Riemann Surfaces and Complex Analysis (MAST90056)

Riemann surfaces arise from complex analysis. They are central in mathematics, appearing in seemingly diverse areas such as differential and algebraic geometry, number theory, integrable systems, statistical mechanics and string theory.

The first part of the subject studies complex analysis. It assumes students have completed a first course in complex analysis so begins with a quick review of analytic functions and Cauchy's theorem, emphasising topological aspects such as the argument principle and Rouche's theorem.

Topics also include: Schwarz's lemma; limits of analytic functions, normal families, Riemann mapping theorem; multiple-valued functions, differential equations and Riemann surfaces. The second part of the subject studies Riemann surfaces and natural objects on them such as holomorphic differentials and quadratic differentials.

Topics may also include: divisors, Riemann-Roch theorem; the moduli space of Riemann surfaces, Teichmueller space; integrable systems.