Algebra, Geometry and Computer Algebra Group


General Information

Below are the English lectures offered by our group in the winter term 2022/2023.

If you would like to attend a seminar or reading course during the winter 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 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 (Lecture)

Algebraic Geometry (Example class)

Click here for the OLAT course.

 

Algorithmic Invariant Theory

Content

  • definition and basic properties of invariant rings,
  • Reynolds operators, Hilbert series, syzygies,
  • the Cohen-Macaulay property of invariant rings
  • algorithmic methods for computing invariants,
  • Hilbert's finiteness theorem,
  • first fundamental theorem of invariant theory.

Contact Time

2 SWS lecture
 

Requirements

Frequency

The lecture takes place irregularly.

Click here for the KIS entry:

Algorithmic Invariant Theory (Lecture)

Click here for the OLAT course:

Link follows

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 (Lecture)

Algorithmic Number Theory (Example Class)

Click here for the OLAT course:

Link follows

Classical Groups

Content

The theory of the classical groups is treated. These groups have proved to be important in a wide variety of venues, ranging from physics to geometry and far beyond. In particular, they play a prominent role in the classification of the finite simple groups.

The following topics are covered:

  • Spaces with forms,
  • Symplectic groups,
  • Orthogonal groups,
  • Clifford algebras,
  • Unitary groups,
  • BN pairs.

Contact Time

2 SWS lecture
 

Requirements

Frequency

The lecture takes place irregularly.

Click here for the KIS entry:

Classical Groups

Click here for the OLAT course:

Link follows

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 (Lecture)

Commutative Algebra (Example Class)

Click here for the OLAT course:

OLAT course

Elliptic Functions and Elliptic Curves

Content

  • elliptic functions, Weierstrass P-function,
  • complex tori,
  • plane geometry and geometric construction of a group structure,
  • elliptic curves,
  • modular forms, modular curves, and classification theory of elliptic curves.

Contact Time

2 SWS lecture
 

Requirements

Fundamentals of Mathematics, Algbraic Structures, Introduction to Algebra, Introduction to Complex Analysis

Frequency

The lecture takes place irregularly.

Click here for the KIS entry:

Elliptic Functions and Elliptic Curves (Lecture)

Click here for the OLAT course:

Link follows.

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 (Lecture)

Representation Theory (Example Class)

Click here for the OLAT course.

 

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