General Information
Below are the English lectures offered by our group in the summer term 2022.
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:
Categories
Content
- Categories, functors, natural transformations,
- duality, Yoneda lemma,
- universal constructions, products, limits,
- adjoint functors,
- Abelian categories, kernels, co-kernels, exact sequences.
Contact Time
2 SWS lecture
Requirements
Frequency
The lecture takes place irregularly.
Click here for the KIS entry:
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:
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:
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:
Quadratic Number Fields
Reflection Groups
Content
Reflection groups are ubiquitous in mathematics. We concentrate on finite reflection groups over a field of characteristic zero and theirs
- central examples (including symmetric groups),
- Structural theory,
- Representation theory.
Contact time
2 SWS Lecture
1 SWS Example Class (optional)
Requirements
Einführung: Algebra, Character Theory of Finite Groups.
Frequency
The lecture is offered irregularly.Click here for the KIS entry:
Click here for the OLAT course:
Link follows
Singularity Theory
Content
- power series, Theorems of Weierstrass,
- analytic algebras,
- elementary theory of coherent sheaves,
- germ of a complex variety,
- local compactness theorem for morphisms,
- invariants of hyper surface singularities,
- finite determinacy,
- deformation theory of complete intersections,
- classification of simple hypersurface singularities.
Contact time
4 SWS Lecture
Requirements
Commutative Algebra and Algebraic Geometry are recommended but not necessary.
Frequency
The lecture takes place irregularly.
Click here for the KIS entry:
Click here for the OLAT course