Department of Mathematics

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.

Mathematikvorlesung an der TUK: Bild aus Hörsaal

Further courses in SS 2021

Information on the courses for undergraduate students (in German language)

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Financial Mathematics

Contents

  • 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)

Contact time

4 SWS lecture
2 SWS tutorials

The lecture is offered every year in the summer term.

Prerequisites with regard to contents

Course "Probability Theory"

Links

Functional Analysis

Content

  • 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)

contact time

4 SWS lecture
2 SWS exercise classes

substantive prerequisites

content of the introductory lecture "Einführung in die Funktionalanalysis" as well as concepts from "Maß- und Integrationstheorie"

Introduction to Online Optimization

Content

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)

Lecturer and staff

Prof. Dr. Sven O. Krumke

M. Sc. Oliver Bachtler

Date

the lecture will be recorded

Tutorials

Friday, 12:00 - 13:30 (virtual room in OLAT)

Materials

OpenOLAT

The access code to the OLAT course will be announced to the participants registered in the URM.

Introduction to Robust Optimization

Content

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.

Lecturer and staff

Prof. Dr. Anita Schöbel

Dr. Philine Schiewe

Date

The lecture will be recorded

Tutorials

Monday, 16:00 - 17:30 (virtual room in OLAT)

Tuesday, 08:00 - 09:30 (virtual room in OLAT)

Materials

OpenOLAT

The access code to the OLAT course is the standard access code of the department.

Introduction to the Theory of Dirichlet Forms

Content

  • resolvents, semigroups, generators (Theorem of Hille and Yosida),
  • coercive bilinear forms (Stampacchia theorem, characterisation by resolvents, semigroups, generators)
  • closed bilinear form,
  • contraction properties (Sub-Markov property, Dirichlet operators, Dirichlet forms).

contact time

4 SWS / 60 h lecture
2 SWS / 30 h exercise classes

substantive prerequisites

Content of the lecture "Functional Analysis"

Life Insurance Mathematics

Contents

  • Elementary financial mathematics (calculation of interest)
  • Mortality
  • Insurance benefits
  • Net premiums and net actuarial reserves
  • Inclusion of costs
  • Life related insurance
  • Various reject causes

Contact time

2 SWS lecture

The lecture is offered every year in the summer term. It takes place during the second half of the semester.

Prerequisites with regard to contents

Course "Stochastic Methods" from the Bachelor's degree programme.

Links

Here you find the KIS entry: Life Insurance Mathematics (lecture)

Here you find the OLAT course: TUK Life Insurance Mathematics SS 21

Nonlinear Optimization

Content

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

Prof. Dr. Stefan Ruzika

M. Sc. Tobias Dietz

Date

Monday, 08:00 - 09:30 (virtual room in OLAT)

Wednesday, 10:00 - 11:30 (virtual room in OLAT)

Übungen

Friday, 08:00 - 09:30 (virtual room in OLAT)

Friday, 14:00 - 15:30 (virtual room in OLAT)

Materials

OpenOLAT

The access code to the OLAT course is the standard access code of the department.

Numerical Methods for Partial Differential Equations I

Contents

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

Contact time

4 SWS Lecture
2 SWS Tutorial

Prerequisites

Informal:

  • "Fundamentals of Mathematics"
  • "Numerics of ODE"
  • "Introduction to PDE"
  • Some functional analysis

Formal:

  • None

Operator Semigroups and Applications to PDE

Content

  • 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).

Contact time

4 SWS Vorlesung
2 SWS Übung

Substantive prerequesites

Lecture "Functional Analysis"

Scientific Computing in Solid Mechanics

Contents

Mathematical modelling, numerical methods, and software for the following topics:

  • elastic bodies
  • special cases of beams and plane strain/stress state
  • finite element space discretisation
  • specific time integration schemes

Contact time

2 SWS Lecture

Prerequisites

Informal:

  • "Fundamentals of Mathematics"
  • "Introduction to Numerical Methods"
  • "Numerics of ODE"
  • "Introduction to PDE"

Formal:

  • None

Frequency

This course is offered irregularly.

Theory of Scheduling Problems

Content

  • 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.

Contact Time

4 SWS / 60 h Lectures
2 SWS / 30 h Exercise Classes

Prerequisites (Content)

Basic lectures in analysis and linear algebra, knowledge in Optimisation and Stochastics (e.g. from courses „Lineare und Netzwerkoptimierung“ and „Stochastische Methoden“)

Frequency

each summer semester

Links/Contact

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