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Quasiconformal methods in calculus of variations

Analytically changing the parameters of von Koch snowflake gives an example of a holomorphic motion. Holomorphic motions, in general, provide a surprising new point of view to old question in the vectorial calculus variations.

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Non-linear Beltrami equations and conformally invariant random structures

A simulation of random domino tiling, with a fixed boundary polygon.
Quasiconformal mappings and non-linear Beltrami equations give strong tools to understand the geometry scaling limits of random domino tilings or, more generally, all dimer models.

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Convex integration in fluid mechanics

In the project we will describe the long expected turbulent solutions to the equations of magnetohydrodynamics which dissipate energy but preserve magnetic helicity. They govern the physics of solar plasma.

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Non-linear Fourier transform, inverse problems and cloaking

In inverse scattering one searches for information of quantum potentials from scattering data. In the picture, we show how the artifacts are better recovered by averaging the measurements. Proofs are based on delicated interplay between complex analysis and non-linear Fourier transforms.