Quasiconformal maps are a certain generalization of analytic maps that have nice distortion properties. They appear in elasticity, inverse problems, geometry (e.g. Mostow's rigidity theorem)... among other places.
In his celebrated paper on area distortion under planar quasiconformal mappings (Acta 1994), Astala proved that if $E$ is a compact set of Hausdorff dimension $d$ and $f$ is $K$-quasiconformal, then $fE$ has Hausdorff dimension at most $d' = \frac{2Kd}{2+(K-1)d}$, and that this result is sharp. He conjectured (Question 4.4) that if the Hausdorff measure $\mathcal{H}^d (E)=0$, then $\mathcal{H}^{d'} (fE)=0$.
UT showed that Astala's conjecture is sharp in the class of all Hausdorff gauge functions. Lacey, Sawyer and UT jointly proved completely Astala's conjecture in all dimensions. The proof uses Astala's 1994 approach, geometric measure theory, and new weighted norm inequalities for Calderon-Zygmund singular integral operators which cannot be deduced from the classical Muckenhoupt $A_p$ theory.
These results are related to removability problems for various classes of quasiregular maps. I will mention sharp removability results for bounded $K$-quasiregular maps (i.e. the quasiconformal analogue of the classical Painleve problem) obtained jointly by Tolsa and UT.
I will further mention recent results related to another conjecture of Astala on Hausdorff dimension of quasicircles obtained jointly by Prause, Tolsa and UT. The talk will be self-contained.