"There are many intersections between complex analysis, geometric measure theory, and harmonic analysis; the interactions between these fields yield many important results and applications. In this work, we focus on two aspects of these connections: the regularity theory of quasiconformal maps and the quantitative study of rectifiable sets. Quasiconformal maps are orientation-preserving homeomorphisms that satisfy certain distortion inequalities; infinitesimally, they map circles to ellipses of uniformly bounded eccentricity. Such maps have many useful geometric distortion properties, and yield a flexible and powerful generalization of conformal mappings. These maps arise naturally in the study of elasticity, in complex dynamics, and in the analysis of partial differential equations. We study the singularities of these maps; in particular, we consider the size and structure of the sets where a quasiconformal map can exhibit given stretching and rotation behavior. We improve the previously known results to give examples of stretching and rotation sets with non-sigma-finite measure at the critical Hausdorff dimension. We further improve this to give examples with positive Riesz capacity at the critical homogeneity, as well as positivity of measure for a broad class of gauged Hausdorff measures at the critical dimension. The local distortion properties of quasiconformal maps also give rise to a certain degree of global regularity and Hölder continuity. We give new lower bounds for the Hölder continuity of these maps, relating both the structure of the underlying partial differential equation for the maps and the geometric distortion they can exhibit; the analysis is based on combining the isoperimetric inequality with a study of the length of quasicircles. Furthermore, the extremizers for Hölder continuity are characterized, and we give a natural application to solutions of elliptic partial differential equations. Finally, given a set in the plane, the average length of its projections in all directions is called the Favard length of a set; it is closely related to the Buffon needle probability of the set. This quantity measures the size and structure of a set, and is closely related to metric and geometric properties of the set such as rectifiability, Hausdorff dimension, and analytic capacity. We develop new geometrically motivated techniques for estimating Favard length. We will give a new proof relating Hausdorff dimension to the decay rate of the Favard length of neighborhoods of a set. We will also show that, for a large class of self-similar one-dimensional sets, the sequence of Favard lengths of the generations of the set is convex; this leads directly to lower bounds on Favard length for various fractal sets."--Pages ii-iii.
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