Complex Analysis: conformal maps and geometry

Practicalities

Lectures will be on Tuesdays from 11am till 1pm.
First lecture on 28th April, last one on 23d June.
There will be no lectures on 16th June

Lecture notes

Here is the first part of the lecture notes: PDF (last updated 8 June 2015).

This is a draft of the lecture notes. It is pretty far from the final version. All comments, suggestions, reports of typos and errors are very welcome. I will update this file as the course goes, so check for updates.

Synopsis

The aim of the course is to teach the principal techniques and methods of analytic and geometric function theory. This is a beautiful subject on its own right but it also have many applications in other areas of mathematics: potential theory, analytic number theory, probability. In the recent years the theory of Loewner equation became a crucial tool in the study of statistical physics lattice models.

This course is a continuation of the basic undergraduate complex analysis course but has much more geometric emphasis. Our main subject will be the theory of conformal maps, their analytical and geometrical properties. We will cover the following topics:

• Riemann uniformization 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 as to connect the geometry of the domain with the analytical properties of the map which sends this domain to the unit disc. Within this section we will discuss
  • Maximum principle and Schwarz lemma, and Möbius transformations
  • Normal families, Hurwitz theorem
  • Proof of Riemann uniformization theorem
  • Constructive uniformization: Christoffel-Schwarz mappings and zipper algorithm
  • Uniformization for multiply-connected domains
  • Applications: Dirichlet problem
• Theory of univalent functions. Univalent function is another term for one-to-one analytical map. We will be mostly interested in their boundary behaviour and how it is related to the geometry of the boundary. This section will cover
• Area theorem and coefficient estimates
• Koebe 1/4 theorem, distortion theorems
• Conformal invariants: extremal length and its applications
• If time will permit we will discuss the Loewner evolution which allows to describe planar curves in a "conformal" way. This technique was originally introduced by Loewner in 1923 in an attempt to prove Bieberbach conjecture about the coefficients of univalent functions. Loewner obtained only partial result, and the conjecture was open until 1985 when de Branges used Loewner evolution and some new ideas to prove the conjecture. In the last decade it was intensively used in the form of Schramm-Loewner evolution to solve numerous problems in statistical physics.

Prerequisites

I will assume that you are familiar with basic complex analysis: analytic functions, Taylor series, contour integration, Cauchy theorem, and residues.

Assessment

The assessment will be in the form of mini-projects. At the end of the course you will be given several topics for a project, most of them will be continuations of some ideas presented in the course, but they will go beyond the course material and will require additional background reading.

Please let me know well in advance if you will need formal assessment.

Bibliography

There are many excellent books on the subject. I would suggest

• W. Rudin, Real and Complex Analysis. A classical book covering all necessary background and some parts of course.
• L. Ahlfors, Complex analysis. This is a very good advanced textbook on Complex analysis. If you are a bit rusty on the basic complex analysis, then you might find everything you need (and a bit more) in Chapters 1—4. We will cover some of the material from chapters 5—6.
• L. Ahlfors, Conformal Invariants. We will cover some topics from Chapters 1—6.
• Ch. Pommerenke, Univalent functions. This book is mostly for further reading. We will discuss some of the results that are covered in Chapters 1,5,6, and 10.
• Ch. Pommerenke, Boundary behaviour of conformal maps. This is an updated version of the previous book. We will be interested in Chapters 1,4, and 8.
• P. Duren, Univalent functions. This is an excellent book about general theory of the univalent functions. We are mostly interested in the first three chapters.
• G. Goluzin, Geometric Theory of Functions of a Complex Variable. This book contains vast amount of information about the geometric function theory. We will cover some of the results from the first four chapters.