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| MAT7167 | Geometry of Discrete Groups | 3+0+0 | ECTS:7.5 | | Year / Semester | Spring Semester | | Level of Course | Third Cycle | | Status | Elective | | Department | DEPARTMENT of MATHEMATICS | | Prerequisites and co-requisites | None | | Mode of Delivery | | | Contact Hours | 14 weeks - 3 hours of lectures per week | | Lecturer | Prof. Dr. Ali Hikmet DEĞER | | Co-Lecturer | | | Language of instruction | Turkish | | Professional practise ( internship ) | None | | | | The aim of the course: | | To study the geometry and topological structure of discrete groups. |
| Programme Outcomes | CTPO | TOA | | Upon successful completion of the course, the students will be able to : | | | | PO - 1 : | learn the topological structure of the discrete groups. | 1,2,8 | 1,3, | | PO - 2 : | learn the geometry of discrete groups. | 1,2,8 | 1,3, | | PO - 3 : | explore the role of discrete groups in non-Euclidean geometry. | 1,3,8 | 1,3, | | 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), PO : Learning Outcome | | |
| Möbius tranformations, structure of topological groups, discontinuous groups, Riemann surfaces, Hyperbolic geometry, Discrete groups, Fundamental regions, finitely generated groups. |
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| Course Syllabus | | Week | Subject | Related Notes / Files | | Week 1 | Preliminaries; Notations, Inequalities, Algebra, Topology, Topological groups, Analysis. | | | Week 2 | Matrices; Non-singular matrices, Metric structure, Discrete groups, Quaternions, Unitary matrices. | | | Week 3 | Möbius transformations in R^n; Möbius group in R^n, Properties of Möbius transformations, Poincare extension, Self mapping of unit ball. | | | Week 4 | The general form of a Möbius transformation, Distortion theorems, Topological group structure, Notes. | | | Week 5 | Complex Möbiüs Transformations, Representations by Quaternions, Representation by Matrices, Fixed points and Conjugacy Classes, Cross ratios, The topology of M, Notes. | | | Week 6 | The elementary groups, Groups with an invariant disk, Discontinuous groups, Jorgensen's inequality, Notes. | | | Week 7 | Riemann surfaces, Quotient spaces, Stable sets. | | | Week 8 | The Hyperbolic geometry; The Hyperbolic plane, The Geodesics. | | | Week 9 | Mid-term exam | | | Week 10 | The İsometries, Convex sets, Angles. | | | Week 11 | Hyperbolic trigonometry; Triangles, Notations, The Sine and Cosine rules, The area of a triangle. | | | Week 12 | Polygons, The geometry of Geodesics, The geometry of isometries. | | | Week 13 | Fuchsian groups; Purely hyperbolic groups, Groups of elliptic elements, Criteria for Discreteness, The Nielsen region, Notes. | | | Week 14 | Fundamental regions, Dirichlet polygon, Poincare's theorem, Finitely generated groups, Points of approximation, Conjugacy classes, The signature of a Fuchsian group, Triangle groups, Notes. | | | Week 15 | Uniformity of discreteness, Hecke groups, Trace inequalities, Canonical regions, and quotient surfaces. | | | Week 16 | End-of term exam | | | |
| 1 | The Geometry of Discrete Groups, Alan F. Beardon, Springer-Verlag Berlin Heidelberg 1983. | | | |
| 1 | Fuchsian Groups, S. Katok, Chicago Lectures in Mathematics. | | | |
| Method of Assessment | | Type of assessment | Week No | Date | Duration (hours) | Weight (%) | | Mid-term exam | 9 | 23.11.2021 | 2 | 50 | | End-of-term exam | 16 | 25.02.2022 | 2 | 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 | 3 | 14 | 42 | | Sınıf dışı çalışma | 9 | 14 | 126 | | Arasınav için hazırlık | 2 | 8 | 16 | | Arasınav | 2 | 1 | 2 | | Dönem sonu sınavı için hazırlık | 3 | 8 | 24 | | Dönem sonu sınavı | 2 | 1 | 2 | | Total work load | | | 212 |
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