GRADUATE SCHOOL

PH.D. In Applied Mathematics and Statistics

MATH 664 | Course Introduction and Application Information

Course Name
Invariant Theory
Code
Semester
Theory
(hour/week)
Application/Lab
(hour/week)
Local Credits
ECTS
MATH 664
Fall/Spring
3
0
3
7.5

Prerequisites
None
Course Language
English
Course Type
Elective
Course Level
Third Cycle
Mode of Delivery -
Teaching Methods and Techniques of the Course -
Course Coordinator -
Course Lecturer(s)
Assistant(s) -
Course Objectives To introduce the fundamental theory of invariants and to provide current research problems and current developements.
Learning Outcomes The students who succeeded in this course;
  • will be able to describe invariants of given representation.
  • will be able to calculate the ring of invariants.
  • will be able to describe syzygy ideal.
  • will be able to analyze algebraic properties.
  • will be able to construct generators.
Course Description This course provides an introduction to the theory of invariants of finite groups. Topics are based both on classical methods and also on new computational methods.

 



Course Category

Core Courses
Major Area Courses
X
Supportive Courses
Media and Management Skills Courses
Transferable Skill Courses

 

WEEKLY SUBJECTS AND RELATED PREPARATION STUDIES

Week Subjects Related Preparation
1 Invariants and Problems “Invariant Theory of Finite Groups” M.D.Neusel, L.Smith
2 Algebraic Finiteness “Invariant Theory of Finite Groups” M.D.Neusel, L.Smith
3 Combinatorial Finiteness “Invariant Theory of Finite Groups” M.D.Neusel, L.Smith
4 Noetherian Finiteness “Invariant Theory of Finite Groups” M.D.Neusel, L.Smith
5 Hilbert's Syzygy Theorem “Invariant Theory of Finite Groups” M.D.Neusel, L.Smith
6 The CohenMacaulay Property “Invariant Theory of Finite Groups” M.D.Neusel, L.Smith
7 Gorenstein and Other Homological Properties “Invariant Theory of Finite Groups” M.D.Neusel, L.Smith
8 Modular Invariant Theory “Invariant Theory of Finite Groups” M.D.Neusel, L.Smith
9 Koszul Complex “Invariant Theory of Finite Groups” M.D.Neusel, L.Smith
10 Pseudoretlection Groups “Invariant Theory of Finite Groups” M.D.Neusel, L.Smith
11 Steenrod Algebra “Invariant Theory of Finite Groups” M.D.Neusel, L.Smith
12 Hopf Algebra Structure of the Steenrod Algebra “Invariant Theory of Finite Groups” M.D.Neusel, L.Smith
13 Inverse Invariant Theory “Invariant Theory of Finite Groups” M.D.Neusel, L.Smith
14 Steenrod Algebra and Dickson Algebra “Invariant Theory of Finite Groups” M.D.Neusel, L.Smith
15 Review
16 Review of the Semester  

 

Course Notes/Textbooks “Invariant Theory of Finite Groups” M.D.Neusel, L.Smith
Suggested Readings/Materials “Polynomial Invariants of finite groups” L. Smith“Computational Invariant Theory” H. Derksen, G. Kemper

 

EVALUATION SYSTEM

Semester Activities Number Weigthing
Participation
Laboratory / Application
Field Work
Quizzes / Studio Critiques
Portfolio
Homework / Assignments
5
20
Presentation / Jury
Project
Seminar / Workshop
Oral Exams
Midterm
1
30
Final Exam
1
50
Total

Weighting of Semester Activities on the Final Grade
50
Weighting of End-of-Semester Activities on the Final Grade
50
Total

ECTS / WORKLOAD TABLE

Semester Activities Number Duration (Hours) Workload
Theoretical Course Hours
(Including exam week: 16 x total hours)
16
3
48
Laboratory / Application Hours
(Including exam week: '.16.' x total hours)
16
0
Study Hours Out of Class
15
6
90
Field Work
0
Quizzes / Studio Critiques
0
Portfolio
0
Homework / Assignments
5
5
25
Presentation / Jury
0
Project
0
Seminar / Workshop
0
Oral Exam
0
Midterms
1
31
31
Final Exam
1
31
31
    Total
225

 

COURSE LEARNING OUTCOMES AND PROGRAM QUALIFICATIONS RELATIONSHIP

#
Program Competencies/Outcomes
* Contribution Level
1
2
3
4
5
1

To develop and deepen his/her knowledge on theories of mathematics and statistics and their applications in level of expertise, and to obtain unique definitions which bring innovations to the area, based on master level competencies,

X
2

To have the ability of original, independent and critical thinking in Mathematics and Statistics and to be able to develop theoretical concepts,

X
3

To have the ability of defining and verifying problems in Mathematics and Statistics,

X
4

With an interdisciplinary approach, to be able to apply theoretical and applied methods of mathematics and statistics in analyzing and solving new problems and to be able to discover his/her own potentials with respect to the application,

X
5

In nearly every fields that mathematics and statistics are used, to be able to execute, conclude and report a research, which requires expertise, independently,

X
6

To be able to evaluate and renew his/her abilities and knowledge acquired in the field of Applied Mathematics and Statistics with critical approach, and to be able to analyze, synthesize and evaluate complex thoughts in a critical way,

X
7

To be able to convey his/her analyses and methods in the field of Applied Mathematics and Statistics to the experts in a scientific way,

X
8

To be able to use national and international academic resources (English) efficiently, to update his/her knowledge, to communicate with his/her native and foreign colleagues easily, to follow the literature periodically, to contribute scientific meetings held in his/her own field and other fields systematically as written, oral and visual.

X
9

To be familiar with computer software commonly used in the fields of Applied Mathematics and Statistics and to be able to use at least two of them efficiently,

X
10

To contribute the transformation process of his/her own society into an information society and the sustainability of this process by introducing scientific, technological, social and cultural advances in the fields of Applied Mathematics and Statistics,

X
11

As having rich cultural background and social sensitivity with a global perspective, to be able to evaluate all processes efficiently, to be able to contribute the solutions of social, scientific, cultural and ethical problems and to support the development of these values,

X
12

As being competent in abstract thinking, to be able to connect abstract events to concrete events and to transfer solutions, to analyze results with scientific methods by designing experiment and collecting data and to interpret them,

X
13

To be able to produce strategies, policies and plans about systems and topics in which mathematics and statistics are used and to be able to interpret and develop results,

X
14

To be able to evaluate, argue and analyze prominent persons, events and phenomena, which play an important role in the development and combination of the fields of Mathematics and Statistics, within the perspective of the development of other fields of science,

X
15

In Applied Mathematics and Statistics, to be able to sustain scientific work as an individual or a group, to be effective in all phases of an independent work, to participate decision-making process and to make and execute necessary planning within an effective time schedule.

X

*1 Lowest, 2 Low, 3 Average, 4 High, 5 Highest

 


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