Fourier Quasicrystals and Lee-Yang varieties

The Poisson summation formula states that the Fourier transform of a lattice’s counting measure is a discrete measure supported on the dual lattice. This fact characterizes periodicity via the Fourier transform and lies at the heart of number theory, harmonic analysis, and mathematical physics. Extending this idea, Fourier Quasicrystals (FQs) are almost periodic sets whose Fourier transform is supported on a discrete set, subject to growth conditions. For two decades, FQs were thought to be periodic or finite unions of periodic sets, a belief supported by Lagarias’s conjecture, proved by Lev and Olevskii in 2016. This changed with Kurasov and Sarnak (2020), who constructed the first genuinely non-periodic FQ. They provided a general construction of one-dimensional FQs by intersecting irrational lines in the torus with zero sets of Lee-Yang polynomials.

In this talk, I will show that all one-dimensional FQs can be constructed via the Kurasov-Sarnak method and present a new framework to generalize it to higher dimensions using a novel class of high-codimension algebraic varieties, termed Lee-Yang varieties.

Based on joint work with Alex Cohen, Cynthia Vinzant, Mario Kummer, and Pavel Kurasov, this presentation explores the interplay between Fourier analysis, algebraic geometry, and is designed for a broad audience without requiring prior expertise.

zoom link:  https://biu-ac-il.zoom.us/j/83293533633?pwd=QuYptGtg3dhheD4DWERu8EaOjzS…;