2017 Advanced topics in Analysis E1

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

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. Teichmüller space is a universal covering of the moduli space. Each point in it is an isomorphism class of 'marked' Riemann surfaces.

This course will be completed with ''Advanced topics of Analysis F1'' in the next quarter. Our goal is to understand Ahlfors’ method
which is the first to derive the complex structure of Teichmüller space. To prepare for it, several basic tools and theorems about Riemann surfaces will be introduced in this course.

Student learning outcomes

At the end of this course, students are expected to:

-- understand the moduli space and the Teichmüller space of the torus

-- be familiar with differential forms on Riemann surfaces


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 Riemann surfaces Details will be provided during each class session.
Class 2 The moduli space of the torus
Class 3 The Teichmüller space of the torus
Class 4 Topology of Riemann surfaces
Class 5 Differential forms
Class 6 Harmonic differetials, holomorphic differetials
Class 7 bilinear relations
Class 8 Period matrices



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.C506 : Advanced topics in Analysis F1

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


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