open problems

How to add shortcuts in the plane Imagine you have a network of roads. One way to judge the efficiency of the roads is in terms of how much longer it takes to travel between two points in your network than if you could just drive off-road in a straight line. This measure of efficiency is called quasiconvexity. Definition: A set $E$ is $C$-quasiconvex meaning that for every $x,y\in \tilde{\Gamma}$, there is a curve $\gamma\subseteq \mathbb{R}^{2}$ connecting $x$ and $y$ so that $\ell(\gamma)\leq C|x-y|$. Not every connected set of finite length is quasiconvex, and this note is about the problem of adding shortcuts to make it so.
7 min read
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.
6 min read
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.
11 min read