- Lecturer: Qian Wang

General Prerequisites:

The only necessary prerequisite is a basic complex analysis course: analytic functions, Taylor series, contour integration, Cauchy theorems, residues, maximum modulus, Liouville's theorem. Some basic results that might not be covered by a basic course (such as argument principle and the Rouche theorem) will be given in the introductory chapter of the lecture notes.

Course Term: Michaelmas

Course Lecture Information: 16 lectures

Course Weight: 1

Course Level: M

Assessment Type: Written Examination

Course Overview:

The theory of conformal maps has a very long and rich history. Its foundations were laid out by Riemann in mid XIX century and it was one of the central areas of the Complex Analysis since then.

The aim of the course is to teach the principal techniques and methods of analytic and geometric function theory. We start by proving the Riemann mapping theorem which claims that all non-trivial simply connected domains are conformal images of the unit disc. This provides a connection between planar domains and 1-to-1, or univalent, analytic functions in the unit disc.

In this course we will study the Riemann mapping theorem and general theory of univalent functions, study how geometric properties of domains are related to analytic properties of functions. We will also study conformal invariants: quantities that do not change under conformal transformations. We will see that many of these invariants have a simple geometric interpretation and could be used to obtain numerous beautiful results.

The aim of the course is to teach the principal techniques and methods of analytic and geometric function theory. We start by proving the Riemann mapping theorem which claims that all non-trivial simply connected domains are conformal images of the unit disc. This provides a connection between planar domains and 1-to-1, or univalent, analytic functions in the unit disc.

In this course we will study the Riemann mapping theorem and general theory of univalent functions, study how geometric properties of domains are related to analytic properties of functions. We will also study conformal invariants: quantities that do not change under conformal transformations. We will see that many of these invariants have a simple geometric interpretation and could be used to obtain numerous beautiful results.

Learning Outcomes:

Students will have been introduced to ideas and techniques of geometric function theory that play important role and have a lot of applications in other areas of analysis. In particular, they will learn the proof of the Riemann mapping theorem and the concept of conformal invariants.

Course Synopsis:

We will cover the following topics:

1. The Riemann mapping theorem. The main goal will be to prove Riemann's theorem which tells us that any non-trivial simply-connected domain can be conformally mapped onto the unit disc. This will be the key result for the entire course since it will allow us to connect the geometry of a domain with the analytical properties of a conformal map which sends this domain to the unit disc. Within this section we will discuss

MÃ¶bius transformations and the Schwarz lemma

Normal families, the Montel and Hurwitz theorems

Proof of Riemann mapping theorem

Constructive uniformization: Christoffel-Schwarz mappings and iterative methods

Boundary correspondence, accessible points and prime ends

Uniformization of multiply-connected domains (without proof)

Applications: the Dirichlet boundary problem

2. Basic theory of univalent functions. In this part we will study the universal estimates that are valid for all conformal maps on the unit disc.

Area theorem and coefficient estimates

The Koebe \(1/4\) theorem, distortion theorems

Conformal invariants: extremal length and its applications

Conformal invariance of Green's function and harmonic measure

Extremal length, its conformal invariance and applications

1. The Riemann mapping theorem. The main goal will be to prove Riemann's theorem which tells us that any non-trivial simply-connected domain can be conformally mapped onto the unit disc. This will be the key result for the entire course since it will allow us to connect the geometry of a domain with the analytical properties of a conformal map which sends this domain to the unit disc. Within this section we will discuss

MÃ¶bius transformations and the Schwarz lemma

Normal families, the Montel and Hurwitz theorems

Proof of Riemann mapping theorem

Constructive uniformization: Christoffel-Schwarz mappings and iterative methods

Boundary correspondence, accessible points and prime ends

Uniformization of multiply-connected domains (without proof)

Applications: the Dirichlet boundary problem

2. Basic theory of univalent functions. In this part we will study the universal estimates that are valid for all conformal maps on the unit disc.

Area theorem and coefficient estimates

The Koebe \(1/4\) theorem, distortion theorems

Conformal invariants: extremal length and its applications

Conformal invariance of Green's function and harmonic measure

Extremal length, its conformal invariance and applications