(Fuglede's tiling-spectrality conjecture. (ERC project
We know by classical Fourier analysis that the unit cube in R^d has an orthogonal basis consisting of exponential functions. Which other domains admit such a basis? Fuglede conjectured (1974) that these so-called "spectral domains" could be characterized geometrically by their possibility to tile the space by translations. I will survey the subject and then discuss some recent results, joint with Rachel Greenfeld, where we focus on the conjecture for convex polytopes.