Module Handbook

  • Dynamischer Default-Fachbereich geändert auf MAT

Module MAT-44-11-M-7

Algorithmic Invariant Theory (M, 4.5 LP)

Module Identification

Module Number Module Name CP (Effort)
MAT-44-11-M-7 Algorithmic Invariant Theory 4.5 CP (135 h)

Basedata

CP, Effort 4.5 CP = 135 h
Position of the semester 1 Sem. irreg.
Level [7] Master (Advanced)
Language [EN] English
Module Manager
Lecturers
+ further Lecturers of the department Mathematics
Area of study [MAT-AGCA] Algebra, Geometry and Computer Algebra
Reference course of study [MAT-88.105-SG] M.Sc. Mathematics
Livecycle-State [NORM] Active

Courses

Type/SWS Course Number Title Choice in
Module-Part
Presence-Time /
Self-Study
SL SL is
required for exa.
PL CP Sem.
2V+1U MAT-44-11-K-7
Algorithmic Invariant Theory
P 42 h 93 h - - PL1 4.5 irreg.
  • About [MAT-44-11-K-7]: Title: "Algorithmic Invariant Theory"; Presence-Time: 42 h; Self-Study: 93 h

Examination achievement PL1

  • Form of examination: oral examination (20-30 Min.)
  • Examination Frequency: each semester
  • Examination number: 86137 ("Algorithmic Invariant Theory")

Evaluation of grades

The grade of the module examination is also the module grade.


Contents

  • 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.

Competencies / intended learning achievements

Upon successful completion of the module, the students have become familiar with fundamental concepts of algorithmic invariant theory and they are able to apply them, e.g. in the context of finite groups. Relevant algorithmic methods are familiar to the students and can be applied by them practically by hand or with computer support. The students are able to name the main propositions of the lecture, classify and explain the illustrated relationships. They understand the proofs presented in the lecture and are able to comprehend and explain them.

By completing the given exercises, the students have developed a skilled, precise and independent handling of the terms, propositions and methods taught in the lecture. They have learnt how to apply the methods to new problems, analyze them and develop solution strategies independently or by team work.

Literature

  • B. Sturmfels: Algorithms in Invariant Theory,
  • H. Derksen, G. Kemper: Computational Invariant Theory.

Materials

Further literature will be announced in the lecture(s); exercise material is provided.

Registration

Registration for the exercise classes via the online administration system URM (https://urm.mathematik.uni-kl.de).

Requirements for attendance (informal)

Knowledge from the course [MAT-41-11-K-7] Computer Algebra is desirable and helpful, but not necessarily required.

Modules:

Courses

Requirements for attendance (formal)

None

References to Module / Module Number [MAT-44-11-M-7]

Module-Pool Name
[MAT-41-MPOOL-7] Specialisation Algebraic Geometry and Computer Algebra (M.Sc.)
[MAT-43-MPOOL-7] Specialisation Algebra and Number Theory (M.Sc.)
[MAT-RM-MPOOL-7] Pure Mathematics (Advanced Modules M.Sc.)