In this talk, we will go over the work of Lev and Matolcsi where they proved the Fuglede conjecture for convex domains is true in all dimensions. The Fuglede conjecture stated that a set in Rd is spectral (i.e. its associated L2 space admits an orthogonal basis of exponential functions) if and only if it tiles Rd by translations.
Although the conjecture was proven false in general, partial results were known for convex domains. Whereas tiling convex domains are spectral, the converse was known only for convex polytopes in R2 or R3. We review Lev and Matolcsi's argument proving that spectral convex domains must be polytopes and tile the space in all dimensions.
https://univienna.zoom.us/j/66031419470?pwd=bXd3V0xEMWM0MTQwS09nWStEV0NnUT09