 definition and basic properties of invariant rings,
 Reynolds operators, Hilbert series, syzygies,
 the CohenMacaulay property of invariant rings
 algorithmic methods for computing invariants,
 Hilbert's finiteness theorem,
 first fundamental theorem of invariant theory.
Module MAT4411M7
Algorithmic Invariant Theory (M, 4.5 LP)
Module Identification
Module Number  Module Name  CP (Effort) 

MAT4411M7  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  [MATAGCA] Algebra, Geometry and Computer Algebra 
Reference course of study  [MAT88.105SG] M.Sc. Mathematics 
LivecycleState  [NORM] Active 
Courses
Type/SWS  Course Number  Title  Choice in ModulePart  PresenceTime / SelfStudy  SL  SL is required for exa.  PL  CP  Sem.  

2V+1U  MAT4411K7  Algorithmic Invariant Theory
 P  42 h  93 h      PL1  4.5  irreg. 
 About [MAT4411K7]: Title: "Algorithmic Invariant Theory"; PresenceTime: 42 h; SelfStudy: 93 h
Examination achievement PL1
 Form of examination: oral examination (2030 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
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.unikl.de).
Requirements for attendance (informal)
Knowledge from the course [MAT4111K7] Computer Algebra is desirable and helpful, but not necessarily required.
Modules:
 [MAT101M2] Fundamentals of Mathematics (M, 28.0 LP)
 [MAT4011M4] Commutative Algebra (M, 9.0 LP)
Courses
 [MAT1211K2] Algebraic Structures (2V+2U, 5.5 LP)
 [MAT1222K3] Introduction to Algebra (2V+1U, 4.5 LP)
Requirements for attendance (formal)
None
References to Module / Module Number [MAT4411M7]
ModulePool  Name  

[MAT41MPOOL7]  Specialisation Algebraic Geometry and Computer Algebra (M.Sc.)  
[MAT43MPOOL7]  Specialisation Algebra and Number Theory (M.Sc.)  
[MATRMMPOOL7]  Pure Mathematics (Advanced Modules M.Sc.) 