Holomorphic motions were introduced by Mane, Sad and Sullivan in the 1980's motivated by applications in holomorphic dynamics. They have since then been applied in various other areas such as Kleinian groups, Teichmüller theory and most notably quasiconformal mappings. In this talk, I will discuss recent results on the variation of several quantities under holomorphic motions, such as logarithmic capacity, Minkowski/Packing/Hausdorff dimension, and area measure. Our method provides a new, unified approach to various celebrated theorems about quasiconformal mappings, including the work of Astala on the distortion of area and dimension under quasiconformal mappings and the work of Smirnov on the dimension of quasicircles.
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