## Geometric Function Theory, a Graduate Course

## Lecturers: Alan Sola (SU) and Fredrik Viklund (KTH)

Email: sola (no spam please) math (dot) su (dot) se, frejo (no spam please) kth (dot) se

**Description**: Geometric function theory is a branch of complex analysis
that seeks to relate analytic properties of conformal maps to geometric
properties of their images. The subject has deep connections with other
areas of mathematics such as potential theory, hyperbolic geometry, and
dynamical systems.
The course aims to introduce students to geometric function theory in a
broad sense, and to define concepts and present techniques required in
modern applications such as the theory of the Schramm-Loewner evolution.

**Content (tentative)**: Review of conformal mapping, Carathéodory's theorem,
Kellogg's theorem, basic potential theory, harmonic measure, extremal
length, Beurling's estimates, Plessner and McMillan's theorems,
Loewner's equation. Further applications may be treated depending on
the audience's interests.

**Textbook**: J.B. Garnett & D.E. Marshall, Harmonic Measure, New Mathematical Monographs, Cambridge University Press, 2005. (Reprint of the 2005 original) Errata

**Further reading**:
- A. Fletcher and V. Markovic, Quasiconformal maps and Teichmuller theory, Oxford University Press, 2007.
- Ch. Pommerenke, Boundary Behavior of Conformal Maps, Springer-Verlag, Berlin, 1992.
- L. Ahlfors, Conformal Invariants: Topics in geometric function theory, AMS Chelsea Publishing, Providence, RI 2010. (Reprint)

**Examination**: Homework problems and oral examination.

**Prerequisites**: Complex Analysis and Advanced Real Analysis I (required); Advanced Real Analysis II and Fourier Analysis (preferred).

**Important notice regarding Covid-19:**Because of the new guidance from the local authorities, we have canceled on-campus meetings for the time being.

**Tentative schedule**: A tentative outline of the course can be found here:
here.

**First meeting: Friday August 28th at 10am, room 34, bld 5, S.U**

**Problem sets**: To be posted here HW.

**Video recordings** can be accessed here and here.
**Lecture notes** Chapter IV here, on QC maps
here.