Bram Petri

Teichmüller Theory

Practical information

Semester: 2018/2019 - summer
Module code: V5D3 - Advanced Topics in Geometry

Times and rooms:
- Tuesdays 10:15 - 12:00 in SR-0.008, Endenicher Allee 60
- Thursdays 12:15 - 14:00 in SR-0.007, Endenicher Allee 60

There will be no more lectures for this course

Exam

The exams will be oral and will take place in office 2.003 at the Endenicher Allee 60. They will take place on July 11 and 12 and August 28, 29 and 30. If you haven't yet picked a time slot, please contact me by email.

Contents

The Teichmüller space of a surface S is the deformation space of complex structures on S and can also be seen as a space of hyperbolic metrics on S. The aim of this course will be to study the geometry and topology of this space and its quotient: the moduli space of hyperbolic metrics on S. In particular, the end goal will be to prove Mirzakhani's recurrence for the Weil-Petersson volumes of moduli spaces.

Preliminaries

Linear algebra, analysis, complex analysis, basic differential geometry, point-set topology.

Lecture notes

I will post my notes here after each lecture. The exercises for every week can be found on the last pages of each section.
DISCLAIMER: I do not guarantee in any way that these notes are correct. I will be happy to hear of any mistakes that are found.

Lecture notes

Literature

Riemann surfaces:

• A. F. Beardon. A primer on Riemann surfaces,volume 78 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1984.
• H. M. Farkas and I. Kra. Riemann surfaces, volume 71 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1992.
• Ernesto Girondo and Gabino González-Diez. Introduction to compact Riemann surfaces and dessins d’enfants volume 79 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2012.
• Phillip Griffiths and Joseph Harris. Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994.

Complex Analysis

• Elias M. Stein and Rami Shakarchi. Complex analysis, Princeton Lectures in Analysis. Princeton University Press, Princeton, NJ, 2003.
• Zeev Nehari. Conformal mapping. Dover Publications, Inc., New York, 1975.

Algebraic topology:

• Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002.

Teichmüller Theory:

• Peter Buser. Geometry and spectra of compact Riemann surfaces. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2010.
• David Dumas. Complex projective structures. In Handbook of Teichmüller theory. Vol. II, volume 13 of IRMA Lect. Math. Theor. Phys., pages 455–508. Eur. Math. Soc., Zürich, 2009.
• Benson Farb and Dan Margalit. A primer on mapping class groups, volume 49 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 2012.
• John Hamal Hubbard. Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Ithaca, NY, 2006.
• Y. Imayoshi and M. Taniguchi. An introduction to Teichmüller spaces. Springer-Verlag, Tokyo, 1992.
• Maryam Mirzakhani. Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. Invent. Math., 167(1):179–222, 2007.

Hyperbolic geometry:

• A. F. Beardon. The geometry of discrete groups, volume 91 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.

Riemannian geometry:

• Jeff Cheeger and David G. Ebin. Comparison theorems in Riemannian geometry. AMS Chelsea Publishing, Providence, RI, 2008.

Symplectic geometry:

• G. Heckman. Symplectic geometry. Lecture notes, available here, 2014.