NICK SALTER

Basic Complex Analysis II: Introduction to Riemann surfaces (Spring 2026)

This course gives an introduction to the theory of Riemann surfaces. Our ultimate aim is to concentrate on the geometry of Riemann surfaces, and to see how this is intimately related to more traditional analytic questions as well as to general themes in algebraic geometry. For background, we will assume only familiarity with a first course in complex analysis, but we will cover the necessary inputs from topology, homological algebra, and projective geometry as we encounter a need for them.

Here is the syllabus.

Homework

• Assignment 3 (Due Feb. 4)
• Assignment 2 (Due Jan. 28)
• Assignment 1 (Due Jan. 21)     Solutions

References

• Notes by Curt McMullen
• Miranda, Algebraic curves and Riemann surfaces
• Griffiths-Harris, Principles of algebraic geometry

Day-by-day references

Suggested background/further reading for each class.

• Jan. 29 (Harmonic and holomorphic differentials): McMullen Ch.5, Miranda Ch. IV
• Jan. 27 (Differential forms on Riemann surfaces): McMullen Ch.5, Miranda IV.1,2
• Jan. 22 (Belyi's theorem): McMullen Ch.2
• Jan. 20 (Branched covers, covering spaces via monodromy): McMullen Ch. 2, Miranda III.4, Hatcher pp. 68-70
• Jan. 15 (Branched covers, Riemann-Hurwitz): McMullen Ch. 2, Miranda Ch. II
• Jan. 13 (Definitions, examples): McMullen Ch. 1, Miranda I.2