DIFFERENTIAL TOPOLOGY
| Title | ΔΙΑΦΟΡΙΚΗ ΤΟΠΟΛΟΓΙΑ / DIFFERENTIAL TOPOLOGY |
| Code | Θ028 |
| Faculty | Sciences |
| School | Mathematics |
| Cycle / Level | 2nd / Postgraduate |
| Teaching Period | Winter/Spring |
| Coordinator | Georgios Raptis |
| Common | Yes |
| Status | Active |
| Course ID | 600025917 |
Programme of Study: PMS Tmīmatos Mathīmatikṓn (2025-2030)
Registered students: 0
| Orientation | Attendance Type | Semester | Year | ECTS |
|---|---|---|---|---|
| THEŌRĪTIKA MATHĪMATIKA | Elective Courses belonging to the selected specialization | Winter | - | 10 |
| Academic Year | 2025 – 2026 |
| Class Period | Winter |
| Faculty Instructors | |
| Weekly Hours | 3 |
| Total Hours | 39 |
| Class ID | 600268389
|
Type Of Offer
- Disciplinary Course
Course Type 2021
Specialization / Direction
Mode of Delivery
- Face to face
- Distance learning
Digital Course Content
- e-Study Guide https://qa.auth.gr/en/class/1/600268389
Erasmus
The course is also offered to exchange programme students.
Language of Instruction
- Greek (Instruction, Examination)
- English (Instruction, Examination)
Prerequisites
General Prerequisites
Basic knowledge of differentiable functions and manifolds at the undergraduate level
Learning Outcomes
Upon successful completion of the course, students will:
1. have advanced knowledge of fundamental notions and useful methods in differential topology
2. know applications of differential topology in other fields
3. have deepened their understanding of differentiable functions, including knowledge of advanced topics such as transversality and Morse theory
4. know approaches to the problem of classifying differentiable structures
5. be familiar with special properties of important differentiable manifolds, such as spheres, projective spaces and surfaces
General Competences
- Make decisions
- Work autonomously
- Work in teams
- Generate new research ideas
- Be critical and self-critical
- Advance free, creative and causative thinking
Course Content (Syllabus)
Differentiable manifolds and tangent bundles. Sard's theorem and applications. The Whitney topology on function spaces. Immersions and embeddings. Vector bundles. Vector fields. Orientations. Transversality theorems. The Euler characteristic. Connected sum and other operations. Elements of Morse theory. Introduction to cobordism theory.
Keywords
differentiable manifold, Sard's theorem, embedding, vector bundle, transversality, orientation, Morse function, cobordism
Educational Material Types
- Notes
- Book
Course Organization
| Activities | Workload | ECTS | Individual | Teamwork | Erasmus |
|---|---|---|---|---|---|
| Lectures | 39 | ||||
| Seminars | 20 | ||||
| Reading Assigment | 183 | ||||
| Project | 20 | ||||
| Written assigments | 35 | ||||
| Exams | 3 | ||||
| Total | 300 |
Student Assessment
Description
1. Written or oral exam (50-70%).
2. Short presentation of a topic and written report (30-50%).
Student Assessment methods
- Written Exam with Extended Answer Questions (Summative)
- Written Assignment (Formative)
- Oral Exams (Summative)
- Performance / Staging (Formative, Summative)
- Written Exam with Problem Solving (Summative)
- Report (Formative, Summative)
Bibliography
Additional bibliography for study
1. Bröcker-Jänich "Introduction to Differential Topology"
2. Guillemin-Pollack "Differential Topology"
3. Hirsch "Differential Topology"
4. Kosinski "Differential Manifolds"
5. Milnor "Topology from the differentiable viewpoint"
Last Update
11-05-2025
