Algebra, Geometry and Computer Algebra Group


General Information

Below are the English lectures offered by our group in the winter term 2020/2021.

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 (open 5th October till 30th October 2020, 12:00)
  • 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 lecture
2 SWS 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:

 Algebraic Geometry

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 Lecture
2 SWS 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 (Vorlesung)

Algorithmic Number Theory (Übung)

Click here for the OLAT course:

Algorithmic Number Theory

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 lecture
2 SWS 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 (Vorlesung)

Commutative Algebra (Übung)

Click here for the OLAT course:

Commutative Algebra

Introduction to Tensor Categories

Contents

  • Categories, functors, additive and Abelian categories,
  • Monoidal categories and functors,
  • Grothendieck ring, categorization, example of the categorization of group rings.

A selection of the following topics:

  • Hopf algebras and quantum groups,
  • Graphical calculus for morphisms,
  • Braided tensor categories, band categories and knot invariants,
  • Modular tensor categories and topological quantum field theories.

Contact Time

2 SWS  Lecture

Requirements

Algebraic Structures (Algebraische Strukturen)

Frequency

The lecture is offered irregularly.Click here for the KIS entry:

Introduction to Tensor Categories (Vorlesung)

Click here for the OLAT course:

Introduction to Tensor Categories

Representation Theory

Content

Modules over 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 Lecture
2 SWS 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.

Frequency

The lecture takes place irregularly.

Click here for the KIS entry:

Representation Theory (Vorlesung)

Representation Theory (Übung)

Click here for the OLAT course:

Representation Theory

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 (Vorlesung)

Click here for the OLAT course:

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