## General Information

Below are the English lectures offered by our group in the summer term 2024.

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**

- global fields,
- modules over Dedekind domains,
- valuations and completions,
- ring of integers and orders.

**Contact Time**

4 SWS lecture

2 SWS example class

**Requirements**

Courses "Einführung: Algebra" and "Commutative Algebra". Knowledge from the "Quadratic Number Fields" module is desirable.

**Frequency**

The lecture takes place irregularly.

**Click here for the KIS entry:**

Example Classes Algebraic Number Theory

**Click here for OLAT course:**

**Content**

- Maschke's theorem,
- character table,
- orthogonality,
- rationality,
- Burnside theorem,
- induced characters,
- Frobenius group.

**Contact Time**

2 SWS lecture

1 SWS example class

**Requirements**

Courses "Algebraische Strukturen" and "Einführung: Algebra".

**Frequency**

The lecture takes place irregularly.

**Click here for the KIS entry:**

Charakter Theory Finite Groups

Example Classes Character Theory of Finite Groups

**Click here for the OLAT course:**

**Content**

- normal forms and standard bases for ideals and modules,
- Syzygies, free resolutions and the proof of the Buchberger-criterion,
- calculation of the normalization of Noetherian rings,
- calculation of the primary decomposition of ideals,
- Hilbert function,
- Ext and Tor.

**Contact Time**

4 SWS lecture

2 SWS example class

**Requirements**

Courses "Einführung in das symbolische Rechnen" and "Commutative Algebra"

**Frequency**

The lecture takes place every summer semester.

**Click here for the KIS entry:**

Example Classes Computer Algebra

**Click here for the OLAT course:**

**Content**

Symmetric cryptosystems:

- stream cipher and block cipher,
- frequency analysis,
- modern ciphers.

Asymmetric cryptosystems:

- factorization of large numbers, RSA,
- primality tests,
- discrete logarithm, Diffie-Hellman key exchange, El-Gamal encoding, Hash function, signature,
- cryptography on elliptic curves (ECC),
- attacks on the discrete logarithm problem,
- factorization algorithms (e.g. quadratic sieve, Pollard ρ, Lenstra).

**Contact Time**

4 SWS lecture

2 SWS example class

**Requirements**

Courses "Algebraische Strukturen" and "Elementare Zahlentheorie"

**Frequency**

The lecture takes place every summer semester.

**Click here for the KIS entry:**

**Click here for the OLAT course:**

**Content**

- theory of schemes (affine, projective and relative schemes),
- structure sheaves and sheaves of modules,
- flat families,
- Grothendieck functor.

**Contact Time**

2 SWS lecture

**Requirements**

Module "Basics of Mathematics", "Commutative Algebra" and "Algebraic Geometry". Knowledge from the course "Algebraic Structures" is desirable and helpful, but not necessarily required.

**Frequency**

The lecture takes place every summer semester.

**Click here for the KIS entry:**

**Click here for the OLAT course:**

**Content**

- construction of p-adic Numbers,
- p-adic integers,
- p-adic Topology,
- Hensel's Lemma,
- algebraic degree,
- Newton polygon,
- inertia and ramification groups.

**Contact Time**

2 SWS lecture

1 SWS example class

**Requirements**

Course "Algebraische Strukturen"; knowledge from the courses "Elementare Zahlentheorie" and "Einführung: Algebra" is beneficial.

**Frequency**

The lecture is offered irregularly.

**Click here for the KIS entry:**

example classes p-adic Numbers

**Click here for the OLAT course:**

**Content**

Mandatory content:

- affine and projective spaces, in particular the projective line and the projective plane,
- plane algebraic curves over the complex numbers,
- smooth and singular points,
- Bézout's theorem for plane projective curves,
- the topological genus of a curve,
- rational maps between plane curves and the Riemann-Hurwitz formula.

A selection of the following topics will be covered:

- polars and Hesse curve,
- dual curves and Plücker formula,
- linear systems and divisors on plane curves,
- real projective curves,
- Puiseux parametrization of plane curve singularities,
- invariants of plane curve singularities,
- elliptic curves,
- further aspects of plane algebraic curves.

**Contact Time**

2 SWS lecture

1 SWS example class

**Requirements**

Course "Algebraische Strukturen"; knowledge from the courses "Einführung: Algebra" and "Einführung: Topologie" is beneficial.

**Frequency**

The lecture takes place every summer semester.

**Click here for the KIS entry:**

Example Classes Plane Algebraic Curves

**Click here for the OLAT course:**