Quasiconformal maps in the plane, 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: Quasiconformal maps were initially considered in the early to mid 20th century in connection with planar mapping problems as generalizations of conformal maps. Since then, a rich theory of qc maps has been developed, in both the planar and higher-dimensional setting, and the qc viewpoint has stimulated research in both analysis and geometry. Quasiconformal theory is now an active subfield in its own right, and many advances have led to important applications in areas of mathematics such as dynamical systems and Teichmüller theory.
The goal of this course is to provide a comprehensive introduction to the theory of quasiconformal maps in the plane, as well as overviews of selected applications.
Content (tentative):
- Definitions of quasiconformality: analytic aspects.
- Definitions of quasiconformality: geometric aspects.
- Quasisymmetric maps.
- The Beltrami equation, the measurable Riemann mapping theorem.
- Applications of QC maps in complex dynamics.
- Introduction to Teichmüller theory.
Textbooks:
- A. Fletcher and V. Markovic, Quasiconformal maps and Teichmuller theory. Oxford University Press, 2007. (Primary text.)
- L. Ahlfors, Lectures on quasiconformal maps, Second edition. American Mathematical Society, Providence, RI 2006. (Reprint, primary text.)
- K. Astala, T. Iwaniec, and G. Martin, Elliptic PDE and quasiconformal mappings in the plane. Princeton University Press, Princeton NJ, 2009. (Secondary source.)
- L. Carleson and T. Gamelin, Complex dynamics, Universitext, Springer-Verlag, 1993. (For dynamical systems background.)
Examination: Homework problems and oral examination.
Prerequisites: Complex Analysis and Advanced Real Analysis I (required); Advanced Real Analysis II and Fourier Analysis (preferred).
Tentative schedule: A tentative outline of the course can be found here:
here.
First meeting: Wednesday 13 September 9am-11am, Cramér Room, S.U.
Problem sets: HW1, HW2.