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 the 1994 paper for which Astala got the Salem prize, he proved a distortion theorem under quasiconformal maps which is sharp at the level of (Hausdorff) dimension. He further conjectured a finer distortion estimate at the level of (Hausdorff) measure.
UT showed in 2008 with a highly non-self similar Cantor set that this finer distortion estimate is sharp. In 2010, Lacey, Sawyer, and UT, jointly proved completely Astala's conjecture in all dimensions. The proof uses Astala's 1994 approach, geometric measure theory, potential theory, and new weighted norm inequalities for Calder\'{o}n-Zygmund singular integral operators which cannot be deduced from the classical Muckenhoupt $A_p$ theory from harmonic analysis.
I will mention related sharp removability results for bounded $K$-quasiregular maps (i.e. the quasiconformal analogue of the classical Painleve problem) obtained jointly by Tolsa and UT.
In various subsequent papers, Lacey, Sawyer, Shen, and UT, furthered the understanding of weighted norm inequalities for Calder\'{o}n-Zygmund singular integral operators, to get the proof of the Nazarov-Treil-Volberg's 2003 conjecture on the T1 characterization of the two-weight norm inequality for the Hilbert transform in a two-part paper by Lacey-Sawyer-Shen-UT and Lacey (Hyt\"{o}nen later removed a technical hypothesis).
This line of work has been further expanded by Sawyer-Shen-UT among others, aiming at analogous T1 type characterizations of the two-weight norm inequality for other Calder\'{o}n-Zygmund singular integral operators. So far only partial results have been obtained.
The talk will be self-contained.