Lior Alon (MIT)

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 some growth conditions. For two decades, it was believed that non-periodic FQs must be a finite union of periodic sets, a belief supported by Lagarias’s conjecture that was proved by Lev and Olevskii in 2016. This changed with Kurasov and Sarnak (2020), who constructed the first non-periodic FQ which is not contained in a union of periodic sets. They provided a general construction of one-dimensional FQs as the zeros of real-rooted trigonometric functions constructed from Lee-Yang polynomials.

In this talk, I will show our result that all one-dimensional FQs arise from the Kurasov-Sarnak method and will present our recent result which generalizes the construction of FQs to any dimension using a novel class of high-codimension algebraic varieties, which we call 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.