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.
Algebraic Number Theory
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 the OLAT course:
Character Theory of Finite Groups
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:
Character Theory of Finite Groups
Example Classes Character Theory of Finite Groups
Click here for the OLAT course:
Computer Algebra
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:
Link coming soon
Cryptography
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:
Geometry of Schemes
Content
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 is offered every year in the winter semester.
Click here for the KIS entry:
Click here for the OLAT course:
Link coming soon
p-adic Numbers
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:
Click here for the OLAT course:
Plane Algebraic Curves
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: