C4.8 Complex Analysis: Conformal Maps and Geometry (2017-18)

General Prerequisites:

The only necessary prerequisite is the basic complex analysis covered in Part A: analytic functions, Taylor series, contour integration, Cauchy theorem, and residues. Integration option is recommended but not necessary.
Course Term: Michaelmas
Course Lecture Information: 16 lectures
Course Weight: 1
Course Level: M
Assessment Type: Written Examination
Course Overview:

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.

Lecturer(s):

Prof. Dmitry Belyaev
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:
  • 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 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

  1. Maximum principle and Schwarz lemma, hyperbolic metric and Möbius transformations
  2. Normal families, Hurwitz theorem
  3. Proof of Riemann uniformization theorem
  4. Constructive uniformization: Christoffel-Schwarz mappings and zipper algorithm (no proofs)
  5. Uniformization for multiply-connected domains (sketch of the proof)
  6. 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
  1. Area theorem and coefficient estimates
  2. Koebe \(1/4\) theorem, distortion theorems
  3. Conformal invariants: extremal length and its applications