GEOMETRY AND REPRESENTATION THEORY
| Title | ΓΕΩΜΕΤΡΙΑ ΚΑΙ ΘΕΩΡΙΑ ΑΝΑΠΑΡΑΣΤΑΣΕΩΝ / GEOMETRY AND REPRESENTATION THEORY |
| Code | Θ032 |
| Faculty | Sciences |
| School | Mathematics |
| Cycle / Level | 2nd / Postgraduate |
| Teaching Period | Winter/Spring |
| Coordinator | Panagiotis Batakidis |
| Common | Yes |
| Status | Active |
| Course ID | 600025921 |
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 | Spring | - | 10 |
| Academic Year | 2025 – 2026 |
| Class Period | Spring |
| Faculty Instructors | |
| Weekly Hours | 3 |
| Total Hours | 39 |
| Class ID | 600268427
|
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/1/600268427
- eLearning: https://elearning.auth.gr/course/view.php?id=9905
Erasmus
The course is also offered to exchange programme students.
Language of Instruction
- Greek (Instruction, Examination)
Prerequisites
General Prerequisites
Knowledge of differential manifolds is required. Basic knowledge of group theory and general topology is desirable.
Learning Outcomes
Upon successful completion of the course, students will:
- they will know the basic definitions of the theory Lie groups their most important families.
- they will know the construction of the Lie group and its basic properties as a manifold, together with the corresponding geometric interpretations with examples.
- they will understand the basic results of the theory of Lie group actions on manifolds and their analysis
- they will know the basic concepts of construction and study of representations of Lie groups
General Competences
- Retrieve, analyse and synthesise data and information, with the use of necessary technologies
- Adapt to new situations
- 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)
Basic matrix groups and topological structures. Introduction to Lie groups and Lie algebras. Homogeneous spaces and examples. Lie group action on a smooth manifold. Types of actions. Orbits and stabilizers. Equivariant Rank Theorem. Lie algebra action on smooth manifold, adjoint and coadjoint actions. Linear representations and types of representations. Orbit Theorem. Introduction to vector bundles. Fibre bundles. Principal bundles. Associated bundles and representations.
Keywords
Manifolds, actions, orbits, flows, Lie algebra and Lie group, Fibre bundle and representations
Educational Material Types
- Notes
- Slide presentations
- 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 | 183 | ||||
| Tutorial | 13 | ||||
| Project | 20 | ||||
| Written assigments | 40 | ||||
| Exams | 5 | ||||
| Total | 300 |
Student Assessment
Description
There will be assignments due by the end of the semester (30% of final grade), together with a small project (40%) and an oral exam or midterm (30%)
Student Assessment methods
- Written Exam with Short Answer Questions (Formative, Summative)
- Written Exam with Extended Answer Questions (Formative, Summative)
- Written Assignment (Formative, Summative)
- Oral Exams (Formative, Summative)
- Performance / Staging (Formative, Summative)
Bibliography
Additional bibliography for study
-Lie Groups, Lie Algebras and Representations, B.C. Hall, Springer, GTM 222, Second Edition, 2015
-Lie Groups, C. Procesi, Springer, Universitext, 2007
-Structure and Geometry of Lie Groups, J. Hilgert & K.-H. Neeb, Springer, Monographs in Mathematics, 2012
-Differential Geometry and Mathematical Physics, G. Rudolph & M. Schmidt, Springer, Theoretical and Mathematical Physics, 2013
Last Update
12-05-2025
