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2016　Special lectures on current topics in Mathematics F

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Shiga Hiroshige  Amano Masanori   
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
Intensive (H201)  
Group
-
Course number
MTH.E636
Credits
2
Academic year
2016
Offered quarter
4Q
Syllabus updated
2016/4/27
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

In this course we will explain the basic theory for Kleinian groups and their deformations. The important concepts is the following: Kleinian group, quasiconformal mapping, Teichmüller space. By viewing the deformation of a Kleinian group as a quasiconformal deformation of a Riemann surface, it is possible to see the relation between the Teichmüller space which is a deformation space of the Riemann surface. To understanding it, first we learn about hyperbolic Riemann surfaces and the hyperbolic geometry. Further, we see a Fuchsian group as the covering transformation group of the Riemann surface and its properties. We also see a Kleinian group, it is a generalized Fuchsian group and consider a relation with a Riemann surface. Then we define quasiconformal mappings which give a deformation of a Riemann surface, and describe the Teichmüller space together with Fuchsian groups. Finally, we give a complex structure of the Teichmüller space, a boundary representation using the Kleinian group, and some applications of the Teichmüller theory.
The aim of this course is the learning of the basic concepts and the theory of Kleinian groups and their deformations. A hyperbolic Riemann surface and a domain of discontinuity of a Kleinian group that are appeared in this course, are related to the hyperbolic geometry and the complex dynamics respectively. In addition, the Teichmüller space theory is an important research field which has a close relationship with the deformation theory of Klein groups, and there are many applications which are known to each other. In this course, in addition to the understanding of the theory, as a theme, we hope that you get to spread views for the researches which are based on low-dimensional complex manifolds.

Student learning outcomes

・Understand that almost all Riemann surfaces can be introduced the hyperbolic metric, and properties of each corresponding Fuchsian group.
・Knowing the definition of Kleinian group, and understand the relation between the domain of discontinuity of it and a Riemann surface.
・Understand two definitions of quasiconfomal mappings, and that they express a deformation of Riemann surfaces.
・Knowing the relation and their applications of the deformation space of Kleinian groups and the Teichmüller space.

Keywords

Hyperbolic geometry, Riemann surface, Fuchsian group, Kleinian group, quasiconformal mapping, Teichmüller space.

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. There will be some assignments.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Möbius transformation Details will be provided during each class session.
Class 2 hyperbolic geometry Details will be provided during each class session.
Class 3 covering theory Details will be provided during each class session.
Class 4 universal covering space and uniformization theorem Details will be provided during each class session.
Class 5 hyperbolic Riemann surface and Fuchsian group Details will be provided during each class session.
Class 6 properties of a Fuchsian group (fundamental domain and hyperbolic metric) Details will be provided during each class session.
Class 7 properties of a Fuchsian group (Shimizu's lemma, trace and hyperbolic length) Details will be provided during each class session.
Class 8 The definition and properties of a Kleinian group Details will be provided during each class session.
Class 9 The limit set and the domain of discontinuity of a Kleinian group Details will be provided during each class session.
Class 10 quasiconformal mapping (geometric definition) Details will be provided during each class session.
Class 11 quasiconformal mapping (analytic definition and Beltrami equation) Details will be provided during each class session.
Class 12 The deformation space of a Kleinian group Details will be provided during each class session.
Class 13 The representation of a Teichmüller space Details will be provided during each class session.
Class 14 Teichmüller space and Kleinian group Details will be provided during each class session.
Class 15 applications of Teichmüller theory Details will be provided during each class session.

Textbook(s)

None required.

Reference books, course materials, etc.

Yoichi Imayoshi and Masahiko Taniguchi. An introduction to Teichmüller spaces. Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors.

Assessment criteria and methods

Assignments (100%).

Related courses

  • MTH.C302 ： Complex Analysis II

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

Students are expected to have passed MTH.C302 ： Complex Analysis II.

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