# Introduction to Riemann Surfaces

## Introduction to Riemann Surfaces מבוא למשטחי רימן

### Spring semester 2017 March 26, 2017 - July 14, 2017

• Lecturer
Bo'az Klartag
Room 264, Ziskind building
Phone: 08-9342844
e-mail:

• Classes
Monday, 11:15 - 13:00, Ziskind 155.

• TA sessions
Wednesday, 16:15 - 17:00, Ziskind 155.

• Syllabus
Riemann surfaces are complex manifolds of one complex dimension or equivalently two real dimensions. They were discovered by Riemann in the 1850s in his studies of multivalued holomorphic functions such as the square root or the logarithm.

The subject has evolved to a rather pretty branch of mathematics, with many connections to other areas. We offer an introductory course on the subject from the point of view complex analysis, geometry and topology.

Here is a rather detailed plan of the course, each topic will probably be discussed  for 1-2 weeks:

1. Holomorphic functions and monodromy
2. Riemann surfaces, examples
3. Degree of a map, algebraic Riemann surfaces
4. Covering maps, Riemann existence theorem
5. Normalization of an algebraic curve
6. Riemann-Hurwitz, holomorphic 1-forms
7. Elliptic functions and integrals
8. Harmonic and subharmonic functions, Harnack's inequality
9. Perron's method for the Dirichlet problem, Green's function, Riemann mapping.
10. Elliptic, parabolic and hyperbolic surfaces, proof of the uniformisation theorem.
• Prerequisites
undergraduate complex analysis (say, Cauchy's theorem).

It will be based on the solution of the homework exercises. The ideas is that the students will be asked to present homework solutions during the TA sessions.

• Related literature
There are tens of books on this classical topic, with great variation in emphasis and style. Among them we recommend:
1. Ahlors, Sario, Riemann Surfaces, 1960.
2. Beardon, A primer on Riemann Surfaces, 1984 (easy to read).
3. Donaldson, Riemann Surfaces, 2011 (another book we will follow).
4. Farkas, Kra, Riemann Surfaces, 1980.
5. Forster, Lectures on Riemann Surfaces, 1981.
6. Schlag, A course in complex analysis and Riemann Surfaces, 2014.
7. For uniformization: Ahlors, Conformal Invariants: Topics in Geometric Function Theory, 1973.