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Information
| Code | MT524 |
| Name | Differential Topology |
| 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 | Doktora Dersi |
| Type | Normal |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator | Prof. Dr. DOĞAN DÖNMEZ |
Course Goal / Objective
To grasp the fundamentals of manifolds, Lie groups and vector bundles.
Course Content
Differentiable Manifolds. Lie groups. Vector Bundles. Characteristic classes
Course Precondition
Pre-requisites None
Resources
Morris W. Hirsch : Differential Topology
Notes
Lecture Notes
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | Can make the definition of differentiable manifold |
| LO02 | Understands the tangent vector bundle. |
| LO03 | Understands vector and principal bundles. |
| LO04 | Understands universal bundles |
| LO05 | Understands characteristic classes |
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. | 4 |
| 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. | 3 |
| PLO06 | Bilgi - Kuramsal, Olgusal | Effectively uses the technical equipment needed to express mathematics | 3 |
| PLO07 | Bilgi - Kuramsal, Olgusal | Sets up original problems in her field and offers different solution techniques | 4 |
| PLO08 | Bilgi - Kuramsal, Olgusal | It 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. | |
| 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. | 3 |
| 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. | 3 |
| PLO13 | Yetkinlikler - Öğrenme Yetkinliği | Adheres to the ethical rules required by its scientific title | 5 |
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | Differentiable manifolds. Parametrization. Atlas | Studying the relevant parts of the course materials. | Öğretim Yöntemleri: Anlatım |
| 2 | Implicit function theorem. Tangent sapce and tangent bundle. | Studying the relevant parts of the course materials. | Öğretim Yöntemleri: Anlatım |
| 3 | Differentiable maps. Maps between tangent bundles | Studying the relevant parts of the course materials. | Öğretim Yöntemleri: Anlatım |
| 4 | Whtiney Embedding theorem. Lie groups. | Studying the relevant parts of the course materials. | Öğretim Yöntemleri: Anlatım |
| 5 | Vector Bundles. Transition functions. | Studying the relevant parts of the course materials. | Öğretim Yöntemleri: Anlatım |
| 6 | Pull-back bundle. Principal bundles | Studying the relevant parts of the course materials. | Öğretim Yöntemleri: Anlatım |
| 7 | Properties of principal bundles. Universal bundles | Studying the relevant parts of the course materials. | Öğretim Yöntemleri: Anlatım |
| 8 | Mid-Term Exam | Solve the homework problems. | Ölçme Yöntemleri: Ödev |
| 9 | Proeties and existence of universal bundles. | Studying the relevant parts of the course materials. | Öğretim Yöntemleri: Anlatım |
| 10 | Special orthogonal group. Stiefel manifolds. | Studying the relevant parts of the course materials. | Öğretim Yöntemleri: Anlatım |
| 11 | Milnos theorem on universal bundles. | Studying the relevant parts of the course materials. | Öğretim Yöntemleri: Anlatım |
| 12 | Classifying spaces and their properties. | Studying the relevant parts of the course materials. | Öğretim Yöntemleri: Anlatım |
| 13 | Cohomology of classifying spaces. Stifel-Whitney classes. | Studying the relevant parts of the course materials. | Öğretim Yöntemleri: Anlatım |
| 14 | Chern classes | Studying the relevant parts of the course materials. | Öğretim Yöntemleri: Anlatım |
| 15 | Pontrayagin classes | Studying the relevant parts of the course materials. | Öğretim Yöntemleri: Anlatım |
| 16 | Term Exams | Solve the homework problems. | Ölçme Yöntemleri: Ödev |
| 17 | Term Exams | Solve the homework problems. | Ölçme Yöntemleri: Ödev |
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 | ||