Preliminary course catalogue - changes and additions are still possible.
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New (and partly not yet in den annotated course catalogue):
Lecturer: Wolfgang Soergel
Assistant: Damian Sercombe
Language: in German
Lecture: Di, Do, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, various dates
Sit-in exam: date to be announced
This lecture continues the linear algebra courses. It treats groups, rings, fields and applications in the number theory and geometry. The highlights of the lecture are the classification of finite fields, the impossibility of the trisection of angles with circle and ruler, the non-existence of a solution formula for the general equations of fifth degree and the quadratic reciprocity law.
Linear Algebra I and II
Elective
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
Elective
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
Elective
Model Theory
Lecturer: Amador Martín Pizarro
Assistant: Charlotte Bartnick
Language: in English
Lecture: Di, Do, 12-14h, SR 404, Ernst-Zermelo-Str. 1
The lecture will probably be held in English.
In this course the basics of geometric model theory will be discussed and concepts such as quantifier elimination and categoricity will be introduced. A theory has quantifier elimination if every formula is equivalent to a quantifier-free formula. For the theory of algebraically closed fields of fixed characteristic, this is equivalent to requiring that the projection of a Zariski-constructible set is again Zariski-constructible. A theory is called \(\aleph_1\)-categorical if all the models of cardinality \(\aleph_1\) are isomorphic. A typical example is the theory of non-trivial \(\mathbb Q\)-vector spaces. The goal of the course is to understand the theorems of Baldwin-Lachlan and of Morley to characterize \(\aleph_1\)-categorical theories.
necessary: Mathematical Logic \
useful: Algebra and Number Theory
Elective
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
Elective
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.
Elective
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.
Elective
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.
Elective
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.
Elective
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
Elective
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: 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
Elective
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.
Elective