GLOBAL DIFFERENTIAL GEOMETRY
| Title | ΟΛΙΚΗ ΔΙΑΦΟΡΙΚΗ ΓΕΩΜΕΤΡΙΑ / GLOBAL DIFFERENTIAL GEOMETRY |
| Code | Θ030 |
| Faculty | Sciences |
| School | Mathematics |
| Cycle / Level | 2nd / Postgraduate |
| Teaching Period | Winter/Spring |
| Coordinator | Manousos Maridakis |
| Common | No |
| Status | Active |
| Course ID | 600025919 |
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 | 600268391
|
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/1/600268391
- eLearning:
Erasmus
The course is also offered to exchange programme students.
Language of Instruction
- Greek (Instruction, Examination)
Prerequisites
General Prerequisites
General background on Differential Manifolds at graduate level.
Learning Outcomes
Upon successful completion of the course, students will be able:
1) to have a basic knowledge of the Laplacian spectrum on compact closed manifolds,
2) to formulate geometric problems using calculus of variations and obtain Euler Lagrange equations in weak formulation,
3) to understand how Sobolev embedding theorems and critical exponents are connected to basic geometric problems
like existence and convergence of geodesics,
4) to draw geometric-topological conclusions from geometric-elliptic problems using Bocner's method,
5) to learn basic geometric-topological applications of the heat kernel.
General Competences
- Apply knowledge in practice
- Work autonomously
- Work in teams
- Generate new research ideas
- Be critical and self-critical
- Advance free, creative and causative thinking
Course Content (Syllabus)
Vector bundles, connection and curvature, differential operators and Sobolev spaces, basic elliptic theory, spectral theory and Hodge theorem, Bochner techique, Calculus of variations and the mountain pass lemma with applications in existence of closed geodesics, spinor bundles and the Dirac operator, the heat kernel on Riemannian manifolds.
Keywords
Curvature, Hodge Theory, Bochner technique, Sobolev embbedings, Elliptic theory, spectral theory, mountain pass lemma, heat kernel
Educational Material Types
- Notes
- Multimedia
- Book
Use of Information and Communication Technologies
Use of ICT
- Use of ICT in Course Teaching
- Use of ICT in Communication with Students
- Use of ICT in Student Assessment
Description
Online exercise hours.
Course Organization
| Activities | Workload | ECTS | Individual | Teamwork | Erasmus |
|---|---|---|---|---|---|
| Lectures | 39 | ||||
| Reading Assigment | 185 | ||||
| Tutorial | 11 | ||||
| Project | 20 | ||||
| Written assigments | 40 | ||||
| Exams | 5 | ||||
| Total | 300 |
Student Assessment
Description
Course grades will be based on homework assignments and on team projects. (The nature of
the projects will be explained in class).
Student Assessment methods
- Written Assignment (Summative)
- Oral Exams (Summative)
- Performance / Staging (Summative)
Bibliography
Course Bibliography (Eudoxus)
Lectures on the Geometry of Manifolds, by Liviu Nicolaescu
Riemannian Geometry and Geometric Analysis, by Jurgen Jost
The Laplacian on Riemannian Manifolds, by Steven Rosenberg.
Additional bibliography for study
Geometric Analysis by Peter Li.
Elliptic Partial Differential Equations by D.Gilbarg and N.Trudinger.
Last Update
12-05-2025
