Lecturer: Patrick Dondl
Assistant: Luciano Sciaraffia
Language: in German
Lecture: Mo, 12-14h, HS II, Albertstr. 23b, Mi, 12-14h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
06.08., 14:00-16:00, HS Rundbau, Albertstr. 21
Attention: Change of time and room!
Linear functional analysis, which is the subject of the lecture, uses concepts of linear algebra such as vector space, linear operator, dual space, scalar product, adjoint map, eigenvalue, spectrum to solve equations in infinite-dimensional function spaces, especially linear differential equations. The algebraic concepts have to be extended by topological concepts such as convergence, completeness and compactness.
This approach was developed at the beginning of the 20th century by Hilbert, among others, and is now part of the methodological foundation of analysis, numerics and mathematical physics, in particular quantum mechanics, and is also indispensable in other mathematical areas.
Linear Algebra I+II, Analysis I–III
Pure Mathematics
Applied Mathematics
Elective
Lecturer: Angelika Rohde
Assistant: Johannes Brutsche
Language: in English
Lecture: Di, Do, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, date to be determined
22.09., 10:00-12:00
The problem of axiomatising probability theory was solved by Kolmogorov in 1933: a probability is a measure of the set of all possible outcomes of a random experiment. From this starting point, the entire modern theory of probability develops with numerous references to current applications.
The lecture is a systematic introduction to this area based on measure theory and includes, among other things, the central limit theorem in the Lindeberg-Feller version, conditional expectations and regular versions, martingales and martingale convergence theorems, the strong law of large numbers and the ergodic theorem as well as Brownian motion.
necessary: Analysis I+II, Linear Algebra I, Elementary Probability Theory I
useful: Analysis III
Applied Mathematics
Elective
Lecturer: David Criens
Assistant: Samuel Adeosun
Language: in English
Lecture: Mi, 14-16h, HS II, Albertstr. 23b, Do, 10-12h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
This lecture builds the foundation of one of the key areas of probability theory: stochastic analysis. We start with a rigorous construction of the It^o integral that integrates against a Brownian motion (or, more generally, a continuous local martingale). In this connection, we learn about It^o's celebrated formula, Girsanov’s theorem, representation theorems for continuous local martingales and about the exciting theory of local times. Then, we discuss the relation of Brownian motion and Dirichlet problems. In the final part of the lecture, we study stochastic differential equations, which provide a rich class of stochastic models that are of interest in many areas of applied probability theory, such as mathematical finance, physics or biology. We discuss the main existence and uniqueness results, the connection to the martingale problem of Stroock-Varadhan and the important Yamada-Watanabe theory.
Probability Theory I and II (Stochastic Processes)
Applied Mathematics
Elective
Mathematics
Concentration Module
Lecturer: Sören Bartels
Assistant: Tatjana Schreiber
Language: in English
Lecture: Mi, 10-12h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
The lecture addresses algorithmic aspects in the practical realization of mathematical methods in big data analytics and machine learning. The first part will be devoted to the development of recommendation systems, clustering methods and sparse recovery techniques. The architecture and approximation properties as well as the training of neural networks are the subject of the second part. Convergence results for accelerated gradient descent methods for nonsmooth problems will be analyzed in the third part of the course. The lecture is accompanied by weekly tutorials which will involve both, practical and theoretical exercises.
Lectures "Numerik I, II" or lecture "Basics in Applied Mathematics"
Applied Mathematics
Elective
Mathematics
Concentration Module
Lecturer: Ernst August v. Hammerstein
Assistant: Sebastian Hahn
Language: in English
Lecture: Mo, 14-16h, HS II, Albertstr. 23b
Exercise session: Mi, 16-18h, SR 403, Ernst-Zermelo-Str. 1
Lévy processes are the continuous-time analogues of random walks in discrete time as they possess, by definition, independent and stationary increments. They form a fundamental class of stochastic processes which has widespread applications in financial and insurance mathematics, queuing theory, physics and telecommunication. The Brownian motion and the Poisson process, which may already be known from other lectures, also belong to this class. Despite their richness and flexibility, Lévy processes are usually analytically and numerically very tractable because their distributions are generated by a single univariate distribution which has the property of infinite divisibility.
The lecture starts with an introduction into infinitely divisible distributions and the derivation of the famous Lévy-Khintchine formula. Then it will be explained how the Lévy processes emerge from these distributions and how the characteristics of the latter influence the path properties of the corresponding processes. Finally, after a short look at the method of subordination, option pricing in Lévy-driven financial models will be discussed.
necessary: Probability Theory I
useful: Probability Theory II (Stochastic Processes)
Applied Mathematics
Elective
Mathematics
Concentration Module
Lecturer: Thorsten Schmidt
Assistant: Simone Pavarana
Language: in English
Lecture: Mi, 12-14h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
In this lecture we will study new and highly efficient tools from machine learning which are applied to stochastic problems. This includes neural SDEs as a generalisation of stochastic differential equations relying on neural networks, transformers as a versatile tool not only for languages but also for time series, transformers and GANs as generator of time series and a variety of applications in Finance and insurance such as (robust) deep hedging, signature methods and the application of reinforcement learning.
The prerequisites are stochastics, for some parts we will require a good understanding of stochastic processes. A (very) short introduction will be given in the lectures – so for fast learners it would be possible to follow the lectures even without the courses on stochastic processes.
Applied Mathematics
Elective
Mathematics
Concentration Module
Lecturer: Patrick Dondl
Assistant: Eric Trébuchon
Language: in English
Lecture: Mi, 14-16h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
This course provides a comprehensive introduction to mathematical modeling. We will learn the systematic process of translating real-world problems into mathematical frameworks, analyzing them using appropriate mathematical tools, and interpreting the results in practical contexts. The course covers both discrete and continuous modeling approaches, with emphasis on differential equations, variational problems, and optimization techniques. Through case studies in physics, biology, engineering, and economics, students will develop skills in model formulation, validation, and refinement. Special attention is given to dimensional analysis, stability theory, and numerical methods necessary for implementing solutions with a focus on numerical methods for ordinary differential equations. The course combines theoretical foundations with hands-on experience in computational tools for model simulation and analysis.
Analysis I, II, Linear Algebra I, II, Numerics I, II
Applied Mathematics
Elective
Mathematics
Concentration Module
Numerical Optimization
Lecturer: Moritz Diehl
Assistant: Léo Simpson
Language: in English
Tutorial / flipped classroom: Di, 14-16h, HS II, Albertstr. 23b
Sir-in Exam: Date to be announced
The aim of the course is to give an introduction into numerical methods for the solution of optimization problems in science and engineering. The focus is on continuous nonlinear optimization in finite dimensions, covering both convex and nonconvex problems. The course divided into four major parts:
The course is organized as inverted classroom based on lecture recordings and a lecture manuscript, with weekly alternating Q&A sessions and exercise sessions. The lecture is accompanied by intensive computer exercises offered in Python (6 ECTS) and an optional project (3 ECTS). The project consists in the formulation and implementation of a self-chosen optimization problem or numerical solution method, resulting in documented computer code, a project report, and a public presentation. Please check the website for further information.
necessary: Analysis I–II, Linear Algebra I–II
useful: Introduction to Numerics
Applied Mathematics
Elective
Mathematics
Concentration Module