COURSE UNIT TITLE

: HYPERBOLIC GEOMETRY

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 5045 HYPERBOLIC GEOMETRY ELECTIVE 3 0 0 7

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

ASSISTANT PROFESSOR CELAL CEM SARIOĞLU

Offered to

Mathematics (English)
Mathematics (English)

Course Objective

This course aims to introduce the beautiful interplay between geometry, algebra and analysis which is involved in a detailed study of the models for hyperbolic spaces.

Learning Outcomes of the Course Unit

1   will be able to explain non-Euclidean geometries.
2   will be able to describe models for hyperbolic plane.
3   will be able to describe the hyperbolic 3-space.
4   will be able to write the isometry groups of each hyperbolic plane models.
5   will be able to find the fundamental domains of discrete groups.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Motivation: Axioms of the Euclides, the birth of Non-Euclidean geometries, a brief history.
2 Euclidean transformations of the Euclidean Plane, the Riemann Sphere, Möbius transformation and the cross ratio.
3 Classification of Möbius transformations, Möbius groups, Discreteness of Möbius groups.
4 The Cayley transformation, Automorphisms of the one point compactificaion of the complex plane C, the upper half space H, and the unit disk D.
5 The Upper half plane model H: geodesics and the hyperbolic metrics. The Disc model D: geodesics and the hyperbolic metric.
6 The Klein model: geodesics and the hyperbolic metric. The hyperboloid model, geodesics and the hyperbolic metric.
7 Hyperbolic 3-space, Higher dimensional hyperbolic spaces.
8 Higher dimensional hyperbolic spaces
9 Hyperbolic area and the Gauss-Bonnet theorem, Hyperbolic trigonometry. Comparison between hyperbolic, spherical and Euclidean trigonometry.
10 Isometries, classification of different type of isometries.
11 Holomorphic functions, The Schwarz lemma, the Riemann mapping theorem.
12 Topology and uniformization, groups generated by side pairing transformations.
13 Fuchsian groups, Fundamental domain of the Modular group, Drichlet regions.
14 Geometry of Fuchsian groups, Arithmetic Fuchsian groups.

Recomended or Required Reading

Textbook:
1. Linda Keen, Nikola Lakic, Hyperbolic Geometry from Local Viewpoint, Cambridge University Press, 2007.
2. Svetlana Katok, Fuchsian Groups, University of Chicago Press, 1992.
Supplementary Books:
3. Ricardo Benedetti, Carlo Petronio, Lectures on Hyperbolic Geometry, Springer 1992.
4. James W. Anderson, Hyperbolic Geometry, Springer, 2008.
5. Birger Iversen, Hyperbolic Geometry, Cambridge University Press, 1992.
6. William M. Goldman, Complex Hyperbolic Geometry, Oxford Mathematical Monographs, Clarendon Press, 1999.
7. Lars V. Ahlfors, Mobius Transformations in Several Dimensions. Minneapolis : University of Minnesota, School of Mathematics, 1981.
8. Saul Stahl, Poincare Half-Plane, Jones&Bartlett Learnning, 1993.
9. Werner Fenchel, Elementary Geometry in Hyperbolic Spave, Walter de Gruyter, 1989.
References:
10. Nicolas Bergeron, The Spectrum of Hyperbolic Surfaces, Springer, 2011.
11. David Borthwick, Spectral Theory of Infinite-Area Hyperbolic Surfaces, 2nd edition, Birkhauser, 2016.
12. Alan F. Beardon, The Geometry of Discrete Groups, Springer-Verlag, 1983.
13. Colin Maclachlan, Alan W. Reid, The Arithmetic of Hyperbolic 3-Manifolds, Springer, 2003.

Planned Learning Activities and Teaching Methods

Lecture notes, Presentation, Problem solving, Homework Assignments

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 ASG ASSIGNMENT
2 FIN FINAL EXAM
3 FCG FINAL COURSE GRADE ASG * 0.50 + FIN * 0.50
4 RST RESIT
5 FCGR FINAL COURSE GRADE (RESIT) ASG * 0.50 + RST * 0.50


*** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

Homework Assignments, Final exam

Assessment Criteria

50% (Ara sınav) +50% (Final sınavı)

Language of Instruction

English

Course Policies and Rules

Attending at least 70 percent of lectures is mandatory.

Contact Details for the Lecturer(s)

Asst. Prof. Dr. Celal Cem SARIOĞLU
E-mail: celalcem.sarioglu@deu.edu.tr
Phone: +90 232 301 8585
Office: B212 (Mathematics Department)

Office Hours

To be announced later.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 14 3 42
Preparations before/after weekly lectures 13 3 39
Preparation for midterm exam 1 25 25
Preparation for final exam 1 30 30
Preparing assignments 1 30 30
Final 1 2 2
TOTAL WORKLOAD (hours) 168

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11
LO.1334434343
LO.2334434343
LO.3334434343
LO.4334434343
LO.5334434343

CONTACT INFORMATION
Dokuz Eylül Üniversitesi Cumhuriyet Bulvarı No: 144 35210 Alsancak / İZMİR
Phone: +90(232) 412 12 12 - Fax: +90 (232) 464 81 35

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