Convex polytopes that tile space with translations: Voronoi domains and spectral sets

Abstract: In this talk I am going to discuss convex d-dimensional polytopes that tile R^d with translations and their properties related to two conjectures. The first conjecture, the Fuglede conjecture, claims that every spectral set in R^d tiles the space with translations; this conjecture was recently settled for convex domains by Lev and Matolcsi. The second conjecture, the Voronoi conjecture, claims that every convex polytope that tiles R^d with translations is the Voronoi domain for some d-dimensional lattice. The conjecture originates from the Voronoi�s geometric theory of positive definite quadratic forms and is related to many questions in mathematical crystallography including Hilbert�s 18th problem. I mostly plan to discuss recent progress in the Voronoi conjecture and the proof of the conjecture for five-dimensional parallelohedra; in the general setting the Voronoi conjecture is still open. The talk is based on a joint work with Alexander Magazinov (Skoltech).

analysis of PDEsmetric geometryprobability

Audience: researchers in the topic

Online asymptotic geometric analysis seminar

Series comments: The link: technion.zoom.us/j/99202255210

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