2017 Advanced topics in Analysis F1

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Academic unit or major
Graduate major in Mathematics
Shiga Hiroshige  Tanabe Masaharu 
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Course description and aims

Lectures are a sequel to ''Advanced topics of Analysis E1'' in the previous quarter. A Riemann surface is a two-real-dimensional manifold with holomorphic coordinate transformations. The moduli space of Riemann surfaces is a geometric space whose points represent classes of conformally equivalent Riemann surfaces. The Teichmüller space is a universal covering of the moduli space.
Each point in it is an isomorphism class of 'marked' Riemann surfaces. Ahlfors was the first to derive the complex structure of Teichmüller space. We will study his method. Topics include Teichmüller spaces, quasiconformal maps, Teichmüller’s theorem,
and the complex structure of Teichmüller space.

Student learning outcomes

At the end of this course, students are expected to:
-- be familiar with quasiconformal maps
-- understand Teichmüller’s theorem
-- understand Ahlfors' approach for the complex structure of Teichmüller space


Riemann surfaces, moduli spaces of Riemann surfaces, Teichmüller spaces

Competencies that will be developed

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

Class flow

Standard lecture course

Course schedule/Required learning

  Course schedule Required learning
Class 1 Teichmüller spaces Details will be provided during each class session.
Class 2 Quasiconformal maps
Class 3 The Teichmüller distance
Class 4 Teichmüller modular groups
Class 5 Quadratic differentials
Class 6 Teichmüller’s theorem
Class 7 Ahlfors' approach I
Class 8 Ahlfors' approach II



Reference books, course materials, etc.

H. M. Farkas and I. Kra, Riemann surfaces, GTM 71, Springer-Verlag
Y. Imayoshi and M. Taniguchi, An Introduction to Teichmüller Spaces, Springer-Verlag
L. V. Ahlfors, The complex analytic structure of the space of closed Riemann surfaces. In Rolf Nevanlinna et. al., editor, Analytic Functions, pages 45-66. Princeton University Press, 1960.

Assessment criteria and methods


Related courses

  • MTH.C301 : Complex Analysis I
  • MTH.C302 : Complex Analysis II
  • MTH.C505 : Advanced topics in Analysis E1

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


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