Preliminary course catalogue - changes and additions are still possible.
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New (and partly not yet in den annotated course catalogue):
Analysis III
Lecturer: Michael Růžička
Assistant: Luciano Sciaraffia
Language: in German
Lecture: Mo, Mi, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, various dates
Sit-in exam: date to be announced
The Analysis III lecture deals with measure and integration theory, with particular emphasis on the Lebesgue measure. These theories are of particular importance for many further lectures in analysis, applied mathematics, stochastics, probability theory and geometry, as well as physics. Main topics are measures and integrals in \(\mathbb R^n\), Lebesgue spaces, convergence theorems, the transformation theorem, surface integrals and Gauss' integral theorem.
Required: Analysis I and II, Linear Algebra I \
Useful: Linear Algebra II
Elective in Data
Introduction to Theory and Numerics of Partial Differential Equations
Lecturer: Patrick Dondl
Assistant: Ludwig Striet, Oliver Suchan
Language: in English
Lecture: Mo, Mi, 12-14h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
The aim of this course is to give an introduction into theory of linear partial differential equations and their finite difference as well as finite element approximations. Finite element methods for approximating partial differential equations have reached a high degree of maturity, and are an indispensable tool in science and technology. We provide an introduction to the construction, analysis, and implementation of finite element methods for different model problems. We will address elementary properties of linear partial differential equations along with their basic numerical approximation, the functional-analytical framework for rigorously establishing existence of solutions, and the construction and analysis of basic finite element methods.
Required: Analysis~I and II, Linear Algebra~I and II as well as knowledge about higher-dimensional integration (e.g. from Analysis~III or from Further Chapters in Analysis) \
Recommended: Numerics for differential equations, Functional analysis
Advanced Lecture in Numerics
Elective in Data
Mathematical Statistics
Lecturer: Ernst August v. Hammerstein
Language: in English
Lecture: Di, Do, 14-16h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
The lecture builds on basic knowledge about Probability Theory. The fundamental problem of statistics is to infer from a sample of observations as precise as possible statements about the data-generating process or the underlying distributions of the data. For this purpose, the most important methods from statistical decision theory such as test and estimation methods are introduced in the lecture.
Key words hereto include Bayes estimators and tests, Neyman-Pearson test theory, maximum likelihood estimators, UMVU estimators, exponential families, linear models. Other topics include ordering principles for reducing the complexity of models (sufficiency and invariance).
Statistical methods and procedures are used not only in the natural sciences and medicine, but in almost all areas in which data is collected and analyzed This includes, for example, economics (“econometrics”) and the social sciences (especially psychology). However, in the context of this lecture, we will focus less on applications, but---as the name suggests---more on the mathematical justification of the methods.
Probability Theory (in particular measure theory and conditional probabilities/expectations)
Advanced Lecture in Stochastics
Elective in Data
Probabilistic Machine Learning
Lecturer: Giuseppe Genovese
Language: in English
Lecture: Di, Do, 12-14h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
The goal of the course is to provide a mathematical treatment of deep neural networks and energy models, that are the building blocks of many modern machine learning architectures. About neural networks we will study the basics of statistical learning theory, the back-propagation algorithm and stochastic gradient descent, the benefits of depth. About energy models we will cover some of the most used learning and sampling algorithms. In the exercise classes, besides solving theoretical problems, there will be some Python programming sessions to implement the models introduced in the lectures.
Probability Theory I \
Basic knowledge of Markov chains is useful for some part of the course.
Advanced Lecture in Stochastics
Elective in Data
Probability Theory II – Stochastic Processes
Lecturer: Angelika Rohde
Assistant: Johannes Brutsche
Language: in English
Lecture: Mo, Mi, 14-16h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
A stochastic process \((X_t)_{t\in T}\) is a family of random variables, where mostly the situation \(T = \mathbb{N}\) or \(T = [0, 1]\) is studied. Basic examples include stationary time series, the Poisson process and Brownian motion as well as processes derived from those. The lecture includes ergodic theory and its applications, Brownian motion and especially the study of its path properties, the elegant concept of weak convergence on Polish spaces as well as functional limit theorems. Finally, we introduce stochastic integration with respect to local martingales, based on the continuous time version of the martingale transform.
Probability Theory I
Advanced Lecture in Stochastics
Elective in Data
Futures and Options
Lecturer: Eva Lütkebohmert-Holtz
Language: in English
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.
Elementary Probability Theory I
Elective in Data
Machine Learning and Mathematical Logic
Lecturer: Maxwell Levine
Language: in English
Lecture: Do, 14-16h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
Developments in artificial intelligence have boomed in recent years, holding the potential to reshape not just our daily routines but also society at large. Many bold claims have been made regarding the power and reach of AI. From a mathematical perspective, one is led to ask: What are its limitations? To what extent does our knowledge of reasoning systems in general apply to AI?
This course is intended to provide some applications of mathematical logic to the field of machine learning, a field within artificial intelligence. The goal of the course is to present a breadth of approachable examples.
The course will include a gentle introduction to machine learning in a somewhat abstract setting, including the notions of PAC learning and VC dimension. Connections to set theory and computability theory will be explored through statements in machine learning that are provably undecidable. We will also study some applications of model theory to machine learning.
The literature indicated in the announcement is representative but tentative. A continuously written PDF of course notes will be the main resource for students.
Background in basic mathematical logic is strongly recommended. Students should be familiar with the following notions: ordinals, cardinals, transfinite induction, the axioms of ZFC, the notion of a computable function, computable and computably enumerable sets (a.k.a. recursive and recursively enumerable sets), the notions of languages and theories and structures as understood in model theory, atomic diagrams, elementarity, and types. The concepts will be reviewed briefly in the lectures. Students are not expected to be familiar with the notion of forcing in set theory.
Elective in Data
Markov Chains
Lecturer: David Criens
Language: in English
Lecture: Mi, 10-12h, 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
Elective in Data
Mathematical Introduction to Deep Neural Networks
Lecturer: Diyora Salimova
Assistant: Ilkhom Mukhammadiev
Language: in English
Lecture: Mi, 12-14h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
The course will provide an introduction to deep learning algorithms with a focus on the mathematical understanding of the objects and methods used. Essential components of deep learning algorithms will be reviewed, including different neural network architectures and optimization algorithms. The course will cover theoretical aspects of deep learning algorithms, including their approximation capabilities, optimization theory, and error analysis.
Analysis I and II, Lineare Algebra I and II
Elective in Data
Measure Theory
Lecturer: Peter Pfaffelhuber
Assistant: Samuel Adeosun
Language: in English
Measure Theory is the foundation of advanced probability theory. In this course, we build on knowledge in analysis and provide all necessary results for later classes in statistics, probabilistic machine learning and stochastic processes. It contains set systems, constructions of measures using outer measures, the integral, and product measures.
Basic courses in analysis, and an understanding of mathematical proofs.
Elective in Data
Numerical Optimal Control
Lecturer: Moritz Diehl
Language: in English
Tutorial / flipped classroom: Di, 14-16h, HS II, Albertstr. 23b
Sit-in exam: date to be announced
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
Elective in Data
Theory and Numerics for Partial Differential Equations – Selected Nonlinear Problems
Lecturer: Sören Bartels
Assistant: Tatjana Schreiber
Language: in English
Lecture: Mo, 12-14h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
The lecture addresses the development and analysis of numerical methods for the approximation of certain nonlinear partial differential equations. The considered model problems include harmonic maps into spheres and total-variation regularized minimization problems. For each of the problems, a suitable finite element discretization is devised, its convergence is analyzed and iterative solution procedures are developed. The lecture is complemented by theoretical and practical lab tutorials in which the results are deepened and experimentally tested.
'Introduction to Theory and Numerics for PDEs' or 'Introduction to PDEs'
Elective in Data
Topological Data Analysis
Lecturer: Mikhail Tëmkin
Language: in English
Information will follow!
Information will follow!
Elective in Data
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, 2025 to October 8, 2025.
Seminar: Computational PDEs – Gradient Flows and Descent Methods
Lecturer: Sören Bartels
Language: Talk/participation possible in German and English
Seminar: Mo, 14-16h, SR 226, Hermann-Herder-Str. 10
Preliminary Meeting 15.07., 12:30, Raum 209, Hermann-Herder-Str. 10
Preparation meetings for talks: Dates by arrangement
The seminar will be devoted to the development of reliable and efficient discretizations of time stepping methods for parabolic evolution problems. The considered model problems either result from minimization problems or dynamical systems and are typically constrained or nondifferentiable. Criteria that allow to adjust the step sizes and strategies that lead to an acceleration of the convergence to stationary configurations will be addressed in the seminar. Specific topics and literature will be assigned in the preliminary meeting.
Elective in Data
Mathematical Seminar
Seminar: Medical Data Science
Lecturer: Harald Binder
Language: Talk/participation possible in German and English
Seminar: Mi, 10:15-11:30h, HS Medizinische Biometrie, 1. OG, Stefan-Meier-Str. 26
Preregistration:
Preliminary Meeting 23.07., 10:15, 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.
Elective in Data
Mathematical Seminar
Seminar: Random Walks
Lecturer: Angelika Rohde
Assistant: Johannes Brutsche
Language: Talk/participation possible in German and English
Seminar: Mo, 16-18h, SR 127, Ernst-Zermelo-Str. 1
Preliminary Meeting 22.07., Raum 232, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Preregistration: If your are interested in the seminar, please write an email to Johannes Brutsche listing your prerequisites in probability and note if you plan to attend the Probability Theory II.
Random walks are stochastic processes (in discrete time) formed by successive summation of independent, identically distributed random variables and are one of the most studied topics in probability theory. Many results that are part of this seminar also carry over to Brownian motion and related processes in continuous time. In particular, the theory for random walks contains many central and elegant proof ideas which can be extended to various other settings. We start the theory at the very beginning but quickly move on to proving local central limit theorems, study Green's function and recurrence properties, hitting times and the Gambler's ruin estimate. Further topics may include a dyadic coupling with Brownian motion, Dirichlet problems, random walks that are not indexed in \(\mathbb{N}\) but the lattice \(\mathbb{Z}^d\), and intersection probabilities for multidimensional random walks (which are processes \(X:\mathbb{N}\rightarrow\mathbb{R}^d\)). Here, we will see that in dimension \(d=1,2,3\) two paths hit each other with positive probability, while for \(d\geq 4\) they avoid each other almost surely.
Probability Theory I \
Some talks only require knowledge of Stochastics I, so if you are interested in the seminar and have not taken part in the probability theory I class, do not hesitate to reach out to us regarding a suitable topic.
Elective in Data
Mathematical Seminar
Seminar: Data-Driven Medicine from Routine Data
Lecturer: Nadine Binder
Note: Only for the degree programme "Mathematics in Data and Technology"
Imagine being able to use routine data such as diagnoses, lab results, and medication plans to answer medical questions in innovative ways and improve patient care. In this seminar, we will learn to identify relevant data, understand suitable analysis methods, and what to consider when applying them in practice. Together, we will analyze scientific studies on routine data and discuss clinical questions, the methods used, and their feasibility for implementation.
What makes this seminar special: Medical and mathematics students collaborate to understand scientific studies from both perspectives. When possible, you will work in pairs (or individually if no pair can be formed) to analyze a study from your respective viewpoints and prepare related presentations. You may test available programming code or develop your own approaches to replicate the methods and apply them to your own questions. The pairs can be formed during the preliminary meeting.
necessary: Basics in Applied Mathematics \
useful: Probability Theory I
Elective in Data
Mathematical Seminar