Link to OLAT page:
TUK Intro Dirichlet Forms SS 21
Courses for Master's students (M.Sc.) offered in Summer Semester 2021
Due to the current situation in connection with the spread of the COVID-19/corona virus, we are unfortunately not yet able to say definitively how the courses will take place in SS 2021.
Here we present the course offerings for the Master's programmes (M.Sc.) of the Department of Mathematics for the summer semester 2021 with links to the respective courses in the handbook of modules (MHB) and continuously updated information (in particular: links to the courses in the online teaching platform OLAT - as soon as they are available).
Most of the courses in the Master's programmes will be offered in digital form - classroom teaching will only be possible in special cases.
The lecture period begins on 19.04.2021 and ends on 24.07.2021.
Further courses in SS 2021
Information on the courses for undergraduate students (in German language)
Seminars and Reading Courses
Seminars offered (see KIS)
Reading Courses offered (see KIS)
More details can be found on the webpages (for teaching) of the Division/Research Groups.
Algebra, Geometry, and Computer Algebra
Analysis and Stochastics (Stochastic Analysis, Image Processing and Data Analysis)
Financial Mathematics, Statistics
Technomathematics (Modelling and Scientific Computing)
Computational Fluid Dynamics (MHB: MAT-81-14-K-7)
- Basics of stochastic analysis (Brownian motion, Itô-integral, Itô-formula, martingale representation theorem, Girsanov theorem, linear stochastic differential equations, Feynman-Kac formula)
- Diffusion model for share prices and trading strategies
- Completeness of market
- Valuation of options with the replication principle, Black-Scholes formula
- Valuation of options and partial differential equations
- Exotic options
- Arbitrage bounds (Put call parity, parity of prices for European and American calls)
- Hahn-Banach theorem and its applications
- Baire category theorem and its applications (uniform boundedness principle, Banach-Steinhaus theorem, open mapping theorem, inverse mapping theorem, closed graph theorem)
- weak convergence (Banach-Alaoglu theorem, reflexive Banach spaces, lemma of Mazur and its applications)
- projections (closed complement theorem)
- bounded operators (adjoint operators, spectrum, resolvent, normal operators)
- compact operators (Fredholm operators, Fredholm alternative and its applications, spectral theorem (Riesz-Schauder) and applications to normal operators)
Introduction to Online Optimization
How should you control an elevator if you don't have any information about future travel orders? Should you buy a Bahncard when the next train journeys are still unknown? What is a good page replacement strategy when caching in virtual storage systems? In classic combinatorial optimization, it is assumed that the data for each problem instance are completely given. In many cases, however, this offline optimization does not adequately model the situations from applications. Numerous problems in practice are naturally online: they require decisions that have to be made immediately and without knowledge of future events. Competitive analysis has established itself as a standard means of evaluating online algorithms, whereby the objective function value of a solution generated by the online algorithm is compared with the value of an optimal offline solution. With the help of competitive analysis algorithms for caching, network routing, scheduling and transport tasks are examined. The weaknesses of competitive analysis are shown and alternative analysis concepts are presented. In addition to the theoretical side, we will also shed light on online optimization in practice, especially in the case of problems with internal logistics. There are a number of online problems associated with controlling automated transport systems. There, further requirements on the algorithms are made. For example, decisions have to be made under strict time restrictions (real-time requirements)
Introduction to Robust Optimization
Real world-data are affected by uncertainty. However, slight changes of the data can have a huge impact on the optimal solution of an optimization problem based on this data. In this lecture we introduce the basic concepts to solve optimization problems with uncertain data and to obtain robust solutions, i.e., solutions that are less sensitive to changes of data. For these concepts formulations and algorithms for different problem classes (e.g., for linear, non-linear, integer, combinatorial optimization problems) and for different uncertainty sets are developed. The following concepts for robust optimization are presented:
- Strict robustness,
- MinMax regret robustness,
- Adjustable robustness,
- Recovery robustness,
- Light robustness.
Introduction to the Theory of Dirichlet Forms
Life Insurance Mathematics
2 SWS lecture
The lecture is offered every year in the summer term. It takes place during the second half of the semester.
Nonlinear optimization problems are optimization problems where the objective function and / or constraints are nonlinear. Such problems that arise in a variety of applications can not be solved by methods known from linear optimization. This lecture covers theoretical background and algorithmic approaches to solve nonlinear optimization problems, both with and without constraints.
Among other things, the following topics are covered:
- one-dimensional and multi-dimensional search,
- Newton and Quasi-Newton procedures,
- convex analysis and separation theorems,
- optimality conditions for convex problems,
- optimality conditions for general problems,
- penalty- and barrier-methods, and
- the SQP-method.
Lecturer and staff
Monday, 08:00 - 09:30 (virtual room in OLAT)
Wednesday, 10:00 - 11:30 (virtual room in OLAT)
Friday, 08:00 - 09:30 (virtual room in OLAT)
Friday, 14:00 - 15:30 (virtual room in OLAT)
The access code to the OLAT course is the standard access code of the department.
Numerical Methods for Partial Differential Equations I
To describe real-world processes, one often makes use of partial differential equations, which, in general, cannot be solved analytically. In this course, we will discuss and study the mathematical techniques required for solving such equations numerically. The focus lies on the discretization of boundary value problems for elliptic differential equations with finite difference or finite element methods. At the end of the course, these ideas will be applied to parabolic differential equations.
The following topics will be covered:
- approximation methods for elliptic problems
- theory of weak solutions
- consistency, stability, and convergence
- approximation methods for parabolic problems
Operator Semigroups and Applications to PDE
- Definitionen, Generatoren, Resolventen, Beispiele,
- Hille-Yosida Theorem, Lumer-Phillips Theorem,
- Kontraktions-Halbgruppen, Analytische Halbgruppen, Operator-Gruppen,
- Approximationen, Störungen,
- Anwendungen auf Partielle Differentialgleichungen (u.a. Wärmeleitungsgleichungen, Wellengleichungen, Schrödinger-Gleichungen).
Link to OLAT page:
Scientific Computing in Solid Mechanics
Links / Contact
Lecturer: Prof. Dr. Bernd Simeon
Scientific Computing in Solid Mechanics
Course in OLAT:
TUK Scientific Computing in Solid Mechanics SS 2021
Theory of Scheduling Problems
- Classification of scheduling problems,
- The link between scheduling and combinatorial optimization problems,
- Single machine problems,
- Parallel machines,
- Job shop scheduling,
- Due-date scheduling,
- Time-Cost tradeoff Problems.
4 SWS / 60 h Lectures
2 SWS / 30 h Exercise Classes
Basic lectures in analysis and linear algebra, knowledge in Optimisation and Stochastics (e.g. from courses „Lineare und Netzwerkoptimierung“ and „Stochastische Methoden“)