## 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).