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Information
| Code | MT002 |
| Name | Introduction to Invariant Theory |
| Term | 2024-2025 Academic Year |
| Term | Spring |
| Duration (T+A) | 3-0 (T-A) (17 Week) |
| ECTS | 6 ECTS |
| National Credit | 3 National Credit |
| Teaching Language | Türkçe |
| Level | Doktora Dersi |
| Type | Normal |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator | Prof. Dr. ŞEHMUS FINDIK |
| Course Instructor |
1 |
Course Goal / Objective
Learning basic knowledge about invariant theory and observing its some applications.
Course Content
Introduction to invariant theory, Symmetric polynomials, Gröbner bases, invariant theory of finite groups, invariants of general linear group
Course Precondition
Algebra
Resources
Algorithms in Invariant Theory, B. Sturmfels
Notes
Lecture Notes
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | Learns term orders on the polynomial algebra. |
| LO02 | Learns symmetrical polynomials. |
| LO03 | Learns elementary polynomial generators |
| LO04 | Knows the fundamental theorem of symmetric groups and its applications |
| LO05 | Learns the invariants of finite groups. |
| LO06 | Learns the invariants of the general linear group. |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|---|---|---|
| PLO01 | Bilgi - Kuramsal, Olgusal | Knows the results of previous research in a special field of mathematics | 5 |
| PLO02 | Bilgi - Kuramsal, Olgusal | Knows in detail the relationship between the results in her area of expertise and other areas of mathematics. | |
| PLO03 | Bilgi - Kuramsal, Olgusal | Establishes new mathematical models with the help of the knowledge gained in the field of specialization. | 3 |
| PLO04 | Bilgi - Kuramsal, Olgusal | Has basic knowledge in all areas of mathematics | 5 |
| PLO05 | Bilgi - Kuramsal, Olgusal | It presents the knowledge gained in different fields of mathematics and their relations with each other in the simplest and most understandable way. | |
| PLO06 | Bilgi - Kuramsal, Olgusal | Effectively uses the technical equipment needed to express mathematics | 4 |
| PLO07 | Bilgi - Kuramsal, Olgusal | Sets up original problems in her field and offers different solution techniques | 2 |
| PLO08 | Bilgi - Kuramsal, Olgusal | It carries out original and qualified scientific studies on the subject related to its field. | 4 |
| PLO09 | Bilgi - Kuramsal, Olgusal | Analyzes existing mathematical theories and develops new theories. | |
| PLO10 | Beceriler - Bilişsel, Uygulamalı | Knows the teaching-learning techniques in areas of mathematics that require expertise and uses these techniques effectively at every stage of education. | 4 |
| PLO11 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | To have foreign language knowledge at a level to be able to follow foreign sources related to the field and to communicate verbally and in writing with foreign stakeholders. | 4 |
| PLO12 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | It presents and publishes its original works within the framework of scientific ethical rules for the benefit of its stakeholders. | |
| PLO13 | Yetkinlikler - Öğrenme Yetkinliği | Adheres to the ethical rules required by its scientific title | 5 |
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | Term orders | Reading the lecture notes | Öğretim Yöntemleri: Anlatım |
| 2 | Applications of Lex, dlex, drlex orders | Reading the lecture notes | Öğretim Yöntemleri: Anlatım |
| 3 | Symmetric polynomials | Reading the lecture notes | Öğretim Yöntemleri: Anlatım |
| 4 | Fundamental theorem of symmetric polynomials | Reading the lecture notes | Öğretim Yöntemleri: Anlatım |
| 5 | Generating elements of algebra of symmetric polynomials of degree n | Reading the lecture notes | Öğretim Yöntemleri: Anlatım |
| 6 | Schur polynomials | Reading the lecture notes | Öğretim Yöntemleri: Anlatım |
| 7 | Problem solving 1 | Reading the lecture notes | Öğretim Yöntemleri: Anlatım |
| 8 | Mid-Term Exam | Reading the lecture notes | Ölçme Yöntemleri: Yazılı Sınav |
| 9 | Gröbner bases | Reading the lecture notes | Öğretim Yöntemleri: Anlatım |
| 10 | Hilbert basis theorem | Reading the lecture notes | Öğretim Yöntemleri: Anlatım |
| 11 | Inariants of finite groups | Reading the lecture notes | Öğretim Yöntemleri: Anlatım |
| 12 | Hilbert finiteness theorem | Reading the lecture notes | Öğretim Yöntemleri: Anlatım |
| 13 | Cohen Macaulay property | Reading the lecture notes | Öğretim Yöntemleri: Anlatım |
| 14 | Invariants of general linear group | Reading the lecture notes | Öğretim Yöntemleri: Anlatım |
| 15 | Problem solving | Reading the lecture notes | Öğretim Yöntemleri: Anlatım |
| 16 | Term Exams | Reading the lecture notes | Ölçme Yöntemleri: Yazılı Sınav |
| 17 | Term Exams | Reading the lecture notes | Ölçme Yöntemleri: Yazılı Sınav |
Student Workload - ECTS
| Works | Number | Time (Hour) | Workload (Hour) |
|---|---|---|---|
| Course Related Works | |||
| Class Time (Exam weeks are excluded) | 14 | 3 | 42 |
| Out of Class Study (Preliminary Work, Practice) | 14 | 5 | 70 |
| Assesment Related Works | |||
| Homeworks, Projects, Others | 0 | 0 | 0 |
| Mid-term Exams (Written, Oral, etc.) | 1 | 15 | 15 |
| Final Exam | 1 | 30 | 30 |
| Total Workload (Hour) | 157 | ||
| Total Workload / 25 (h) | 6,28 | ||
| ECTS | 6 ECTS | ||