### MAT 626 - Topics in Complex Analysis: Quasiconformal Mapping, Sobolev Spaces, and Removability of Fractals - Fall 2021

Course Schedule (updated weekly)

The class meets every Monday and Wednesday at 9:45-11:05am in Math 4130. (Ignore the lecture time shown on Solar.)

### Office hours

Office hours (on zoom): | Monday 2:00pm-3:00pm |

Wednesday 2:00pm-3:00pm | |

Also by appointment | |

MLC hour: | Tuesday 3:00-4:00pm |

### Description

The problem of removability of a set, in general, asks whether one can glue functions of a given class along that set and obtain a function lying in the same class. Removability of sets for the class of (quasi)conformal mappings has applications in many problems of Analysis. We, therefore, seek geometric conditions on sets that guarantee their removability. In principle, sets with "simple" topology and "good" geometry are removable. However, much less is known for sets that have "bad" geometry or sets whose complement has infinitely many connected components.

In the first part of this course we will introduce the basic theory of Sobolev spaces and quasiconformal mappings, focusing on the properties that are relevant for the problem of removability. We will prove classical results such as the removability of rectifiable sets and of sets with "good" geometry. In the second part, we will discuss fractals with "complicated" topology, such as the Sierpiński carpet and the Sierpiński gasket. As time permits, we will study connections to the problem of rigidity of circle domains and applications in Complex Dynamics.

No textbook is required. Suggested literature will be posted.

Grading is based on attendance.

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