PMATH950: Topics in Analysis: Riemann surfaces

Lectures: MW 2:30--3:50 (the lectures will be delivered live via zoom until at least 24 January 2022).
Virtual Office hours: Th 2:30--3:50, or by appointment.
Registration: Zoom link for lectures and office hours.
Course outline: Outline.

Overview: Riemann surfaces can be defined in several different, equivalent ways, for example as one-dimensional complex manifolds, or as oriented two-dimensional real manifolds. In addition, any compact Riemann surface can be embedded in projective space, thus giving it the structure of an algebraic curve. Riemann surfaces therefore appear in many areas of mathematics, from complex analysis, algebraic and differential geometry, to algebraic topology and number theory. This course will cover fundamentals of the theory of compact Riemann surfaces from an analytic and topological perspective.

Prerequisites: The course should be accessible to students who have taken PMATH 352 (Complex Analysis) or an equivalent course.

Topics will include: Riemann surfaces (definitions and examples, algebraic curves, quotients, modular curves); holomorphic maps; elliptic functions (Weierstrass and theta functions); sheaves and analytic continuation; maps between Riemann surfaces (basic properties, covering maps, monodromy and the Riemann Existence Theorem); holomorphic and meromorphic forms; de Rham and Dolbeault cohomology; harmonic forms and the Hodge decomposition; cohomology of sheaves; Riemann-Roch; Serre duality; maps to projective space; Riemann-Hurwitz formula; curves and their Jacobian; factors of automorphy and line bundles; the Uniformisation Theorem (time permitting).