Algebra, Geometry and Computer Algebra Group


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.
 

Important Links

  • KIS: All dates of lectures/seminars
  • URM: Apply for an example class 
  • OpenOLAT: Material and example sheets and more informations (Login in the first lecture)

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:

Algebraic Number Theory

Example Classes Algebraic Number Theory

Click here for the OLAT course:

Algebraic Number Theory

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:

Categories

Click here for the OLAT course:

Categories

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:

Character Theory of Finite Groups

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:

Cryptography

Example Classes Cryptography

Click here for the OLAT course:

Cryptography

 

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:

Plane Algebraic Curves

Example Classes Plane Algebraic Curves

Click here for the OLAT course:

Plane Algebraic Curves

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:

Reflection Groups

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:

Singularity Theory

 

Click here for the OLAT course

Singularity Theory

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