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Information
| Code | MT521 |
| Name | Introduction to Differential Geometry |
| Term | 2024-2025 Academic Year |
| Semester | . Semester |
| Duration (T+A) | 3-0 (T-A) (17 Week) |
| ECTS | 6 ECTS |
| National Credit | 3 National Credit |
| Teaching Language | Türkçe |
| Level | Yüksek Lisans Dersi |
| Type | Normal |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator | Prof. Dr. ALİ ARSLAN ÖZKURT |
Course Goal / Objective
To teach the concept of differentiable manifolds and submanifolds. To teach the concept of vector field on a diferentiable manifold and Lie groups
Course Content
Differentiability for functions of several variables, vector fields on open subsets of an Euclidean space, the rank of a mapping, Definition of differentiable manifolds and examples, Rank of a differentiable function defined on a differentiable manifold, immersions and submersions, submanifolds, Lie groups and vector fields
Course Precondition
Basic knowledge of derivatives and integrals is required.
Resources
Diferansiyel Denklemler ve Uygulamaları, mehmet Aydın, Gönül Gündüz,Beno Kuryel
Notes
Differentiel Equuations, L.Shipley Ross Differentiel Equuations, Frank Ayres Differential Equations, Lester R. Ford
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | Knows the concept of differentiable manifold and gives some examples |
| LO02 | Makes some analysis on differentiable manifolds |
| LO03 | Knows the concept of submanifold |
| LO04 | Knows the concept of a Lie group |
| LO05 | Can investigate actions of a Lie group on a differentiable manifold |
| LO06 | Knows vector fields on differentiable manifolds |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|---|---|---|
| PLO01 | Bilgi - Kuramsal, Olgusal | Knows in detail the relationship between the results in her area of expertise and other areas of mathematics. | 5 |
| PLO02 | Bilgi - Kuramsal, Olgusal | Knows in detail the relationship between the results in his 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. | |
| 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. | 5 |
| PLO06 | Bilgi - Kuramsal, Olgusal | Effectively uses the technical equipment needed to express mathematics. | 5 |
| PLO07 | Bilgi - Kuramsal, Olgusal | poses original problems related to field and presents different solution techniques. | 4 |
| PLO08 | Bilgi - Kuramsal, Olgusal | carries out original and qualified scientific studies on the subject related to its field. | |
| PLO09 | Bilgi - Kuramsal, Olgusal | Analyzes existing mathematical theories and develops new theories. | 4 |
| 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 knowledge of a foreign language 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 | 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 | Topological manifolds | Read the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 2 | Analysis on the functions of several variables | Read the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 3 | Differentiable manifolds | Read the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 4 | Differentiable maps between differantiable manifolds | Read the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 5 | Derivative maps | Read the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 6 | The rank of a mapping | Read the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 7 | Submanifolds of a differentiable manifold | Read the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 8 | Mid-Term Exam | Review the topics discussed in the lecture notes and source | Ölçme Yöntemleri: Ödev |
| 9 | Lie operator | Read the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 10 | Lie groups | Read the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 11 | Actions of a Lie group on a manifold | Read the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 12 | The tangent space at a point of a manifold | Read the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 13 | Tangent bundle | Read the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 14 | Vector fields on Manifolds | Read the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 15 | One parameter groups acting on a manifold | Read the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 16 | Term Exams | Review the topics discussed in the lecture notes and source | Ölçme Yöntemleri: Ödev, Performans Değerlendirmesi |
| 17 | Term Exams | Review the topics discussed in the lecture notes and source | Ölçme Yöntemleri: Ödev, Performans Değerlendirmesi |
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 | ||