEs,
• generative diffusion models,
• advection equations,
• kinetic transport equations,
• Boltzmann equation of gas dynamics,
• Fokker-Planck reaction-diffusion equation,
• Patlack-Keller-Segel model of bacterial motion and chemotaxis,
• Burgers’ equation and conservation laws in fluid dynamics,
• opinion dynamics,
• Lotka-Volterra predator-prey systems,
• chemical reaction dynamics.

Description: This course establishes fundamental concepts in real and functional analysis that are vital for many topics and applications in mathematics, physics, computing, and engineering. Functional analysis combines ideas from geometry, linear algebra, and analysis to study infinite-dimensional spaces of functions and their properties and relations.  We put particular emphasis on the behaviour of linear operators acting on function spaces. Such operators serve as a basis for aspects of several important branches of applied mathematics, including Fourier series, integral and differential equations, numerical analysis, and data science. The aim of this course is to provide a working knowledge of the subject, oriented toward applications. Emphasis is placed both on rigorous mathematical development and on applications to control theory, data analysis and partial differential equations.
The following concepts form the cornerstones of the course:
• Banach spaces, Hilbert spaces, Linear Operators, and Duality
• Hahn-Banach Theorem
• Open Mapping and Closed Graph Theorem
• Uniform Boundedness Principle
We will see applications of these ideas to the following topics:
• The Fourier transform
• Sobolev spaces, Sobolev embedding theorem, Trace theorem
• Contraction Mapping Principle, with applications to the Implicit Function Theorem and ODEs
• Calculus of Variations (optimization over function spaces)
• Applications to inverse problems and high-dimensional estimation

This class focuses on the mathematics behind many algorithms and techniques used in machine learning. Besides mathematical theory, we will also look at applied machine learning. - After successfully completing this course the student will be able to find and implement the machine learning technique appropriate for a given applied problem at hand.

This course addresses master students of mathematics / mathematical physics / computational mathematics.

The lecture requires knowledge from the courses Analysis 1,2 and 3 from the Bachelor studies as well as basic knowledge in stochastics.