Invariant Sets of the Double Hilbert Transform

We study planar sets whose indicator functions are invariant under the double Hilbert transform. After showing that bounded sets can be approximately, though never exactly, invariant, we describe several families of exactly invariant sets, from bent strips to sets of positive density built from three families of parallel lines, and discuss what is known about their classification. In the talk we will also mention connections with Sylvester-Gallai configurations, tomography, and superpositions of traveling waves. This is joint work with E. Abakumov, K. Domelevo and S. Petermichl.