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2018　Advanced topics in Analysis E

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Shiga Hiroshige 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
Thr3-4(H117)  
Group
-
Course number
MTH.C501
Credits
1
Academic year
2018
Offered quarter
1Q
Syllabus updated
2018/3/20
Lecture notes updated
-
Language used
Japanese
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Course description and aims

This course will give the fundamental theory of Teichmullar space, Kleinian groups, and their deformation spaces. The Teichmuller space is a deformation space of a Riemann surface and a Kleinian group is a discontinuous group of linear fractional transformations. The deformation space of a Kleinian group is a generalization of the Teichmuller space. Those theories are important for complex dynamics, topology as well as complex analysis. In this course, those theories will treated consistently by using quasiconformal maps.
This course will be completed with "Advanced topics in Analysis F" in the next quarter.

The students will become familiar with fundamentals of quasicormal mappings, Teichmuller spaces, and Kleinian groups, and be ready for learning deep understandings of them.

Student learning outcomes

At the end of this course, students are expected to:
-- Understad fundamentals of Riemann surfaces and Fuchsian groups
-- understand fundamentals of Kleinian groups

Keywords

Riemann surfaces, Fuchsian groups, quasiconformal mappings, Teichmuller spaces, Kleinian groups.

Competencies that will be developed

✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

This is a standard lecture course. Homework will be assigned occasionally

Course schedule/Required learning

  Course schedule Required learning
Class 1 Introduction to linear fractional transformations and Kleinian groups Details will be provided in class.
Class 2 Kleinian groups and Riemann surfaces Details will be provided in class.
Class 3 Uniformization theorem and hyperbolic geometry Details will be provided in class.
Class 4 Universal coverings and Fuchsian groups Details will be provided in class.
Class 5 Properties of Fuchsian groups (1 : fundamental domains and the hyperbolic metric) Details will be provided in class.
Class 6 Properties of Fuchsian groups (2 : Shimizu's lemma and hyperbolic length) Details will be provided in class.
Class 7 Limit sets and region of discontinuities of Kleinian groups Details will be provided in class.
Class 8 Hyperbolic geometry and Kleinian groups Details will be provided in class.

Textbook(s)

None required

Reference books, course materials, etc.

Ahlfors, "Lectures on Quasifoncormal mappings", AMS
Hubbard, "Teichmuller Theory, Vol. 1"

Assessment criteria and methods

assignments 100%.

Related courses

  • MTH.C502 ： Advanced topics in Analysis F

Prerequisites (i.e., required knowledge, skills, courses, etc.)

None required

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