Detailed information can be found in the course descriptions and in the module handbooks (in German only).
Lecturer: Eva Lütkebohmert-Holtz
Assistant: Hongyi Shen
Language: in English
Lecture: Mo, 10-12h, HS 3042, KG III
Exercise session: Di, 8-10h, HS 1015, KG I
Sit-in exam (resit) 14.08., 15:00-18:00
This course covers an introduction to financial markets and products. Besides futures and standard put and call options of European and American type we also discuss interest-rate sensitive instruments such as swaps.
For the valuation of financial derivatives we first introduce financial models in discrete time as the Cox--Ross--Rubinstein model and explain basic principles of risk-neutral valuation. Finally, we will discuss the famous Black--Scholes model which represents a continuous time model for option pricing.
Required: Elementary Probability Theory~I
Supplementary Module in Mathematics
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: David Criens
Assistant: Dario Kieffer
Language: in English
Lecture: Do, 12-14h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
The class of Markov chains is an important class of (discrete-time) stochastic processes that are used frequently to model for example the spread of infections, queuing systems or switches of economic scenarios. Their main characteristic is the Markov property, which roughly means that the future depends on the past only through the current state. In this lecture we provide the mathematical foundation of the theory of Markov chains. In particular, we learn about path properties, such as recurrence and transience, state classifications and discuss convergence to the equilibrium. We also study extensions to continuous time. On the way we discuss applications to biology, queuing systems and resource management. If the time allows, we also take a look at Markov chains with random transition probabilities, so-called random walks in random environment, which is a prominent model in the field of random media.
Required: Elementary Probability Theory~I \ Recommended: Analysis~III, Probability Theory~I
Supplementary Module in Mathematics
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Mathematical Physics
Lecturer: Wolfgang Soergel
Language: in German
Lecture: Di, 16-18h, SR 403, Ernst-Zermelo-Str. 1
Introduction to classic mechanics from the point of view of mathematics. We start with the mathematical modelling of space and time. Then we discuss Newton's equations of movement, physical systems with compulsory conditions, the D'Alembert principle, the Hamilton formalism and its derivation from the Newton's equations and applications of Hamilton formalism.
Required: Analysis III
Supplementary Module in Mathematics
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Students in the B.Sc. Mathematics programme can choose to give a talk, in which case the course counts as a seminar. Usability and requirements as for the seminar ‘Theory of Non-Commutative Algebras’.
Lecturer: Diyora Salimova
Assistant: Ilkhom Mukhammadiev
Language: in English
Lecture: Di, Fr, 12-14h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
Computer exercise: 2 hours, date to be determined
Oral exam 06.12.
This course takes only place in the first half of the semester, until end of November.
The aim of this course is to enable the students to carry out simulations and their mathematical analysis for stochastic models originating from applications such as mathematical finance and physics. For this, the course teaches a decent knowledge on stochastic differential equations (SDEs) and their solutions. Furthermore, different numerical methods for SDEs, their underlying ideas, convergence properties, and implementation issues are studied.
Required: Probability and measure theory, basic numerical analysis and basics of MATLAB programming.
Supplementary Module in Mathematics
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Moritz Diehl
Assistant: Florian Messerer
Language: in English
Tutorial / flipped classroom: Di, 14-16h, HS II, Albertstr. 23b
The aim of the course is to give an introduction to numerical methods for the solution of optimal control problems in science and engineering. The focus is on both discrete time and continuous time optimal control in continuous state spaces. It is intended for a mixed audience of students from mathematics, engineering and computer science.
The course covers the following topics:
The lecture is accompanied by intensive weekly computer exercises offered both in MATLAB and Python (6~ECTS) and an optional project (3~ECTS). The project consists in the formulation and implementation of a self-chosen optimal control problem and numerical solution method, resulting in documented computer code, a project report, and a public presentation.
Required: Analysis~I and II, Linear Algebra~I and II \ Recommended: Numerics I, Ordinary Differential Equations, Numerical Optimization
Supplementary Module in Mathematics
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Computer exercises for Introduction to Theory and Numerics of Partial Differential Equations
Lecturer: Sören Bartels
Assistant: Vera Jackisch
Language: in English
The computer tutorial accompanies the lecture with programming exercises.
See the lecture – additionally: programming knowledge.
Supplementary Module in Mathematics
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Computer exercises in Numerics
Lecturer: Sören Bartels
Assistant: Tatjana Schreiber
Language: in German
In the computer tutorial accompanying the Numerics (first term) lecture the algorithms developed and analyzed in the lecture are put into practice and and tested experimentally. The implementation is carried out in the programming languages Matlab, C++ and Python. Elementary programming knowledge is assumed.
See the lecture {\em Numerics I} (which should be attended in parallel or should already have been completed). \ Additionally: Elementary programming knowledge.
Supplementary Module in Mathematics
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Please note the registration modalities for the individual seminars published in the comments to the course catalog: As a rule, places are allocated after pre-registration by e-mail at the preliminary meeting at the end of the lecture period of the summer semester. You must then register online for the exam; the registration period runs from August 1, 2024 to October 9, 2024.
Lecturer: Ernst August v. Hammerstein
Language: in German
A knot can be mathematically defined relatively simply as a closed curve in the three-dimensional space \(\mathbb{R}^3\). From everyday life, one is certainly already familiar with different types of knots, e.g, surgeons knot, sailors knots, and many more. The aim of mathematical knot theory is to find characteristic quantities for the description and classification of knots and thus possibly also to be able to decide whether two knots are equivalent, i.e., if they can be transformed into one another through certain operations.
Ropes, cords or wires can be used to illustrate knots as well as interlacings. Prospective teachers can use these not only in this seminar, but perhaps also later in the classroom to display different results in a very practical way.
Required: Basic Mathematics courses. \ Possibly a little knowledge in topology in addition.
Supplementary Module in Mathematics
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Thorsten Schmidt
Assistant: Moritz Ritter
Language: Talk/participation possible in German and English
Seminar: Fr, 10-12h, SR 125, Ernst-Zermelo-Str. 1
Preregistration:
Preliminary Meeting 18.10.
Preparation meetings for talks: Dates by arrangement
This seminar will focus on theoretical machine learning results, including modern universal approximation theorems, approximation of filtering methods through transformes, application of machine learning methods in financial markets and possibly other related topics. Moreover, we will cover topics in stochastic analysis, like fractional Ito calculus, uncertainty, filtering and optimal transport. You are also invited to suggest related topics.
Required: Basic Probability and either Machine Learning or Probability Theory II (Stochastic Processes).
Supplementary Module in Mathematics
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Machine-Learning Methods in the Approximation of PDEs
Lecturer: Sören Bartels
Assistant: Tatjana Schreiber
Language: Talk/participation possible in German and English
Machine-learning methods have recently been used to approximate solutions of partial differential equations. While in some cases they lead to advantages over classical approaches, their general superiority is widely open. In the seminar we will review the main concepts and recent developments.
Introduction to Theory and Numerics for PDEs
Supplementary Module in Mathematics
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Medical Data Science
Lecturer: Harald Binder
Language: Talk/participation possible in German and English
Seminar: Mi, 10-11:30h, HS Medizinische Biometrie, 1. OG, Stefan-Meier-Str. 26
Preregistration:
Preliminary Meeting 17.07., HS Medizinische Biometrie, 1. OG, Stefan-Meier-Str. 26
To answer complex biomedical questions from large amounts of data, a wide range of analysis tools is often necessary, e.g. deep learning or general machine learning techniques, which is often summarized under the term ``Medical Data Science''. Statistical approaches play an important rôle as the basis for this. A selection of approaches is to be presented in the seminar lectures that are based on recent original work. The exact thematic orientation is still to be determined.
Good knowledge of probability theory and mathematical statistics.
Supplementary Module in Mathematics
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Guofang Wang
Assistant: Xuwen Zhang
Language: Talk/participation possible in German and English
Seminar: Mi, 16-18h, SR 125, Ernst-Zermelo-Str. 1
Preliminary Meeting 17.07., 16:00
Preparation meetings for talks: Dates by arrangement
Minimal surfaces are surfaces in space with a “minimal” area and can be described using holomorphic functions. They occur, for example in the investigation of soap skins and the construction of stable objects (e.g. in architecture). In the investigation of minimal surfaces elegant methods from various mathematical fields such as function theory, calculus of variations, differential geometry and partial differential equations. are applied.
Required: Analysis III or knowledge about multidimensional integration and complex analysis. \ Recommended: Elementary knowledge about differential geometry.
Supplementary Module in Mathematics
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Sebastian Goette
Assistant: Mikhail Tëmkin
Language: Talk/participation possible in German and English
Seminar: Di, 14-16h, SR 125, Ernst-Zermelo-Str. 1
Preliminary Meeting 16.07., SR 125, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
We will discuss advanced topics in algebraic topology. Depending on the interest of the participants we could work on one of the following topics---if you have other topic suggestions, please contact the lecturer.
Algebraic Topology~I and II
Supplementary Module in Mathematics
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Annette Huber-Klawitter
Assistant: Xier Ren
Language: Talk/participation possible in German and English
Seminar: Fr, 8-10h, SR 404, Ernst-Zermelo-Str. 1
Preregistration:
Preliminary Meeting 15.07., 11:00, SR 318, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
In this seminar, we are going to study finite dimensional (unital, possibly non-commutative) algebras over a (commutative) field \(k\). Prototypes are the rings of square matrices over \(k\), finite field extensions, or the algebra \(k^n\) with diagonal multiplication.
We will concentrate on path algebras of finite quivers (German: Köcher). Modules over them are equivalently described as representations of the quiver. Many algebraic properties can be directly understood from properties of the quiver.
Required: Linear Algebra \ Recommended: Algebra and Number Theory, Commutative Algebra and Introduction to Algebraic Geometry
Supplementary Module in Mathematics
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.