## General Information

Below are the English lectures offered by our group in the winter term 2024/2025.

If you would like to attend a seminar or reading course during the winter semester, please contact the respective supervisor by e-mail. Appointments are then determined in consultation with the participants.

If you would like to write a thesis in our working group, simply contact the desired supervisor directly.

**Content**

Mandatory content:

- Affine and projective varieties (especially: dimension, morphisms, smooth and singular points, blow-ups of points, applications and examples),
- Sheaves and sheaf cohomology with applications (Riemann-Roch's theorem for curves, projective embedding of curves).

It also covers a selection of the following topics:

- Schemes,
- Differential forms,
- other aspects of algebraic geometry.

**Contact Time**

4 SWS lecture

2 SWS example class

**Requirements**

Module "Commutative Algebra".

Knowledge from the course "Plane Algebraic Curves" is desirable and helpful, but not necessarily required.

**Frequency**

The lecture is offered every year in the winter semester.

**Click here for the KIS entry:**

Algebraic Geometry (Example Class)

**Hier geht es zum OLAT-Kurs:**

**Content**

- LLL algorithm,
- Number fields, ring of integers, units, class group,
- Decomposition behavior of primes,
- Algorithmic calculation of these quantities.

**Contact Time**

4 SWS lecture

2 SWS example class

**Requirements**

Courses "Algebraic Structures" and "Introduction: Algebra";

In addition, basic properties of Dedekind rings from the course "Commutative Algebra" are used. Knowledge from the "Quadratic Number Fields" module is desirable and helpful.

**Frequency**

The lecture takes place irregularly.

**Click here for the KIS entry:**

Vorlesung Algorithmic Number Theory

Übungen Algorithmic Number Theory

**Hier geht es zum OLAT-Kurs:**

**Content**

- Rings, modules, localization, lemma of Nakayama,
- Noether / Artin rings and modules,
- Primary decomposition,
- Krull's Principal Ideal Theorem, dimension theory
- Integral ring extensions, going-up, going-down, normalization,
- Noether normalization, Hilbert's Nullstellensatz,
- Dedekind rings, invertible ideals.

**Contact Time**

4 SWS lecture

2 SWS example class

**Requirements**

Lecture "Algebraische Strukturen".

The lecture "Introduction: Algebra" is nice to have, but not necessary.

**Frequency**

The lecture is offered every year in the winter semester.

**Click here for the KIS entry:**

Commutative Algebra (Example Class)

**Click here for the OLAT course:**

**Content**

- Algorithms for computing orbits, transversals, stabilisers; applications thereof,
- Fundamental Algorithms for permutation groups, e.g. stabiliser chains, Schreier-Sims algorithm, backtrack search,
- Algorithms for finitely presented groups, e.g. Tietze transformations, coset enumeration, Abelian invariants, subgroup presentations.

**Contact Time**

2 SWS lecture

1 SWS example class

**Requirements**

Modul "Grundlagen der Mathematik"

Lectures "Algebraische Strukturen" and "Einführung: Algebra"

**Frequency**

The lecture takes place irregularly.

**Click here for the KIS entry:**

Algorithmic Group Theory (Vorlesung)

Algorithmic Group Theory (Übung)

**Click here for the OLAT course:**