Riemann Surfaces and Mapping Class Groups

When and where: Sunday 15:15-17:00 Room 155 Ziskind Building (On Sunday 3/11 the lesson is moved to Room 208 Goldsmith Building. There will be no lesson on Sunday 29/12/19 and 05/01/20).
A tentative synopsis can be found on the course page of the FGS.

Lecture notes

These are the notes that I wrote. Comments are always welcome.

• Chapter 1: Planar Domains (v2.1)
• Chapter 2: Structures on surfaces (v2)
• Chapter 3: Riemann surfaces (v2)
• Chapter 4: Topology and geometry of surfaces (v1.1)
• Chapter 5: Teichmüller and Thurston (v1.1)

Exercises

These are the problem sheets. You do not have to solve all the exercises.

• Exercises - Week 1
• Exercises - Week 2
• Exercises - Week 3
• Exercises - Week 4
• Exercises - Week 5
• Exercises - Week 6
• Exercises - Week 7
• Exercises - Week 8
• Exercises - Week 9
• Exercises - Week 10
• Exercises - Week 11
• Exercises - Week 12 - End Projects

Some extra material

These are some extra notes that you might find useful. They are not going to be mentioned in class.

• Planar differential geometry. Here we introduce a few more ideas of Riemannian geometry in the context of planar domains.

References

These are some books and papers that I found useful while preparing this course:

• (Recommended)H.M. Farkas, I. Kra: Riemann Surfaces. Springer (1980)
• W. Rudin: Real and complex analysis. McGraw-Hill (1986)
• (Recommended)B. Martelli: An Introduction to Geometric Topology. Independent Publication (2016)
• L. Ahlfors, L. Sario: Riemann Surfaces. Princeton University press (1960)
• H. Zieschang, E. Vogt, H.D. Coldewey: Surfaces and Planar Discontinuous Groups. Springer (1980)
• (Recommended)B. Farb, D. Margalit: A primer on mapping class groups. Princeton University press (2012)
• H. Cartan: Elementary Theory of Analytic Functions of One or Several Complex Variables Dover Books in Mathematics (1973).
• F. Bonahon: The geometry of Teichmiiller space via geodesic currents. Invent. Math. 92,139-162 (1988)
• W. Thurston (S. Levy edit.): Three-Dimensional Geometry and Topology. Princeton University Press (1997)
• W. Thurston: The Geometry and Topology of Three-Manifolds. Lecture notes
• M. do Carmo: Differential Geometry of Curves and Surfaces. Prentice-Hall (1976)
• S. Gallot, D. Hulin, J. Lafontaine: Riemannian Geometry. Springer (1987)
• E. Moise: Geometric Topology in Dimensions 2 and 3. Springer (1977)
• M. Spivak: A comprehensive introduction to Differential Geometry. Publish or Perish, 3rd Edition (1999)
• A. Hatcher Algebraic topology. Cambridge University Press (2002) pdf

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