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Quasiconformal mappings (MATS225)

Lectures: Tuesdays 12.15–14.00 (MaD355) and Thursdays 14.15–16.00 (MaD380)
Start: Tuesday, March 17, 2026
Exercises: Tuesdays, 14.15–16.00 (MaD380)
Credit points: 5
Lecturer: Katrin Fässler
Course assistant: Abhishek Pandey
See also SISU page.
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​What is... a quasiconformal mapping? (text by Juha Heinonen in AMS Notices)
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​Study materials
[K] P. Koskela: Lectures on quasiconformal and quasisymmetric mappings (link)
 
Program

  1. Conformal mappings [K, 11.1, Lem. 8.4, 11.2], [GMP, Thm 3.1.1], [M, 1]                (Lecture 1) (Lecture 2) 

  2. Metric definition of quasiconformality [K, 1], [GMP, 6.2.1, Thm 6.6.29/31], [V, 16.1, 16.2]
    (Lecture 1) (Lecture 2) -     updated 26.3.2026

  3. From metrically quasiconformal to locally quasisymmetric [K, 2.1, 2.2]                    (Lecture 1) (Lecture 2)

  4. Tools from real and harmonic analysis [K, 2.3, 2.4, 2.5, 2.6]                                   (Lecture 1) (Lecture 2)

  5. Basic properties of quasisymmetric mappings [K, 3, 10.4-10.9], [H, 10-12]              (Lecture 1) (Lecture 2)

  6. Gehring's lemma [K, 4.1, 4.2, 4.3]                                                                                (Lecture 1) (Lecture 2)

  7. The analytic definition of quasiconformality [K, 5]                                                      (Lecture 1) (Lecture 2)

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Exercises

  1. Sheet 1 (Solutions by Abhishek)

  2. Sheet 2

  3. Sheet 3

  4. Sheet 4

  5. Sheet 5

  6. Sheet 6

  7. Sheet 7

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Evaluation
There is no written course exam. The grade will be based on a final project and the exercises
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Topics for final presentations
See here for more information.

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  • Sobolev spaces and convergence of quasiconformal mappings (Erte)

  • Quasisymmetries and conformal dimension (Yibo)

  • Quasiconformal mappings and Cantor sets (Mikko)

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Further literature

  • [GeMaPa] F. Gehring, G. J. Martin, B. P. Palka: An Introduction to the Theory of Higher-Dimensional Quasiconformal Mappings; American Mathematical Soc., 2017

  • [H] J. Heinonen: Lectures on Analysis on Metric Spaces; Springer, 2012

  • [M] G. Martin: The Theory of Quasiconformal Mappings in Higher Dimensions, I

  • [V] J. Väisälä: Lectures on n-Dimensional Quasiconformal Mappings; Springer, 2006

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