General Information
Below are the English lectures offered by our group in the summer term 2023.
If you would like to attend a seminar or reading course during the summer 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:
Link coming soon
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:
Kleinian Singularities
Content
Kleinian singularities are complex surfaces with an isolated singularity. They can be realized as the orbit space of a finite subgroup of the 2x2 special linear group. Their geometry is deeply intertwined with the representation theory of the corresponding group. The theory is classical but it is very beautiful, rich, explicit, and instructive. It is the basis of active modern research and illustrates how representation theory (non-commutative algebra) and algebraic geometry (commutative algebra) can interact.
Contact Time
2 SWS
Requirements
Course "Commutative Algebra";
knowledge from the courses "Character Theory of Finite Groups" and "Algebraic Geometry" are beneficial.
Frequency
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:
Quadratic Number Fields
Content
- structure of imaginary quadratic fields,
- ideals and ideal class group,
- ideals as geometric lattices,
- finiteness of the class group.
Contact Time
2 SWS lecture
1 SWS example class
Requirements
Course "Algebraische Strukturen"; knowledge from the courses "Elementare Zahlentheorie" and "Einführung: Algebra" are beneficial.
Frequency
The lecture takes place irregularly.
Click here for the KIS entry:
Quadratic Number Fields
Example Classes Quadratic Number Fields
Click here for the OLAT course:
Link coming soon
Riemann Surfaces
Content
Riemann surfaces and complex manifolds, fundamental groups, differential forms and integration, sheaves, cohomology, divisors, theorem by Riemann-Roch, Serre duality
Contact Time
2 SWS
Requirements
Course "Einführung in die Funktionentheorie"
Frequency
The lecture takes place irregularly.
Click here for the KIS entry:
Click here for the OLAT course: