Detailed information can be found in the course descriptions and in the module handbooks (in German only).
Lecture: Di, Do, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, various dates
25.02., 13:00-16:00, HS Rundbau, Albertstr. 21
Sit-in exam (resit) 09.04., 09:00-12:00
Teacher: Wolfgang Soergel
Assistant: Damian Sercombe
Language: in German
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.
Required: Linear Algebra~I and II
Pure Mathematics
Elective
Algebraic Number Theory
Lecture: Di, Do, 12-14h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
Teacher: Abhishek Oswal
Assistant: Andreas Demleitner
Language: in English
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
Pure Mathematics
Elective
Mathematics
Concentration Module
Lecture: Mo, Mi, 14-16h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
Teacher: Sebastian Goette
Assistant: Mikhail Tëmkin
Language: in German
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
Pure Mathematics
Elective
Mathematics
Concentration Module
Lecture: Mo, Mi, 12-14h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
Teacher: Guofang Wang
Assistant: Christine Schmidt, Xuwen Zhang
Language: in German
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})
Pure Mathematics
Elective
Mathematics
Concentration Module
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
Teacher: David Criens
Assistant: Eric Trébuchon
Language: in German
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
Pure Mathematics
Elective
Set Theory – Independence Proofs
Lecture: Di, Do, 12-14h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
Teacher: Maxwell Levine
Assistant: Hannes Jakob
Language: in English
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
Pure Mathematics
Elective
Mathematics
Concentration Module
Lecture: Di, Do, 10-12h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
Teacher: Annette Huber-Klawitter, Amador Martín Pizarro
Assistant: Christoph Brackenhofer
Language: in German
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
Pure Mathematics
Elective
Mathematics
Concentration Module
Lecture: Mo, 14-16h, SR 127, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
Teacher: Xuwen Zhang
Language: in English
We will study functions of bounded variation, which are functions whose weak first partial derivatives are Radon measures. This is essentially the weakest definition of a function to be differentiable in the measure-theoretic sense. After discussing the basic properties of them, we move on to the study of sets of finite perimeter, which are Lebesgue measurable sets in the Euclidean space whose indicator functions are BV functions. Sets of finite perimeter are fundamental in the modern Calculus of Variations as they generalize in a natural measure-theoretic way the notion of sets with regular boundaries and possess nice compactness, thus appearing in many Geometric Variational problems. If time permits, we will discuss the (capillary) sessile drop problem as one important application.
Required: Basic knowledge in measure theory and analysis is required.
Pure Mathematics
Elective
Mathematics
Concentration Module
Lecture: Do, 14-16h, SR 404, Ernst-Zermelo-Str. 1
Exercise session: Di, 8-10h, SR 127, Ernst-Zermelo-Str. 1
Teacher: Maximilian Stegemeyer
Language: in English
Lie groups and operations of Lie groups play a central role in geometry and topology. They can be used to describe continuous symmetries, one of the most important concepts of mathematics and physics. Exploiting symmetries, e.g. when describing homogeneous spaces, makes it easier to solve many specific problems and often provides a deeper insight into the structures examined. In addition, the geometry and topology of Lie groups and homogeneous spaces is of great interest.
In this lecture, we start with introducing the basic theory of Lie groups and Lie algebras, especially with insights into the structure theory of Lie algebras. In the second part we will look at homogeneous spaces with a special focus on Riemannian symmetric spaces. The latter form an important class of examples of Riemannian manifolds. In addition to the Lie-theoretical aspects, a special focus will always be on the homogeneous Riemannian metrics of the respective spaces.
Required: Differential geometry~I
Pure Mathematics
Elective
Mathematics
Concentration Module