Lecturer: Maximilian Stegemeyer
Language: in German

Time and place

Lecture: Di, Do, 10-12h, SR 404, Ernst-Zermelo-Str. 1
Exercise session: Mi, 14-16h, SR 403, Ernst-Zermelo-Str. 1

Requirements on examinations, assessments and coursework will be described in the supplements of the module handbooks to be published as part of the course cataloque by end of October.

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

Lecturer: Michael Růžička
Assistant: Luciano Sciaraffia
Language: in German

Time and place

Lecture: Mo, Mi, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, various dates
Sit-in exam 10.02., 10:00-12:00

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

Lecturer: Yuchen Bi
Language: in English

Time and place

Lecture: Di, Do, 12-14h, SR 226, Hermann-Herder-Str. 10, to be confirmed
Tutorial: 2 hours, date to be determined and announced in class

This course offers an introduction to differential geometry with a focus on the structure of smooth manifolds. Key topics include the construction and properties of vector fields, differential forms, and their applications. The course will also include an introduction to Riemannian metrics if time permits, though the treatment will remain at an introductory level.

Required: Analysis~I–III, Lineare Algebra~I and II \ Prior exposure to curves and surfaces (“Kurven und Flächen”) and topology is beneficial.

Mathematical Concentration

Lecturer: Stefan Kebekus
Assistant: Xier Ren
Language: in German

Time and place

Lecture: Di, Do, 8-10h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined and announced in class
Sit-in exam: date to be announced

Analysis I and II, Linear Algebra I

Mathematical Concentration

Lecturer: Patrick Dondl
Assistant: Ludwig Striet, Oliver Suchan
Language: in English

Time and place

Lecture: Mo, Mi, 12-14h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined and announced in class

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

Lecturer: Guofang Wang
Assistant: Florian Johne
Language: in German

Time and place

Lecture: Mo, Mi, 10-12h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined and announced in class

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