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.
Homework
- Assignment 9 (Due Apr. 8)
- Assignment 8 (Due Apr. 1) Solutions
- Assignment 7 (Due Mar. 25) Solutions
- MIDTERM (Due FRIDAY Mar. 6) Solutions
- Assignment 6 (Due Feb. 25) Solutions
- Assignment 5 (Due Feb. 18) Solutions
- Assignment 4 (Due Feb. 11) Solutions
- Assignment 3 (Due Feb. 4) Solutions
- Assignment 2 (Due Jan. 28) Solutions
- Assignment 1 (Due Jan. 21) Solutions
- Notes by Curt McMullen
- Miranda, Algebraic curves and Riemann surfaces
- Griffiths-Harris, Principles of algebraic geometry
Suggested background/further reading for each class.
- Apr. 2 (Classification of Riemann surfaces in low genus): Miranda VII.2.
- Mar. 31 (Weierstrass points, automorphisms of Riemann surfaces): McMullen Ch. 12, Miranda VII.4, Griffiths-Harris pp. 273-277.
- Mar. 24, 26 (Embeddings, canonical map): McMullen Ch. 11, 12, Miranda V.4, VII.2.
- Mar. 17,19 (Projective geometry, linear systems): McMullen Ch. 11, Miranda III.5, V.4.
- Mar. 5 (Riemann surfaces and algebraic curves): McMullen Ch. 4, Miranda VI.1.
- Mar. 3 (Mittag-Leffler, Serre duality): McMullen Ch. 9, 10, Miranda VI.2,3.
- Feb. 24 (Basics of projective geometry): Miranda III.5, V.4
- Feb. 19 (Riemann-Roch): McMullen Ch. 8, Miranda Ch. VI
- Feb. 17 (Dolbeault cohomology, divisors, Riemann-Roch): McMullen Ch. 7, 8, Miranda Ch. V
- Feb. 12 (Sheaves of differentials, smooth and holomorphic): McMullen Chs. 6,7, Griffiths-Harris Ch. 0 secs. 1,3
- Feb. 10 (Sheaf cohomology): McMullen Ch. 6, Miranda Ch. IX, Griffiths-Harris Ch. 0 sec. 3
- Feb. 3,5 (More differentials; introduction to sheaves): McMullen Ch. 5; McMullen Ch. 3, Miranda Ch. IX
- 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