| Date | Topic | Reading |
|---|---|---|
| Week 1 Aug 23-27 |
Course outline Sobolev spaces |
W.P. Ziemer, Weakly Differentiable Functions, Chapter 2 |
| Week 2 Aug 30-Sept 3 |
Absolute continuity on lines Sobolev removability in dimension \(n=1\) |
J.J. Benedetto and W. Czaja, Integration and Modern Analysis, Section 4.6 |
| Sept 6 | Labor Day No classes in session |
|
| Week 3 Sept 8 |
Quasiconformal mappings The analytic definition |
General literature: O. Lehto and K.I. Virtanen, Quasiconformal Mappings in the Plane L.V. Ahlfors, Lectures on Quasiconformal Mappings J. Väisälä, Lectures on \(n\)-Dimensional Quasiconformal Mappings S. Hencl and P. Koskela, Lectures on Mappings of Finite Distortion |
| Week 4 Sept 13-17 |
Differentiability Change of variables |
O. Lehto and K.I. Virtanen, Quasiconformal Mappings in the Plane, Section III.3 |
| Week 5 Sept 20-24 |
Lusin \((N)\) property Modulus |
S. Hencl and P. Koskela, Lectures on Mappings of Finite Distortion, Section 1.1 |
| Week 6 Sept 27-Oct 1 |
Absolutlely continuous paths Geometric definition of quasiconformality |
J. Väisälä, Lectures on \(n\)-Dimensional Quasiconformal Mappings, Chapter 1 |
| Week 7 Oct 4-8 |
Quadrilaterals Equivalence of definitions of quasiconformality |
|
| Oct 11 | Fall break No classes in session |
|
| Week 8 Oct 13 |
Removability in dimension \(n\geq 2\) Removability for sets of \(\sigma\)-finite Hausdorf \(1\)-measure |
A. S. Besicovitch, On Sufficient Conditions for a Function to be Analytic, and on Behaviour of Analytic Functions in the Neighbourhood of Non-Isolated Singular Points [mathscinet] J. Väisälä, Lectures on \(n\)-Dimensional Quasiconformal Mappings, Chapter 4 |
| Week 9 Oct 18-22 |
Preliminaries on the quasihyperbolic distance | D. Ntalampekos, A removability theorem for Sobolev functions and detour sets [arXiv] D. Ntalampekos and M. Younsi, Rigidity theorems for circle domains [arXiv] |
| Week 10 Oct 25-29 |
Quasihyperbolic conditions for removability | P.W. Jones and S.K. Smirnov, Removability theorems for Sobolev functions and quasiconformal maps [link] |
| Week 11 Nov 1-5 |
John domains Non-removable sets |
W. Smith and D.A. Stegenga, Hölder domains and Poincaré domains [link] |
| Week 12 Nov 8-12 |
Non-removability of the gasket for Sobolev spaces | D. Ntalampekos, Non-removability of the Sierpiński gasket [arXiv] |
| Week 13 Nov 15-19 |
Non-removability of the gasket for quasiconformal maps | |
| Week 14 Nov 22 |
Uniformization of metric spaces | M. Bonk and B. Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres [mathscinet] K. Rajala, Uniformization of two-dimensional metric surfaces [mathscinet] D. Ntalampekos and M. Romney, Polyhedral approximation of metric surfaces and applications to uniformization [arXiv] |
| Nov 24 | Thanksgiving Break No classes in session |
|
| Nov 29 | Guest lecture: Chris Bishop Title: Conformal removability is hard (On zoom) |
Abstract: Suppose E is a compact set in the complex plane and U is its complement. The set E is called removable for a property P, if any holomorphic function on U with this property extends to be holomorphic on the whole plane. This is an important concept with applications in complex analysis, dynamics and probability. Tolsa famously characterized removable sets for bounded holomorphic functions, but such a characterization remains unknown for conformal maps on U that extend homeomorphically to the boundary. We offer an explanation for why the latter problem is actually harder: the collection of removable sets for bounded holomorphic maps is a G-delta set in the space of compact planar sets with the Hausdorff metric, but the collection of conformally removable sets is not even a Borel subset of this space. These results follow from known facts, but they suggest a number of new questions about fractals, removable curves and conformal welding. |
| Dec 1 | Guest lecture: Matthew Romney Title: The uniformization problem for surfaces (In person) |
Abstract: The uniformization problem concerns the relationship between the geometry of a metric space and the existence of geometrically well-behaved parametrizations, for example by quasiconformal maps. It is closely related to Plateau's problem, which asks for the existence and regularity of minimal surfaces spanning a given boundary curve. We will survey recent progress on the uniformization problem in the non-fractal setting and indicate directions for future research. |
| Week 16 Dec 6 |
Non-removability of carpets | D. Ntalampekos, Non-removability of Sierpiński carpets [arXiv] |
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