In this post I describe an open problem concerning harmonic measure and its relationship with Hausdorff measure of non-integral dimension. For background on the definition, notation, and some related results, one can read through (or watch) the material for Week 10 of my geometric measure theory class.
We discuss an open problem about parametrising boundaries of semi-uniform domains, a class of domains introduced by Aikawa and Hirata. The question is essentially whether one can generalise the result that curves of bounded turning in the plane are quasi symmetric images of the real line, or they are bi-Lipschitz images if additionally they are Ahlfors regular. We first give a general background of the problem.
In Week 5 of my Geometric Measure Theory course, we’ll be studying Lipschitz functions and various useful techniques for working with them. In this note I’m going to discuss one of my favorite results from this week and show how I illustrated it using Numpy and Matplotlib. Neither the theorem nor the code are difficult, but I’ve always wanted to visualize this theorem, and coding it is a nice exercise.