Algebra, Geometry and Computer Algebra Group


General Information

Below are the English lectures offered by our group in the summer term 2021.

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

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

Cohomology of Groups

Content

  • Semi-direct products,
  • homological algebra,
  • free, projective and injective resolutions,
  • homology and cohomology of groups,
  • group extensions and cohomology,
  • Schur multiplier,
  • central extensions,
  • projective representations.

Contact Time

4 SWS lecture
2 SWS example class

Requirements

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

Frequency

The lecture takes place irregularly.

Click here for the KIS entry:

Cohomology of Groups

Example Classes Cohomology of Groups

Click here for the OLAT course:

Cohomology of Groups

Complex Manifolds and Hodge Theory

Content

  • complex manifolds, sub varieties,
  • Vector bundle, cuts, cohomology,
  • Applications, e.g. Divisors and straight bundles,
  • Differential forms,    
  • Serre-duality.

Contact time

2 SWS / 30 h Lecture

Requirements

Course "Introduction: Function Theory" and Module "Algebraic Geometry".

Frequency

The lecture takes place irregularly.

Click here for the KIS entry:

Complex Manifolds and Hodge Theory

Click here for the OLAT entry:

Complex Manifolds and Hodge Theory

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:

Computer Algebra

Example Classes Computer Algebra

Click here for the OLAT course:

Computer Algebra

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

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