Course Name Code Semester T+U Hours Credit ECTS
Introduction To Fuchsian Groups MAT 604 0 3 + 0 3 6
Precondition Courses
Recommended Optional Courses
Course Language Turkish
Course Level Doctorate Degree
Course Type Optional
Course Coordinator Prof.Dr. REFİK KESKİN
Course Lecturers
Course Assistants
Course Category Other
Course Objective

Fuchsian Groups is one of the topics of mathematics that contains algebra, topology and analysis. Aim of this lesson is to give the fundamental knowledge about this topic.

Course Content

Topological groups, group action, discrete groups, Steographic projection and expanded complex plane, Mobius Transformations, classifying Mobius Transformations, PSL(2,R) and its subgroups, Fuchsian groups, Parabolic class number, modular groups, subgroups of modular groups, orbitals and factor space, fundamental domain, suborbital graphs.

# Course Learning Outcomes Teaching Methods Assessment Methods
1 He/she has knowledge about group action. Lecture, Drilland Practice, Problem Solving, Testing, Homework,
2 He/she remembers the subjects of steographic projection and expanded complex plane. Lecture, Drilland Practice, Problem Solving, Testing, Homework,
3 He/she recognizes Mobius transformations. Lecture, Drilland Practice, Problem Solving, Testing, Homework,
4 He/she classifies the Mobius transformations and gain the ability to relate them. Lecture, Drilland Practice, Problem Solving, Testing, Homework,
5 He/she recognizes Fuchsian groups and learns their structure. Lecture, Drilland Practice, Problem Solving, Testing, Homework,
6 He/she investigates modular group and its special subgroups. Lecture, Drilland Practice, Problem Solving, Testing, Homework,
7 He/she researches suborbital graphs on some Fuchsian groups. Lecture, Drilland Practice, Problem Solving, Testing, Homework,
Week Course Topics Preliminary Preparation
1 Topological groups
2 Group action
3 Discrete groups
4 Steographic projection and expanded complex plane
5 Mobius Transformations
6 Classifying Mobius Transformations
7 PSL(2,R) and its subgroups
8 Fuchsian groups
9 Parabolic class number
10 Modular groups
11 Subgroups of modular groups
12 Orbitals and factor space
13 Fundamental domain
14 Suborbital graphs.
Resources
Course Notes
Course Resources

1- G. A. Jones and D. Singerman, Complex functions: An algebraic and geometric viewpoint, Cambridge University Press, 1987.
2- T. Miyake, Modular forms, Springer Verlag, 1989.

Order Program Outcomes Level of Contribution
1 2 3 4 5
1 At a master´s degree level, student reaches new knowledge via scientific researches, the use of knowledge of the same field as him/her or of different field from him/her, and the use of knowledge based on the competence in his/her field; s/he interprets the knowledge and prospects for the fields of application.
1 At a master´s degree level, student reaches new knowledge via scientific researches, the use of knowledge of the same field as him/her or of different field from him/her, and the use of knowledge based on the competence in his/her field; s/he interprets the knowledge and prospects for the fields of application. X
2 Student completes the missing or limited knowledge by using the scientific methods. X
2 Student completes the missing or limited knowledge by using the scientific methods.
3 Student freely poses a problem of his/her field, develops a solution method, solves the problem, and evaluates the result.
3 Student freely poses a problem of his/her field, develops a solution method, solves the problem, and evaluates the result. X
4 Student conveys, orally or in writing, his/her studies or the current developments in his/her field to the people in or out of his/her field. X
4 Student conveys, orally or in writing, his/her studies or the current developments in his/her field to the people in or out of his/her field.
5 Student finds a solution to the unforeseen complex problems in his/her studies by developing new approaches.
5 Student finds a solution to the unforeseen complex problems in his/her studies by developing new approaches.
6 At a doctorate degree level, student prepares at least one scientific article of his/her field to be published in an international indexed journal, and s/he extends its popularity. X
6 At a doctorate degree level, student prepares at least one scientific article of his/her field to be published in an international indexed journal, and s/he extends its popularity.
7 Student analyzes the works that have been published before, approaches the same subjects with different proof methods, or determines the open problems about the current subject matters.
7 Student analyzes the works that have been published before, approaches the same subjects with different proof methods, or determines the open problems about the current subject matters.
8 Student looks for the scientists studying on the same field as him/her, and s/he gets in touch with them for a collaborative work. X
8 Student looks for the scientists studying on the same field as him/her, and s/he gets in touch with them for a collaborative work.
9 Student knows enough foreign language to do a collaborative work with the scientists studying on the same field as him/her abroad.
9 Student knows enough foreign language to do a collaborative work with the scientists studying on the same field as him/her abroad. X
10 Student follows the necessary technological developments in his/her field, and s/he uses them. X
10 Student follows the necessary technological developments in his/her field, and s/he uses them.
11 Student looks out for the scientific and ethic values while gathering, interpreting and publishing data.
Evaluation System
Semester Studies Contribution Rate
1. Ara Sınav 100
Total 100
1. Yıl İçinin Başarıya 40
1. Final 60
Total 100
ECTS - Workload Activity Quantity Time (Hours) Total Workload (Hours)
Course Duration (Including the exam week: 16x Total course hours) 16 3 48
Hours for off-the-classroom study (Pre-study, practice) 16 3 48
Mid-terms 1 30 30
Assignment 1 10 10
Final examination 1 10 10
Total Workload 146
Total Workload / 25 (Hours) 5.84
dersAKTSKredisi 6