Geometric Methods In Mathematical Physics
Physics
Institute of Natural and Applied Sciences
Third Cycle (Doctorate Degree)
| Course Unit Title | Course Unit Code | Type of Course Unit | Level of Course Unit | Year of Study | Semester | ECTS Credits |
|---|---|---|---|---|---|---|
| Geometric Methods In Mathematical Physics | FIZ614 | Elective | Doctorate degree | 1 | Spring | 8 |
Name of Lecturer(s)
Associate Prof. Dr. Oktay CEBECİOĞLU
Learning Outcomes of the Course Unit
1) Recognise differential geometric structures in Mathematics and Physics, formulate them in the language of Differential Geometry, and analyse them using the methods and tools of invariant differential calculus
2) Describe, construct and analyze differential manifolds, vector bundles, tensor fields and linear connections
3) Apply the techniques of invariant tensor calculus to basic problems in geometry and physics
4) Use the techniques of vector bundles and connections to problems in geometry and physics
5) Familiar with covariant derivative, geodesic curves and Cartan’s structure equations
6) Calculate the components of the Riemann curvature tensor from a given line element
Program Competencies-Learning Outcomes Relation
| Program Competencies | ||||||||||||
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | ||
| Learning Outcomes | ||||||||||||
| 1 | Low | Low | Low | Low | Low | Low | Low | Low | Low | Low | Low | |
| 2 | Low | Middle | Low | Low | Middle | Middle | Middle | Low | Low | Low | Low | |
| 3 | Low | Low | Middle | Middle | Middle | Middle | Middle | Low | Low | Middle | Low | |
| 4 | Middle | Low | High | Low | Low | Middle | High | High | Middle | Low | Low | |
| 5 | Low | Low | Middle | Middle | High | High | Middle | Middle | Middle | High | High | |
| 6 | Middle | Middle | Low | Low | Middle | High | Middle | Low | Middle | High | Middle | |
Mode of Delivery
Face to Face
Prerequisites and Co-Requisites
None
Recommended Optional Programme Components
yok
Course Contents
Candidates are provided with profound knowledge on differential forms, Grassmann Algebra: Interior Product and Wedge Product, Hodge Duality, exterior differentiation, behaviour under pull-back, Stokes’ Theorem, action of groups on manifolds, definition and elementary properties of group actions, homogeneous spaces and co-set spaces, left and right actions on groups, representations of groups ,geometry of Lie groups ,left and right invariant vector fields, exponential map, Cartan-Maurer Equations, connections and Metrics on Lie Groups ,geodesics and auto-parallels on Lie Groups, fibre bundles, definition of fibre bundles, principal Bundles, vector bundles, associated bundles, connections on bundles, curvature and Cartan’s Equations.
Weekly Schedule
1) Differential Forms2) Grassmann Algebra: Interior Product and Wedge Product
3) Hodge Duality. Exterior Differentiation. Behaviour under pull-back. Stokes’ Theorem
4) Action of Groups on Manifolds. Definition and Elementary Properties of Group Actions
5) Homogeneous Spaces and Co-set spaces. Left and Right Actions on Groups
6) Representations of groups
7) Geometry of Lie Groups. Left and right invariant vector fields
8) midterm
9) Exponential representation
10) Maurer- Cartan Equations
11) Connections and Metrics on Lie Groups
12) Geodesics and auto-parallels on Lie Groups
13) Fibre Bundles. Definition of Fibre bundles
14) Principal Bundles, Vector Bundles, Associated Bundles
15) Connections on Bundles, Curvature and Cartan’s Equations
16) Connections on Bundles, Curvature and Cartan’s Equations
Recommended or Required Reading
1- The geometry of Physics. An introduction,.Theodore Frankel, Cambridge University Press.2011
2- Geometry, Topology and Physics, M. Nakahara, Institute of Physics Publ.2003
3- Differential Geometry, Gauge Theories, and Gravity, M. Goeckeler and T. Schuecker, Cambridge University Press 1987.
Planned Learning Activities and Teaching Methods
1) Lecture
2) Discussion
3) Group Study
4) Problem Solving
Assessment Methods and Criteria
Contribution of Midterm Examination to Course Grade |
40% |
|---|---|
Contribution of Final Examination to Course Grade |
60% |
Total |
100% |
Language of Instruction
Turkish
Work Placement(s)
Not Required