Algebra, Geometry and Computer Algebra Group


General Information

Below are the English lectures offered by our group in the summer semester 2019.

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.

The events, which take place irregularly, are planned for the coming semesters as follows:

At least one of the courses "Character Theory of Finite Group", "p-adic Numbers" or "Quadratic Number Fields" is offered every summer semester.

Important Links

  • KIS: All dates of lectures/seminars
  • URM: Apply for an example class (open til April  18th 2019, 12:00)
  • OpenOLAT: Material and example sheets and more informations (Login in the first lecture)

Algebraic Groups

Content

  • Linear algebraic groups,
  • classical groups,
  • nilpotent and solvable groups,
  • Borel-subgroups,
  • root systems,
  • classification theorem,
  • structure of semisimple groups.

Contact Time

4 SWS / 60 h lecture

Requirements

Courses "Einführung: Algebra" and "Commutative Algebra". Knowledge from the course "Algebraic Geometry" is desirable.

Frequency

The lecture takes place every summer semester.

Click here for the KIS entry:
Algebraic Groups (Vorlesung)
Algebraic Groups (Übung)

Click here for the OLAT course:
TUK: Algebraic Groups SS19

Algebraic Number Theory

Content

  • global fields,
  • modules over Dedekind domains,
  • valuations and completions,
  • ring of integers and orders.

Contact Time

4 SWS / 60 h lecture
2 SWS / 30 h 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 (Vorlesung)
Algebraic Number Theory (Übung)

Click here for the OLAT course:
TUK Algebraic Number Theory SS 19

Character Theory of Finite Groups

Content

  • Maschke's theorem,
  • character table,
  • orthogonality,
  • rationality,
  • Burnside theorem,
  • induced characters,
  • Frobenius group.

Contact Time

2 SWS / 30 h lecture
1 SWS / 15 h 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 (Vorlesung)
Character Theory of Finite Groups (Übung)

Click here for the OLAT course:
TUK Character Theory of Finite Groups SS 19

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 / 60 h lecture
2 SWS / 30 h 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 (Vorlesung)
Computer Algebra (Übung)

Click here for the OLAT course:
TUK: Computer Algebra SS19

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 / 60 h lecture
2 SWS / 30 h example class

Requirements

Courses "Algebraische Strukturen" and "Elementare Zahlentheorie"

Frequency

The lecture takes place every summer semester.

Click here for the KIS entry:
Cryptographie (Vorlesung)
Cryptographie (Übung)

Click here for the OLAT course:
TUK Cryptographie SS 19

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 / 30 h lecture
1 SWS / 15 h 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 (Vorlesung)
Plane Algebraic Curves (Übung)

Click here for the OLAT course:
TUK Plane Algebraic Curves SS 19

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 / 30 h lecture
1 SWS / 15 h 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 (Vorlesung)
Quadratic Number Fields (Übung)

Click here for the OLAT course:
TUK Quadratic Number Fields SS 19

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