Quasiconformal Methods in Analysis and Applications

ended

The use of delicate quasiconformal methods, in conjunction with convex integration and/or nonlinear Fourier analysis, will be the common theme of the proposal. A number of important outstanding problems are susceptible to attack via these methods. First and foremost, Morrey's fundamental question in two dimensional vectorial calculus of variations will be considered as well as the related conjecture of Iwaniec regarding the sharp $L^p$ bounds for the Beurling transform. Understanding the geometry of conformally invariant random structures will be one of the central goals of the proposal. Uhlmann's conjecture regarding the optimal regularity for uniqueness in Calder\'on's inverse conductivity problem will also be considered, as well as the applications to imaging. Further goals are to be found in fluid mechanics and scattering, as well as the fundamental properties of quasiconformal mappings, interesting in their own right, such as the outstanding deformation problem for chord-arc curves.