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Quasiconformal mappings in the plane, spring 2016

 

Teacher: Istvan Prause 

Scope: 10 cr (5+5)

Type: Advanced studies

Teaching: Weeks 3-9: Monday 12-14 and Wednesday 14-16 in room B322.

Weeks 11-18: Monday 12-14 in room C122 (changed room!)

Topics: The course will consist of two parts. Period III will be an introduction to the theory of planar  quasiconfomal mappings, their geometric properties and their interactions with PDE's.

Period IV will be based on project work / student presentations. This will include covering some omitted topics from the lectures and working through some recent papers, especially on holomorphic motions and holomorphic interpolation.

Prerequisites: Real analysis, basic complex analysis, Sobolev spaces (useful but not required)

News

  •  The first lecture is on January 18, first Exercise class on January 28 (changed!).

Teaching schedule

Weeks 3-9 on Monday 12-14 and Wednesday 14-16 in room B322.

Weeks 11-18 on Monday 12-14 in room C122 (changed room!).

Easter holiday 24.-30.3. 

Exams

The credit can be obtained by doing the exercises (at least 75%) for period III and by giving presentations in period IV.

Course material

The lectures in period III will be based on Kari Astala's course (Spring, 2013). The study folder is kept up to date on the 3rd floor (pdf, final version - updated on March 3).

For a comprehensive treatment, see the book (available as e-book from HY university network)

Astala-Iwaniec-Martin: Elliptic PDE's and Quasiconformal mappings in the plane. Princeton University, 2009.  

Topics to be covered (period III): distortion theorems for conformal maps, quasisymmetry vs quasiconformality, basic properties of quasiconformal maps and the measurable Riemann mapping theorem.

Topics for period IV (student presentations):

  • hyperbolic metric and Schwarz lemma
  • holomorphic motions and the lambda-lemma
  • Examples of holomorphic motions, change of Hausdorff dimension
  • real and complex interpolation of L^p-spaces
  • Interpolation lemma and higher integrability for quasiconformal maps
  • Distortion of area and Hausdorff dimension under quasiconformal maps
  • Dimension of quasicircles

Registration

No need to register in advance, just come to the first class.
Did you forget to register? What to do?

Exercises

Assignments

The solutions are to be submitted to Oleg Ivrii (mailbox on the 3rd floor or oleg.ivrii@helsinki.fi) before each Exercise class.

Exercise classes (during period III)

GroupDayTimeRoomInstructor
1.to14-16B321Oleg Ivrii

Course feedback

Course feedback can be given at any point during the course. Click here.

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