Algebra, Geometry and Computer Algebra Group


General Information

Below are the English lectures offered by our group in the winter semester 2018/2019.

If you would like to attend a seminar or reading course during the winter semester, please contact the respective supervisor by e-mail by 30th September. 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 Oct 26 th 2018)
  • OpenOLAT: Material and example sheets and more informations (Login in the first lecture)

Algebraic Geometry

Content

Mandatory content:

  • Affine and projective varieties (especially: dimension, morphisms, smooth and singular points, blow-ups of points, applications and examples),
  • Sheaves and sheaf cohomology with applications (Riemann-Roch's theorem for curves, projective embedding of curves).

It also covers a selection of the following topics:

  • Schemes,
  • Differential forms,
  • other aspects of algebraic geometry.

contact time

4 SWS / 60 h lecture
2 SWS / 30 h example class

Requirements

Module "Commutative Algebra". Knowledge from the course "Plane Algebraic Curves" is desirable and helpful, but not necessarily required.

 

 

Frequency

The lecture is offered every year in the winter semester.Click here for the KIS entry:

Algebraic Geometry (Vorlesung)
Algebraic Geometry (Übung)

Click here for the OLAT course:

TUK Algebraic Geometry WS 18/19

Algorithmic Number Theory

Content

  • LLL algorithm,
  • Number fields, ring of integers, units, class group,
  • Decomposition behavior of primes,
  • Algorithmic calculation of these quantities.

Contact time

4 SWS / 60 h Lecture
2 SWS / 30 h Example Class

Requirements

Courses "Algebraic Structures" and "Introduction: Algebra"; In addition, basic properties of Dedekind rings from the course "Commutative Algebra" are used. Knowledge from the "Quadratic Number Fields" module is desirable and helpful.

Frequency

The lecture takes place irregularly.

Click here for the KIS entry:


Algorithmic Number Theory (Lecture)

Algorithmic Number Theory (Example Class)

Click here for the OLAT course:
TUK Algorithmic Number Theory WS 18/19

Commutative Algebra

Content

  • Rings, modules, localization, lemma of Nakayama,
  • Noether / Artin rings and modules,
  • Primary decomposition,
  • Krull's Principal Ideal Theorem, dimension theory
  • Integral ring extensions, going-up, going-down, normalization,
  • Noether normalization, Hilbert's Nullstellensatz,
  • Dedekind rings, invertible ideals.

Contact time

4 SWS / 60 h lecture
2 SWS / 30 h example class

Requirements

Lecture "Algebraische Strukturen". The lecture "Introduction: Algebra" is nice to have, but not necessary.

Frequency

The lecture is offered every year in the winter semester.

Click here for the KIS entry:

Commutative Algebra (Lecture)

Commutative Algebra (Example Class)

Click here for the OLAT course:
TUK Commutative Algebra WS 18/19

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 (Vorlesung)

Click here for the OLAT entry:
TUK Complex Manifolds and Hodge Theory WS 18/19

Representation Theory

Content

Modules about rings and algebras:

  • the theorems of Wedderburn, Jordan-Hölder and Krull-Schmidt.

Modules via group algebras:

  • Induction and restriction,
  • the Mackey formula,
  • Clifford theory,
  • projective representations,
  • Blocks.

Representation theory of symmetric groups.

Contact time

4 SWS / 60 h Lecture
2 SWS / 30 h Example Class

Requirements

Course "Introduction: Algebra" and module "Commutative Algebra"; Knowledge from the module "Character Theory of Finite Groups" is desirable and helpful, but not necessarily required.

Frequence

The lecture takes place irregularly.

Click here for the KIS entry:
Representation Theory (Lecture)
Representation Theory (Example Class)

Click here for the event page:
Representation Theory WS 18/19

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