Core Courses

The BMS Core Course sequences in RTA 7 cover the fundamental concepts, results, methods of...

Functional Analysis
◦ Metric and Normed Spaces
◦ Linear Operators and Dual Spaces
◦ Fundamental Theorems of Functional Analysis such as Hahn-Banach and Banach-Steinhaus
◦ Weak Convergence and Weak Topology
◦ Hilbert Spaces and the Riesz Representation Theorem
◦ Spectral Theory for Compact and Self-Adjoint Operators

Partial Differential Equations
◦ Scalar first order equations
◦ Elementary PDEs: heat equation, wave equation, Laplace equation
◦ Sobolev spaces
◦ Strong and weak solutions
◦ Elliptic problems via Lax–Milgram theory
◦ Additional topics like distributions, Galerkin schemes, monotone operators, semilinear equations, etc.

Mathematical Modelling with PDEs
◦ General principles of continuum mechanics and thermodynamics
◦ Symmetries and conservation laws
◦ Variational principles
◦ Derivation and discussion of models from hydrodynamics, solid mechanics, thermoelasticity, geodynamics, climate research or quantum mechanics

 

Spectrum of Advanced Courses
Regularly offered advanced courses include Nonlinear Functional Analysis, Evolution Equations, Calculus of Variations, Nonlinear Partial Differential Equations, Nonlinear Dynamical Systems, and Bifurcation Theory, and Rough paths and Regularity Structures. More specialized courses on current topics complement these choices.

Core Courses

The scope of numerical mathematics ranges from the design of numerical methods, their analysis, the incorporation of optimization techniques, to the use of complex methods for simulating problems from applications (scientific computing). The BMS Core Course sequences in RTA 6 introduce major concepts for the design of numerical methods and their analysis concerning stability, convergence, and efficiency. Advanced methods, which are derived from the basic ones, have been and will be utilized in many projects. This area provides a strong link to partners from engineering, the natural sciences, and industry.

Nonlinear Optimization
This course presents the mathematical foundations and numerical methods for several classes of nonlinear optimization problems. Both, unconstrained and constrained problems are covered.

Numerical Methods for ODEs and Numerical Linear Algebra
This course covers numerical methods for initial value problems with ordinary differential equations. It discusses, e.g., explicit and implicit Runge-Kutta schemes, stability analysis, and multi-step methods. In addition, advanced topics from numerical linear algebra will be presented, e.g., iterative solvers for linear systems of equations or numerical methods for differential-algebraic equations.

Numerical Methods for PDEs
This course discusses numerical methods for boundary value problems with partial differential equations. Important aspects are approaches like finite difference, finite element, and finite volume methods and their numerical analysis.

 

Spectrum of Advanced Courses
Advanced Finite Element Methods, Uncertainty Quantification, Numerical Analysis of PDAEs, Optimal Control of PDEs, Control Theory, Theory and Numerics of Non-Smooth Optimization, Variational Inequalities, Inverse Problems

Core Courses

The BMS Core Course sequences in RTA 5 cover the fundamental concepts, results, and methods in discrete geometry, understood in a broad sense. Adjacent fields with mutual cross-fertilization include combinatorics, algebraic and differential geometry, topology, functional analysis, and optimization. Application areas include computational biology, physics, economics, and more.

Classical Geometries
This course serves as an introduction to discrete differential geometry. Key topics include projective geometry, spherical and hyperbolic geometry, Möbius geometry, Lie sphere geometry.

Discrete Differential Geometry and Visualization
This course is a second course in discrete differential geometry. Key topics include discrete surfaces and polyhedral complexes, discretizations of mean and Gauss curvature, Delaunay triangulations, subdivision surfaces, discrete Hodge theory, geometry processing.

Discrete Geometry I + II
These two courses cover the foundations of polytope theory, making the connection to linear optimization. Key topics include Minkowski-Weyl-Theorem, McMullen's Uppper Bound Theorem, Dehn-Sommerville equations, computing convex hulls, Voronoi digrams, regular subdivisions of point configurations.

 

Spectrum of Advanced Courses
Lattice polytopes, toric geometry, tropical geometry, combinatorial algebraic geometry, solving systems of polynomial equations, high dimensional convex geometry, geometry of numbers, isoperimetric problems, combinatorial topology.

Core Courses

The BMS Core Course sequences in RTA 3 cover the fundamental concepts, results, methods of stochastic processes.

Stochastic Processes I: Discrete Time
Stochastic processes I develops the tools and methods to describe and study stochastic processes evolving in discrete time. Key notions are martingales and Markov chains. Brownian motion is constructed and discussed as the prime example of a stochastic process evolving in continuous time.

Stochastic Processes II: Continuous Time
Stochastic processes II is devoted to the study of stochastic processes in continuous time. The course introduces to Stochastic Analysis and explains how its tools such as Ito’s formula can be used to study the dynamics of continuous-time stochastic processes. Stochastic Differential Equations are analyzed and illlustrated as models for real-world applications from Finance, Biology, or Physics.

 

Spectrum of Advanced Courses

Courses are regularly offered on various advanced topics from the spectrum of our interests, including statistics of stochastic processes, mathematical finance, stochastic partial differential equations, rough paths and regularity structures, interacting particle systems, stochastic pro-cesses in evolution, stochastic processes in neuroscience, spatial stochastic point processes, stochastic control, Markov processes, Lévy processes, computational finance, large deviations.