Kaisa_2012_3_photo by Veikko Somerpuro
The course is an introduction to the theory of planar quasiconformal mappings, their analytic and geometric properties and their interactions with PDE's.
Schedule: Lectures Tue 10-12 B322, Wed 10-12 C323, Exercise class: Mon 12-14 C322 (N.B. no exercise class on 3.9.2018, first lecture on 4.9.2018!)
Aikataulu
Materiaalit
Here you will find some relevant study material such as lecture notes and links to literature.
A comprehensive treatment of the subject is the monograph Elliptic partial differential equations and quasiconformal mappings in the plane by K. Astala, T. Iwaniec and G. Martin (link to e-book accessible from HY network). Parts of Chapters 1-5 are relevant for this course.
Luentomateriaalit
Tehtävät
Exercise set 1
This is the first exercise set. We will discuss the solutions on the September 17 Exercise class. Return your answers before the class.
Exercise set 2
The second exercise set is due by September 24.
Exercise set 3
Exercise set 3 is due by 1.10.2018
Exercise set 4
The Exercise set 4 is due by October 8.
Exercise set 5
The final exercise set is due by October 15.
Kurssin suorittaminen
The course can be completed by solving the exercises. The solutions should be returned to Istvan Prause (mailbox on the 3rd floor or by email istvan.prause@helsinki.fi) by the indicated deadline. You can also hand in the solutions at the beginning of the exercise class.
Kuvaus
| Real analysis |
| Complex analysis I, Functional analysis |
| Introduction to the theory of planar quasiconformal mappings, their analytic and geometric properties and their interactions with PDE's |
| 1. or 2. year |
| Distortion theorems for conformal maps. Quasisymmetry versus quasiconformality, geometric versus analytic properties. Basic properties of quasiconformal maps, Lusin's condition N, Hölder continuity. Beurling transform, Beltrami equation and the measurable Riemann mapping theorem. |
| Exam and exercises, other methods will be described later |
| Lecture notes |
| Lectures and exercise classes |
| Exam and excercises, Course will be graded with grades 1-5 |
| Exam and exercises, other methods will be described later |