Algebra and Number Theory
Lecture: Di, Do, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, various dates
Sir-in Exam: Date to be announced
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
Algebra and Number Theory (2HfB21, MEH21)
Compulsory Elective in Mathematics (BSc21)
Introduction to Algebra and Number Theory (MEB21)
Algebra and Number Theory (MEdual24)
Pure Mathematics (MSc14)
Elective (MSc14)
Elective (MScData24)
Algebraic Topology
Lecture: Di, Do, 10-12h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
Teacher: Maximilian Stegemeyer
Language: in German
Elective (Option Area) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Mathematical Concentration (MEd18, MEH21)
Pure Mathematics (MSc14)
Mathematics (MSc14)
Concentration Module (MSc14)
Elective (MSc14)
Elective (MScData24)
Linear Algebraic Groups
Lecture: Mi, 14-16h, SR 125, Ernst-Zermelo-Str. 1
Teacher: Abhishek Oswal
Undergraduate seminar: Elementary Number Theory
Seminar: Mi, 8-10h, SR 404, Ernst-Zermelo-Str. 1
Preregistration:
Preliminary Meeting
Preparation meetings for talks: Dates by arrangement
Teacher: Annette Huber-Klawitter
Assistant: Christoph Brackenhofer
Undergraduate Seminar (2HfB21, BSc21, MEH21, MEB21)
Seminar
Seminar: Di, 14-16h, SR 127, Ernst-Zermelo-Str. 1
Teacher: Wolfgang Soergel
Assistant: Damian Sercombe
Elective (Option Area) (2HfB21)
Mathematical Seminar (BSc21)
Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)
Mathematical Seminar (MSc14)
Elective (MSc14)
Elective (MScData24)
Lecture: Di, Do, 8-10h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
Teacher: Wolfgang Soergel
Assistant: Xier Ren
Language: in German
In linear algebra you studied linear systems of equations. In commutative algebra, we study polynomial equation systems such as \(x^2+y^2 = \) 1 and their solution sets, the algebraic varieties. It will turn out that such a variety is closely related to the ring of the restrictions of polynomial functions on that variety, and that we can extrapolate this relationship to a geometric understanding of any commutative rings, in particular the ring of the integers. Commutative algebra, algebraic geometry, and number theory grow together in this conceptual building. The lecture aims to introduce into this conceptual world. We will especially focus on the dimension of algebraic varieties and their cutting behavior, which generalizes the phenomena known from the linear algebra on the case of polynomial equation systems.
necessary: Linear Algebra I+II
useful: Algebra and Number Theory
Elective (Option Area) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Pure Mathematics (MSc14)
Mathematics (MSc14)
Concentration Module (MSc14)
Elective (MSc14)
Elective (MScData24)
Lecture: Mi, 14-16h, SR 125, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
Teacher: Amador Martín Pizarro
Assistant: Charlotte Bartnick
Language: in German
Groups without any non-trivial normal subgroup are called simple groups. Similar to prime numbers for the natural numbers, simple groups form the building blocks for finite groups. It is easy to see that Abelian finite simple groups are cyclic. Non-Abelian examples are alternating groups and Lie-type groups.
The classification of finite simple groups is far beyond the scope of this course. However, we will illustrate some of the recurring ideas of classification and, in particular, prove the following result of Brauer and Fowler:
Theorem: Let G be a finite group of even order such that the centre is of odd order. Then there is an element \(g \neq 1_G\) with \(|G| < |C_G (g)|^3\) .
This theorem had a particularly large impact on the classification of finite simple groups, as it suggests that these could be classified by examining the centralisers of elements of order 2.
Algebra and Number Theory
Elective (Option Area) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)
Pure Mathematics (MSc14)
Mathematics (MSc14)
Concentration Module (MSc14)
Elective (MSc14)
Elective (MScData24)
Seminar: Do, 14-16h, SR 404, Ernst-Zermelo-Str. 1
Preregistration: until 30.01. to Vivien Vogelmann
Preliminary Meeting 04.02., 12:00, SR 318, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Teacher: Susanne Knies
Assistant: Vivien Vogelmann
Language: in German
The seminar deals with statements that are known from finite-dimensional vector spaces but no longer apply in the infinite-dimensional case. What are the consequences of this? What applies instead? With which additional conditions can one possibly save oneself?
For more detailed information see the webpage!
Analysis I, II and Linear Algebra I, II
Undergraduate Seminar (2HfB21, BSc21, MEH21, MEB21)
Undergraduate seminar: Lattice Theory
Seminar: Mo, 14-16h, SR 127, Ernst-Zermelo-Str. 1
Preregistration: until 31.01. to Markus Junker
Preliminary Meeting 07.02., 11:15, SR 318, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Teacher: Markus Junker
Language: in German
Lattices are similarly basic mathematical structures as orderings or groups. A lattice is a set with two associative and commutative operations \(\cap\) and \(\cup\) that satisfy the absorption laws \(a \cap (a \cup b) = a\) and \(a \cup (a \cap b) = a\). For example, the subsets of a fixed set form a lattice; or the sub vector spaces of a fixed vector space if \(\cup\) is interpreted as the sub vector space generated by the set-theoretic union . Lattices with special additional properties are, for example, Boolean algebras,
In this seminar, we will look at what can be said about arbitrary lattices and then at some results about more specialised lattices.
Linear Algebra I and II, Analysis I
Undergraduate Seminar (2HfB21, BSc21, MEH21, MEB21)
Seminar: Do, 10-12h, SR 403, Ernst-Zermelo-Str. 1
Preregistration: by e-mail to Wolfgang Soergel
Preliminary Meeting 28.01., 14:15, SR 127, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Teacher: Wolfgang Soergel
Assistant: Damian Sercombe
Language: Talk/participation possible in German and English
This seminar is intended to introduce to the theory of linear algebraic groups. Linear algebraic groups are generalizations of the matrix groups known from linear algebra.
I imagine a format in which I or Sercombe lecture and in between the seminar participants give talks. The seminar is a reasonable addition to the commutative algebra lecture. Reference to that lecture increase in the course of the seminar.
Algebra and Number Theory (where the details of Galois theory and field theory are less relevant than the general theory of groups and rings) and Linear Algebra.
Elective (Option Area) (2HfB21)
Mathematical Seminar (BSc21)
Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)
Mathematical Seminar (MSc14)
Elective (MSc14)
Elective (MScData24)
Seminar on p-adic Geometry
Seminar: Mo, 10-12h, SR 404, Ernst-Zermelo-Str. 1
Preliminary Meeting 13.02., 14:30, SR 404, Ernst-Zermelo-Str. 1, Please email Abhisehk Oswal, and Ben Snodgrass if you are interested in the seminar but cannot make it to the preliminary meeting.
Teacher: Abhishek Oswal
Assistant: Ben Snodgrass
Language: in English
It has become clear over the last several decades that \(p\)-adic techniques play an indispensable role in arithmetic geometry. At an elementary level, \(p\)-adic numbers provide a compact and convenient language to talk about congruences between integers. Concretely, just as the field of real numbers \(\mathbb R\) arise as the completion of the field \(\mathbb Q\) of rational numbers with respect to the usual notion of distance on \(\mathbb Q\), the field \(\mathbb Q_p\) of \(p\)-adic numbers arise as the completion of \(\mathbb Q\) with respect to an equally natural \(p\)-adic metric. Roughly, in the \(p\)-adic metric, an integer \(n\) is closer to \(0\), the larger the power of the prime number \(p\) that divides it. A general philosophy in number theory is then to treat all these completions \(\mathbb R\), \(\mathbb Q_p\) of the field \(\mathbb Q\) on an equal footing. As we shall see in this course, familiar concepts from real analysis (i.e. notions like analytic functions, derivatives, measures, integrals, Fourier analysis, real and complex manifolds, Lie groups...), have completely parallel notions over the \(p\)-adic numbers.
While the Euclidean topology of \(\mathbb R^n\) is rather well-behaved (so one may talk meaningfully about paths, fundamental groups, analytic continuation, ...), the \(p\)-adic field \(\mathbb Q_p\) on the other hand is totally disconnected. This makes the task of developing a well-behaved notion of global \(p\)-adic analytic manifolds/spaces rather difficult. In the 1970s, John Tate’s introduction of the concept of rigid analytic spaces, solved these problemsand paved the way for several key future developments in \(p\)-adic geometry.
The broad goal of this course will be to introduce ourselves to this world of \(p\)-adic analysis and rigid analytic geometry (due to Tate). Along the way, we shall see a couple of surprising applications of this circle of ideas to geometry and arithmetic. Specifically, we plan to learn Dwork’s proof of the fact that the zeta function of an algebraic variety over a finite field is a rational function.
Field theory, Galois theory and Commutative algebra.
Some willingness to accept unfamiliar concepts as black boxes. Prior experience with algebraic number theory, or algebraic geometry will be beneficial but not necessary.
Elective (Option Area) (2HfB21)
Mathematical Seminar (BSc21)
Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)
Mathematical Seminar (MSc14)
Elective (MSc14)
Elective (MScData24)
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
Algebra and Number Theory (2HfB21, MEH21)
Compulsory Elective in Mathematics (BSc21)
Introduction to Algebra and Number Theory (MEB21)
Algebra and Number Theory (MEdual24)
Pure Mathematics (MSc14)
Elective (MSc14)
Elective (MScData24)
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
Elective (Option Area) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Mathematical Concentration (MEd18, MEH21)
Pure Mathematics (MSc14)
Mathematics (MSc14)
Concentration Module (MSc14)
Elective (MSc14)
Elective (MScData24)
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
Elective (Option Area) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Mathematical Concentration (MEd18, MEH21)
Pure Mathematics (MSc14)
Mathematics (MSc14)
Concentration Module (MSc14)
Elective (MSc14)
Elective (MScData24)
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
Elective (Option Area) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Pure Mathematics (MSc14)
Mathematics (MSc14)
Concentration Module (MSc14)
Elective (MSc14)
Elective (MScData24)
Seminar: Di, 14-16h, SR 127, Ernst-Zermelo-Str. 1
Preregistration:
Preparation meetings for talks: Dates by arrangement
Teacher: Wolfgang Soergel
Assistant: Damian Sercombe
Language: in German
In this proseminar we will discuss topics that are found in various textbooks and scripts for basic lectures in linear algebra but which are not part of the standard material. The lectures build on each other only slightly.
Linear Algebra ~I and II, Analysis~I and II.
Undergraduate Seminar (2HfB21, BSc21, MEH21, MEB21)
Seminar: Di, 14-16h, SR 125, Ernst-Zermelo-Str. 1
Preliminary Meeting 16.07., SR 125, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Teacher: Sebastian Goette
Assistant: Mikhail Tëmkin
Language: Talk/participation possible in German and English
We will discuss advanced topics in algebraic topology. Depending on the interest of the participants we could work on one of the following topics---if you have other topic suggestions, please contact the lecturer.
Algebraic Topology~I and II
Elective (Option Area) (2HfB21)
Mathematical Seminar (BSc21)
Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)
Mathematical Seminar (MSc14)
Elective (MSc14)
Elective (MScData24)
Seminar: Fr, 8-10h, SR 404, Ernst-Zermelo-Str. 1
Preregistration:
Preliminary Meeting 15.07., 11:00, SR 318, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Teacher: Annette Huber-Klawitter
Assistant: Xier Ren
Language: Talk/participation possible in German and English
In this seminar, we are going to study finite dimensional (unital, possibly non-commutative) algebras over a (commutative) field \(k\). Prototypes are the rings of square matrices over \(k\), finite field extensions, or the algebra \(k^n\) with diagonal multiplication.
We will concentrate on path algebras of finite quivers (German: Köcher). Modules over them are equivalently described as representations of the quiver. Many algebraic properties can be directly understood from properties of the quiver.
Required: Linear Algebra \ Recommended: Algebra and Number Theory, Commutative Algebra and Introduction to Algebraic Geometry
Elective (Option Area) (2HfB21)
Mathematical Seminar (BSc21)
Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)
Mathematical Seminar (MSc14)
Elective (MSc14)
Elective (MScData24)