Detailed information can be found in the course descriptions and in the module handbooks (in German only).
Algebraic Number Theory
Lecturer: Abhishek Oswal
Assistant: Andreas Demleitner
Language: in English
Lecture: Di, Do, 12-14h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
Short description of topics: Number fields, Prime decomposition in Dedekind domains, Ideal class groups, Unit groups, Dirichlet's unit theorem, local fields, valuations, decomposition and inertia groups, introduction to class field theory.
Required: Algebra and Number Theory
Mathematical Concentration
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Patrick Dondl
Assistant: Oliver Suchan
Language: in German
Lecture: Mo, 12-14h, HS Rundbau, Albertstr. 21, Mi, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, various dates
19.02., 10:15-11:45, HS Rundbau, Albertstr. 21
Sit-in exam (resit) 28.04., 10:00-11:30, SR 226, Hermann-Herder-Str. 10
Lebesgue measure and measure theory, Lebesgue integral on measure spaces and Fubini's theorem, Fourier series and Fourier transform, Hilbert spaces. Differential forms, their integration and outer derivative. Stokes' theorem and Gauss' theorem.
Required: Analysis I and II, Linear Algebra I
Mathematical Concentration
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Sebastian Goette
Assistant: Mikhail Tëmkin
Language: in German
Lecture: Mo, Mi, 14-16h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
Differential geometry, especially Riemannian geometry, deals with the geometric properties of curved spaces. Such spaces also occur in other areas of mathematics and physics, for example in geometric analysis, theoretical mechanics and the general theory of relativity.
Required: Analysis~I–III, Lineare Algebra~I and II \ Recommended: Analysis of Curves and Surfaces ("Kurven und Flächen"), Topology
Mathematical Concentration
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Guofang Wang
Assistant: Christine Schmidt, Xuwen Zhang
Language: in German
Lecture: Mo, Mi, 12-14h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
A large number of different problems from the natural sciences and geometry lead to partial differential equations. Consequently, there can be no talk of an all-encompassing theory. Nevertheless, there is a clear picture for linear equations, which is based on three prototypes: the potential equation \(-\Delta u = f\), the heat equation \(u_t - \Delta u = f\) and the wave equation \(u_{tt} - \Delta u = f\), which we will examine in the lecture.
Required: Analysis III \ Recommended: Complex Analysis ({\em Funktionentheorie})
Mathematical Concentration
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Sören Bartels
Assistant: Vera Jackisch
Language: in English
Lecture: Di, Do, 10-12h, SR 226, Hermann-Herder-Str. 10
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 Extensions of Analysis) \ Recommended: Numerics for differential equations, Functional analysis
Mathematical Concentration
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: David Criens
Assistant: Eric Trébuchon
Language: in German
Lecture: Di, Mi, 16-18h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
24.02., 14:00-16:00, HS II, Albertstr. 23b
Complex analysis deals with functions \(f : \mathbb C \to \mathbb C\) , which map complex numbers to complex numbers. Many concepts of Analysis~I can be directly transferred to this case, e.\,g. the definition of differentiability. One might expect that this would lead to a theory analogous to Analysis~I but much more is true: in many respects you get a more elegant and simpler theory. For example, complex differentiability on an open set implies that a function is even infinitely often differentiable, and this is further consistent with analyticity. For real functions, all these notions are different. However, some new ideas are also necessary: For real numbers \(a\), \(b\) one integrates for \[\int_a^b f(x) \mathrm dx\] over the elements of the interval \([a, b]\) or \([b, a]\). However, if \(a\), \(b\) are complex numbers, it is no longer so clear clear how such an integral is to be calculated. One could, for example, in the complex numbers along the line that connects \(a, b \in \mathbb C\), or along another curve that leads from \(a\) to \(b\). Does this lead to a well-defined integral term or does such a curve integral depend on the choice of the curve?
Required: Analysis I+II, Linear Algebra I
Mathematical Concentration
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Set Theory – Independence Proofs
Lecturer: Maxwell Levine
Assistant: Hannes Jakob
Language: in English
Lecture: Di, Do, 12-14h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
How does one prove that something cannot be proved? More precisely, how does one prove that a particular statement does not follow from a particular collection of axioms?
These questions are often asked with respect to the axioms most commonly used by mathematicians: the axioms of Zermelo-Fraenkel set theory, or ZFC for short. In this course, we will develop the conceptual tools needed to understand independence proofs with respect to ZFC. On the way we will develop the theory of ordinal and cardinal numbers, the basics of inner model theory, and the method of forcing. In particular, we will show that Cantor's continuum hypothesis, the statement that \(2^{\aleph_0}=\aleph_1\), is independent of ZFC.
Required: Mathematical Logic
Mathematical Concentration
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Annette Huber-Klawitter, Amador Martín Pizarro
Assistant: Christoph Brackenhofer
Language: in German
Lecture: Di, Do, 10-12h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
Semi-algebraic geometry is about properties of subsets of \(**R**^n\), which are given by inequalities of the form [ f(x1, \dots, xn)\geq 0] for polynomials \(f\in**R**[X_1,\dots,X_n]\).
The theory has many different facets. On the one hand, it can be seen as a version of algebraic geometry over \(\mathbf{R}\) (or even more generally over so-called real closed fields). On the other hand, the properties of these fields are a central tool for the model-theoretic proof of Tarski-Seidenberg's theorem on quantifier elimination in real closed fields. Geometrically, this is interpreted as a projection theorem.
From this theorem, a proof of Hilbert's 17th problem easily follows, which was solved by Artin in 1926.
\textit{Is every real polynomial \(P \in \mathbf{R}[x_1, \dots, x_n]\), which takes a non-negative value for every \(n\)-tuple in \(\mathbf{R}^n\), a sum of squares of rational functions (i.e., quotients of polynomials)?}
In the lecture, we will explore both aspects. Necessary tools from commutative algebra or model theory will be discussed according to the prior knowledge of the audience.
Required: Algebra and Number Theory \ Recommended: Knowledge in commutative algebra and algebraic geometry (cf. Kommutative Algebra und Einführung in die algebraische Geometrie), model theory
Mathematical Concentration
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.