MTK754 - HYPERBOLIC MANIFOLDS

Course Name Code Semester Theory
(hours/week)		 Application
(hours/week)		 Credit ECTS
HYPERBOLIC MANIFOLDS MTK754 2nd Semester 3 0 3 12
Prequisites
Course languageTurkish
Course typeElective 
Mode of DeliveryFace-to-Face 
Learning and teaching strategiesLecture
Discussion
Question and Answer
Problem Solving
 
Instructor (s)Prof.Dr. YaÅŸar Sözen 
Course objectiveThe aim of the course is to give an introductory course on hyperbolic N-manifolds, especially to introduce hyperbolic surfaces and hyperbolic 3-manifolds, also provide the necessary background for applications of hyperbolic geometry on low-dimensional (N=2,3) manifolds.  
Learning outcomes
  1. Understands 2 and 3-dimensional hyperbolic geometry, can prove identities and properties of hyperbolic geometry and trigonometry.
  2. Learns the disk and upper half space models for 2 and 3-dimensional hyperbolic spaces, also can do computations in these models.
  3. Understands how Möbius transformations act on 2 and 3-dimensional hyperbolic spaces as isometries of these spaces.
  4. Understands the actions of discrete subgroups of 2 and 3-dimensional hyperbolic spaces on these spaces and also by using this action learns how to obtain 2 and 3-dimensional hyperbolic manifolds.
  5. Proves how compact orientable Riemann surfaces without boundary are hyperbolic surfaces, learns Teichmüller spaces of such surfaces and also basic facts of Teichmüller space.
  6. Learns Geometrization Theorems on 3-dimensions, and Mostow Rigidity Theorem.
Course ContentHyperbolic plane/space,
Disk and Upper Half Space models for hyperbolic plane/space,
Hyperbolic arc length,
Hyperbolic trigonometry,
Geodesics,
Isometries of Hyperbolic space,
Discrete Groups (Fuchsian and Kleinian groups),
Fundamental domains,
Hyperbolic Manifold,
Topology of compact surfaces,
Hyperbolic surface,
Teichmüller space,
Hyperbolic knots,
Geometrization Theorems in 3-dimension,
Mostow Rigidity Theorems 
References1. Francis Bonahon, Low-dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots, American Mathematical Society, 2009, ISBN: 0821884654, 9780821884652.\\
2. Riccardo Benedetti, Carlo Petronio, Lectures on Hyperbolic Geometry, Universitext - Springer, 1992, ISBN 978-3-642-58158-8.\\
3. John, Ratcliffe, Foundations of Hyperbolic Manifolds, 2006, Springer, Graduate Texts in Mathematics, Vol. 149, ISBN 978-0-387-47322-2.\\
4. Richard Canary, Albert Marden, David Epstein, Fundamentals of Hyperbolic Manifolds: Selected Expositions, 328, London Mathematical Society Lecture Note Series, Cambridge University Press, 2006, ISBN : 113944719X, 9781139447195.\\
5. Michael Kapovich, Hyperbolic Manifolds and Discrete Groups, Modern Birkhäuser classics Progress in mathematics (Boston, Mass.) v. 183, Springer Science & Business Media, 2001, ISBN : 0817639047, 9780817639044
 

Course outline weekly

WeeksTopics
Week 1Euclidean plane: Euclidean length and metric, shortest curves, isometries 2-dimensional sphere: spherical length and metric, shortests curves, isometries
Week 2Hyperbolic plane: disc and upper half plane models, hyperbolic length and metric, geodesics, isometries, Möbius transforms, hyperbolic triangles, hyperbolic trigonometry
Week 3Euclidean surfaces, Spherical surfaces
Week 4Hyperbolic surfaces
Week 5Teichmüller space
Week 6Midterm
Week 7Shortest curves in 3-dimensional hyperbolic space
Week 8Isometries of 3-dimensional hyperbolic space
Week 9Kleinian groups and limit sets, Fuchsian groups
Week 10The Geometrization Theorem for knot complements
Week 11Geometrization Theorem for surfaces
Week 12Geometrization Theorem for 3-dimensional manifolds
Week 13Mostow Rigidity Theorem
Week 14Preparation for Final Exam
Week 15Final Exam

Assesment methods

Course activitiesNumberPercentage
Attendance00
Laboratory00
Application00
Field activities00
Specific practical training00
Assignments630
Presentation00
Project00
Seminar00
Midterms130
Final exam140
Total100
Percentage of semester activities contributing grade succes060
Percentage of final exam contributing grade succes040
Total100

WORKLOAD AND ECTS CALCULATION

Activities Number Duration (hour) Total Work Load
Course Duration (x14) 14 3 42
Laboratory 0 0 0
Application000
Specific practical training000
Field activities000
Study Hours Out of Class (Preliminary work, reinforcement, ect)1412168
Presentation / Seminar Preparation000
Project000
Homework assignment620120
Midterms (Study duration)11515
Final Exam (Study duration) 11515
Total Workload3665360

Matrix Of The Course Learning Outcomes Versus Program Outcomes

D.9. Key Learning OutcomesContrubition level*
12345
1. Deepens the concepts of mathematics in the level of expertise.    X
2. Grasps the inter-disciplinary interaction related to the area; reaches original results by using the specialist knowledge in analyzing and evaluating new ideas.  X  
3. Gains the ability to think independently and develops theoretical concepts.     X
4. Develops original mathematical models by using interrelations between mathematics and other disciplines and applies them to other disciplines.     
5. Uses high level research methods in studies in the area.   X 
6. Develops a new idea, method and/or application independently, finds a solution, and contributes to the progress in the area by carrying out original studies.     
7. Fulfills the leader role in the environments where solutions are thought for the area and/or inter-disciplinary problems.     
8. Develops continually the skills of creativity, decision making and problem solving.    X
9. Defends original opinions by communicating with experts in the area.     
10. Uses a foreign language- at least C1 Level-, communicates with foreign colleagues and follows the international literature.   X 
11. Follows the latest developments in the information and communication technologies and uses them in the area.     
12. Does research in national and international research groups.     
13. Makes strategic decision in the solution of problems in the area.     
14. Protects the rights of other researchers in regards to ethics, privacy, ownership and copyright.  X  

*1 Lowest, 2 Low, 3 Average, 4 High, 5 Highest