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| MAT4004 | Discrete Groups | 4+0+0 | ECTS:6 | | Year / Semester | Spring Semester | | Level of Course | First Cycle | | Status | Elective | | Department | DEPARTMENT of MATHEMATICS | | Prerequisites and co-requisites | None | | Mode of Delivery | Face to face | | Contact Hours | 14 weeks - 4 hours of lectures per week | | Lecturer | Öğr. Gör. Dr Süleyman UZUN | | Co-Lecturer | PROF. DR. MEHMET AKBAŞ, ÖĞR.GÖR.DR. SÜLEYMAN UZUN
| | Language of instruction | Turkish | | Professional practise ( internship ) | None | | | | The aim of the course: | | To provide a bridge through groups, topology and complex analysis, to build a No-Euclidean Geometry (Hiperbolic Geometry) and to get a combinatorial structure. |
| Learning Outcomes | CTPO | TOA | | Upon successful completion of the course, the students will be able to : | | | | LO - 1 : | understand the basic spaces, a model for the Hyperbolic plane, the Riemann Sphere and the boundary at infinity of the upper half-plane. | 5,7 | 1, | | LO - 2 : | understand the group of Möbiüs transformations and transitivity properties, the cross ratio, classification of Möbiüs transformations, a matric representation, reflections, the conformality of elements of Möbiüs, preserving the upper half-plane. | 5,7 | 1 | | LO - 3 : | understand topological groups,topological transformation groups,coverings,PSL(2,R) group and diskrete subgroups of its. | 5,7 | 1 | | LO - 4 : | understand hyperbolic length and distance in the upper half-plane, hyperbolic polygons, hyperbolic area and Gauss-Bonnet formula, Fuchsian groups and algebraic properties, fundamental fundamental domains. | 5,7 | 1 | | CTPO : Contribution to programme outcomes, TOA :Type of assessment (1: written exam, 2: Oral exam, 3: Homework assignment, 4: Laboratory exercise/exam, 5: Seminar / presentation, 6: Term paper), LO : Learning Outcome | | |
| The Basic Spaces; A Model for the Hyperbolic Plane, Riemann Sphere, The Boundary at Infinity of the upper half-plane, The General Möbiüs Group, The Group of Möbiüs Transformations and Transitivity Properties of its, The Cross Ratio, Classification of Möbiüs Transformations, A Matrix Representation, Reflections, The Conformality of Elements of Möb, Preserving the upper half-plane, Topological groups, topological transformation groups, coverings, PSL (2, R) group and diskrete subgroups of its, Hyperbolic Length and Distance in the upper half-plane, hyperbolic polygons, hyperbolic area, Gauss-Bonnet formula, Fuchsian groups and algebraic properties, fundamental domains. |
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| Course Syllabus | | Week | Subject | Related Notes / Files | | Week 1 | Basic spaces, and the model for the Hyperbolic plane, | | | Week 2 | The Riemann Sphere, and the boundary at infinity of the upper half-plane, | | | Week 3 | The group of Möbiüs transformations, | | | Week 4 | Transitivity properties, and the cross ratio, | | | Week 5 | Classification of Möbiüs transformations, and the matric representation of a Möbiüs transformation, | | | Week 6 | Reflections, the conformality of a Möbiüs transformation, preserving the upper half-plane, | | | Week 7 | The structure of topological group, | | | Week 8 | Mid-term exam | | | Week 9 | Topological group and properties, | | | Week 10 | Topological transformation groups, and coverings, | | | Week 11 | PSL(2,R) group, and diskrete subgroups of PSL(2,R) group, | | | Week 12 | Hyperbolic length and distance in the upper half-plane, | | | Week 13 | Hyperbolic polygons, hyperbolic area and Gauss-Bonnet formula, | | | Week 14 | Fuchsian groups and algebraic properties, | | | Week 15 | Fundamental domains. | | | Week 16 | End-of-term exam | | | |
| 1 | Anderson, James W. 2005; Hyperbolic Geometry, Springer-Verlag, London | | | |
| 1 | Jones, G.A., Singerman, D. 1987; Complex Functions: an algebraic and geometric viewpoint, Gambridge University Press, Gambridge | | | |
| Method of Assessment | | Type of assessment | Week No | Date | Duration (hours) | Weight (%) | | Mid-term exam | 9 | 17/04/2024 | 1,20 | 50 | | End-of-term exam | 16 | 06/06/2024 | 1,20 | 50 | | |
| Student Work Load and its Distribution | | Type of work | Duration (hours pw) | No of weeks / Number of activity | Hours in total per term | | Yüz yüze eğitim | 4 | 14 | 56 | | Sınıf dışı çalışma | 5 | 14 | 70 | | Arasınav için hazırlık | 8 | 1 | 8 | | Arasınav | 1.3 | 1 | 1.3 | | Dönem sonu sınavı için hazırlık | 10 | 1 | 10 | | Dönem sonu sınavı | 1.3 | 1 | 1.3 | | Total work load | | | 146.6 |
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