Preliminary course catalogue - changes and additions are still possible.
Click on the course title for more information!
New (and partly not yet in den annotated course catalogue):
Further Chapters in Analysis
Lecturer: Ernst August v. Hammerstein
Language: in German
Lecture: Mi, 8-10h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, various dates
Sit-in exam: date to be announced
This compulsory lecture for teacher training students in the M.Ed. builds on the basic lectures Analysis I and II and supplements them with the following two main topics:
\textit{Multidimensional integration:} The one-dimensional Riemann integral known from Analysis I is generalized to real-valued functions of several variables, for which a suitable instrument for measuring the content/volume of multidimensional sets is first introduced with the Jordan content. Then the classical integral theorems (transformation theorem, Fubini's theorem) are derived, and path and surface integrals are considered. With the help of the divergence and rotation of vector fields, the two aforementioned integral types can be related to each other using the integral theorems of Gauß and Stokes, which considerably simplifies the calculations in practical applications.
\textit{Complex Analysis:} In contrast to Analysis I, here the (complex) differentiability of functions of a complex variable is examined. As will be shown, complex differentiable, so-called holomorphic functions are subject to much stricter rules and laws than their real-valued counterparts, which leads to both beautiful and surprising results. To this end, we will prove Cauchy's intergal theorem and Cauchy's integral formula and take a closer look at applications and conclusions from these.
Analysis~I and II, Linear Algebra~I and II
Further Chapters in Analysis
Lecturer: Maximilian Stegemeyer
Language: in German
Lecture: Di, Do, 10-12h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
Algebraic topology studies topological spaces by assigning algebraic objects, e.g. groups, vector spaces or rings, to them in a particular way. This assignment is usually done in a way which is invariant under homotopy equivalences. Therefore one often speaks of homotopy invariants and algebraic topology can be seen as the study of the construction and the properties of homotopy invariants.
In this lecture we will first recall the notion of the fundamental group of a space and study its connection to covering spaces. Then we will introduce the singular homology of a topological space and study it extensively. In the end, we will consider cohomology and homotopy groups and explore their relation to singular homology. We will also consider various applications of these invariants to topological and geometric problems.
Topology
Mathematical Concentration
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
Mathematical Concentration
Complex Analysis
Lecturer: Stefan Kebekus
Assistant: Andreas Demleitner
Language: in German
Lecture: Di, Do, 8-10h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
Sit-in exam: date to be announced
Analysis I and II, Linear Algebra I
Mathematical Concentration
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
Mathematical Concentration
Calculus of Variations
Lecturer: Guofang Wang
Assistant: Florian Johne
Language: in German
Lecture: Mo, Mi, 10-12h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
The aim of the calculus of variations is to minimise or maximise certain mathematically treatable quantities. More precisely, we consider \(\Omega \subset {\mathbb R}^n\) functionals or variation integrals of the form \[F (u) = \int_\Omega f(x,u (x ),Du (x))dx, \quad \hbox{ f\"ur } u : \Omega\to {\mathbb R}\] on \(\Omega \subset {\mathbb R}^n\).
Examples are arc length and area, as well as energies of fields in physics. The central question is the existence of minimisers. After a brief introduction to the functional analysis tools, we will first familiarise ourselves with some necessary and sufficient conditions for the existence of minimisers. We will see that compactness plays a very important role. We will then introduce some techniques that help us to get by without compactness in special cases: The so-called compensated compactness and the concentrated compactness.
necessary: Functional Analysis \
useful: PDE, numerical PDE
Mathematical Concentration
Reading courses
Lecturer: All professors and 'Privatdozenten' of the Mathematical Institute
Language: Talk/participation possible in German and English
In a reading course, the material of a four-hour lecture is studied in supervised self-study. In rare cases, this may take place as part of a course; however, reading courses are not usually listed in the course catalog. If you are interested, please contact a professor or a private lecturer before the start of the course; typically, this will be the supervisor of your Master's thesis, as the reading course ideally serves as preparation for the Master's thesis (both in the M.Sc. and the M.Ed. programs).
The content of the reading course, the specific details, and the coursework requirements will be determined by the supervisor at the beginning of the lecture period. The workload should be equivalent to that of a four-hour lecture with exercises.
Reading Course
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
Supplementary Module in Mathematics
Linear Algebraic Groups
Lecturer: Abhishek Oswal
Assistant: Damian Sercombe
Language: in English
Lecture: Mo, 14-16h, SR 125, Ernst-Zermelo-Str. 1
There is no information available yet.
There is no information available yet.
Supplementary Module in Mathematics
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.
Supplementary Module in Mathematics
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
Supplementary Module in Mathematics
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
Supplementary Module in Mathematics
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
Supplementary Module in Mathematics
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'
Supplementary Module in Mathematics
Topics in Mathematical Physics
Lecturer: Chiara Saffirio
Language: in English
Lecture: Mo, 12-14h, SR 404, Ernst-Zermelo-Str. 1
This course provides an introduction to analytical methods in mathematical physics, with a particular emphasis on many-body quantum mechanics. A central focus is the rigorous proof of the stability of matter for Coulomb systems, such as atoms and molecules. The key question - why macroscopic objects made of charged particles do not collapse under electromagnetic forces - remained unresolved in classical physics and lacked even a heuristic explanation in early quantum theory. Remarkably, the proof of stability of matter marked the first time that mathematics offered a definitive answer to a fundamental physical and stands as one of the early triumphs of quantum mechanics.
Content:
Analysis III and Linear Algebra are required. \
No prior knowledge of physics is assumed; all relevant physical concepts will be introduced from scratch.
Supplementary Module in Mathematics
Topological Data Analysis
Lecturer: Mikhail Tëmkin
Language: in English
Information will follow!
Information will follow!
Supplementary Module in Mathematics
Mathematics Education ‒ Functions and Analysis
Lecturer: Katharina Böcherer-Linder
Language: in German
Do, 9-12h, SR 226, Hermann-Herder-Str. 10
Exemplary implementations of the theoretical concepts of central mathematical thought processes such as concept formation, modeling, problem solving and reasoning for the content areas of functions and analysis. \\ Barriers to understanding, pre-concepts, basic ideas, specific difficulties for the content areas of functions and analysis. \\ Fundamental possibilities and limitations of media, in particular of computer-aided mathematical tools mathematical tools and their application for the content areas of functions and analysis. Analysis of individual mathematical learning processes and errors as well as development individual support measures for the content areas of functions and analysis.
Introduction to Mathematics Education \
Knowledge about analysis and numerics
Mathematics Education for Specific Areas of Mathematics
Mathematics Education ‒ Probability Theory and Algebra
Lecturer: Frank Reinhold
Language: in German
Mi, 11-14h, SR 404, Ernst-Zermelo-Str. 1
Exemplary implementations of the theoretical concepts of central mathematical thought processes such as concept formation, modeling, problem solving and reasoning for the content areas of stochastics and algebra. \\ Barriers to understanding, pre-concepts, basic ideas, specific difficulties for the content areas of stochastics and algebra.\ Basic possibilities and limitations of media, especially computer-based mathematical tools and their mathematical tools and their application for the content areas of stochastics and algebra. and algebra. \\ Analysis of individual mathematical learning processes and errors as well as development individual support measures for the content areas of stochastics and algebra.
Introduction to Mathematics Education \ knowledge from stochastics and algebra
Mathematics Education for Specific Areas of Mathematics
Mathematics education seminar: Media Use in Teaching Mathematics
Lecturer: Jürgen Kury
Language: in German
Seminar: Mi, 15-18h, SR 404, Ernst-Zermelo-Str. 1
Recommended: Basic courses in mathematics
GeoGebra Account (can be created in the seminar)
Supplementary Module in Mathematics Education
Mathematics education seminars at Freiburg University of Education
Lecturer: Lecturers of the University of Education Freiburg
Language: in German
Supplementary Module in Mathematics Education
Module "Research in Mathematics Education"
Lecturer: Lecturers of the University of Education Freiburg
Language: in German
Part 1: Seminar 'Development Research in Mathematics Education ‒ Selected Topics': Mo, 14-16h, Raum noch nicht bekannt, PH Freiburg
Part 2: Seminar 'Research Methods in Mathematics Education': Mo, 16-19h, Raum noch nicht bekannt, PH Freiburg
Part 3: Master's thesis seminar: Development and Optimisation of a Research Project in Mathematics Education
Registration: see course descriptions
The three related courses of the module prepare students for an empirical Master thesis in mathematics didactics. The course is jointly designed by all professors at the PH with mathematics didactics research projects at secondary levels 1 and 2 and is carried out by one of these researchers. Afterwards, students have the opportunity to start Master thesis with one of these supervisors - usually integrated into larger ongoing research projects.
The first course of the module provides an introduction to strategies of empirical didactic research (research questions, research status, research designs). Students deepen their skills in scientific research and the evaluation of subject-specific didactic research. In the second course (in the last third of the semester) students are introduced to central qualitative and quantitative research methods through concrete work with existing data (interviews, student products, experimental data), students are introduced to central qualitative and quantitative research methods. The third course is an accompanying seminar for the Master thesis.
The main objectives of the module are the ability to receive mathematics didactic research in order to didactic research to clarify questions of practical relevance and to plan an empirical mathematics didactics Master thesis. It will be held as a mixture of seminar, development of research topics in groups and active work with research data. Recommended literature will be depending on the research topics offered within the respective courses. The parts can also be attended in different semesters, for example part~1 in the second Master semester and part~2 in the compact phase of the third Master semester after the practical semester.
Research in Mathematics Education
Learning by Teaching
Organisation: Susanne Knies
Language: in German
What characterizes a good tutorial? This question will be discussed in the first workshop and tips and suggestions will be given. Experiences will be shared in the second workshop.
Supplementary Module in Mathematics
School Mathematical Aspects of Analysis and Linear Algebra
Lecturer: Katharina Böcherer-Linder, Markus Junker
Language: in German
Mo, 14-16h, SR 404, Ernst-Zermelo-Str. 1
Supplementary Module in Mathematics
Computer exercises for 'Introduction to Theory and Numerics of Partial Differential Equations'
Lecturer: Patrick Dondl
Language: in English
The computer tutorial accompanies the lecture with programming exercises.
See the lecture – additionally: programming knowledge.
Supplementary Module in Mathematics
Computer exercises in Numerics
Lecturer: Patrick Dondl
Assistant: Jonathan Brugger
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
Computer exercises for 'Theory and Numerics of Partial Differential Equations – Selected Nonlinear Problems'
Lecturer: Sören Bartels
Assistant: Tatjana Schreiber
Language: in English
In the practical exercises accompanying the lecture 'Theory and Numerics for Partial Differential Equations – Selected Nonlinear Problems', the algorithms developed and analyzed in the lecture are implemented and tested experimentally. The implementation can be carried out in the programming languages Matlab, C++ or Python. Elementary programming knowledge is assumed.
see lecture
Supplementary Module in Mathematics
Please note the registration modalities for the individual proseminars 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 examination; the registration period runs from August 1, 2025 to October 8, 2025; if you would like to attend a proseminar but have not been allocated a place, please contact the degree program coordinator immediately.
Proseminar: Mathematik im Alltag
Lecturer: Susanne Knies
Language: in German
Remaining places in the M.Ed. seminar after the school practical semester can be allocated as undergraduate seminar places. For more information see there!
Supplementary Module in Mathematics
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.
Lecturer: Susanne Knies
Assistant: Jonah Reuß
Language: in German
The seminar is preferably intended for M.Ed. students. Remaining places can be allocated as undergraduate seminar places.
Supplementary Module in Mathematics
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.
Supplementary Module in Mathematics
Lecturer: Wolfgang Soergel
Language: Talk/participation possible in German and English
Seminar: Di, 14-16h, SR 127, Ernst-Zermelo-Str. 1
Preregistration: In case of interest, please email to Wolfgang Soergel
Preliminary Meeting 17.07., 12:15
Structure of noncommutative rings with applications to representations of finite groups.
necessary: Linear Algebra I and II \
useful: Algebra and Number Theory
Supplementary Module in Mathematics
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.
Supplementary Module in Mathematics
Seminar: Minimal Surfaces
Lecturer: Guofang Wang
Language: Talk/participation possible in German and English
Seminar: Mi, 16-18h, SR 125, Ernst-Zermelo-Str. 1
Preliminary Meeting 30.07., SR 125, Ernst-Zermelo-Str. 1
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 appear, for example, in the investigation of soap skins and the construction of stable objects (e.g. in architecture). Elegant methods from various mathematical fields such as complex analysis, calculus of variations, differential geometry, and partial differential equations are used to analyse minimal surfaces.
Supplementary Module in Mathematics
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.
Supplementary Module in Mathematics