Abstract: The Poisson summation formula tells us that a lattice and its dual lattice are related through Fourier transform: the Fourier transform of the counting measure of one is the counting measure of the other. The search for non-periodic analogues of Poisson summation—discrete sets whose Fourier transform is again a discrete measure—has appeared independently across number theory, harmonic analysis, and crystallography. In the physical setting, the Fourier transform of the counting measure of an atomic structure is exactly the diffraction pattern observed experimentally. The term quasicrystal refers to a non-periodic set whose diffraction is a pure point measure, and the discovery of such materials by Dan Shechtman in 1984 profoundly altered crystallography and led to a Nobel Prize.
A major mystery that remained unresolved until very recently was whether one could find a quasicrystal whose Fourier transform is supported on a discrete (rather than merely countable) set. These objects are now called Fourier quasicrystals (FQs), and until 2020 only the trivial periodic examples were known. Kurasov and Sarnak constructed the first genuinely non-periodic one-dimensional FQ using Lee–Yang polynomials, multivariate polynomials whose zeroes avoid prescribed regions of the complex plane.
In our recent work, we show that every one-dimensional Fourier quasicrystal arises from the Kurasov–Sarnak construction via Lee–Yang polynomials, yielding a complete classification in dimension one. We then extend this framework to all dimensions by introducing Lee–Yang varieties, high-codimension algebraic varieties whose geometric and analytic properties govern the Fourier structure of the associated quasicrystals.
The talk will introduce Fourier quasicrystals, Lee–Yang polynomials, and Lee–Yang varieties, and explain the geometric mechanism of generating summation formulas from algebraic varieties and how it is related to real-rootedness of trigonometric equations. No prior background will be assumed.
Joint work with Alex Cohen, Cynthia Vinzant, Mario Kummer, and Pavel Kurasov.
Fourier Quasicrystals and Lee–Yang Varieties
Fri, Feb. 27 2:30pm (Math 3…
Probability
Lior Alon (MIT)
X
Motivated by Berry’s random wave model for chaotic domains, high-energy eigenfunctions are expected, at the scale of the wavelength, to behave like random waves. Accordingly, the size of nodal sets is determined to first order by the Weyl law, while its second-order fluctuations are expected to be universal. In analogy with spectral statistics, we refer to these fluctuations of the nodal set size as nodal statistics.
Beginning with work of Berry, and later of Blum–Gnutzmann–Smilansky and Bogomolny–Schmit, it was conjectured that in chaotic systems nodal statistics are asymptotically Gaussian. While striking mathematical results support this picture on the sphere (Nazarov–Sodin), arithmetic symmetries on the torus prevent chaoticity and lead to a breakdown of universality, as shown by Wigman and collaborators and by Marinucci, Rossi, and Peccati.
In this talk I will follow Smilansky’s insight and focus on nodal statistics in graph-based models. For quantum graphs and discrete operators on graphs, Berkolaiko’s nodal magnetic theorem relates nodal statistics to the stability of eigenvalues under magnetic perturbations. This connection yields Gaussian limiting statistics for quantum graphs with disjoint cycles (Alon–Band–Berkolaiko), and, combined with Morse inequalities, extends to random discrete operators on finite graphs with disjoint cycles and to complete graphs with strong on-site disorder (Alon–Goresky).
I will conclude with recent joint work on nodal statistics for random matrices (Alon–Mikulincer–Urschel), showing that nodal statistics for GOE matrices obey a semicircle law rather than the conjectured Gaussian behaviour. If time permits, I will briefly discuss ongoing work on the Rosenzweig–Porter model, where adding a random on-site potential to a GOE matrix leads to nodal statistics interpolating from semicircle behaviour at low disorder to Gaussian behaviour at strong disorder, suggesting a phase transition between universality classes and a connection to localization–delocalization phenomena.
Nodal statistics from quantum graphs to random matrices: universality and phase transitions
Fri, Feb. 27 11:10pm (MATH …
Math Phys
Lior Alon (MIT)
X
The Poisson summation formula tells us that a lattice and its dual lattice are related through Fourier transform: the Fourier transform of the counting measure of one is the counting measure of the other. The search for non-periodic analogues of Poisson summation — discrete sets whose Fourier transform is again a discrete measure — has appeared independently across number theory, harmonic analysis, and crystallography. In the physical setting, the Fourier transform of the counting measure of an atomic structure is exactly the diffraction pattern observed experimentally. The term quasicrystal refers to a non-periodic set whose diffraction is a pure point measure, and the discovery of such materials by Dan Shechtman in 1984 profoundly altered crystallography and led to a Nobel Prize. A major mystery that remained unresolved until very recently was whether one could find a quasicrystal whose Fourier transform is supported on a discrete (rather than merely countable) set. These objects are now called Fourier quasicrystals (FQs), and until 2020 only the trivial periodic examples were known. Kurasov and Sarnak constructed the first genuinely non-periodic one-dimensional FQ using Lee–Yang polynomials, multivariate polynomials whose zeroes avoid prescribed regions of the complex plane.
In our recent work, we show that every one-dimensional Fourier quasicrystal arises from the Kurasov–Sarnak construction via Lee–Yang polynomials, yielding a complete classification in dimension one. We then extend this framework to all dimensions by introducing Lee–Yang varieties, high-codimension algebraic varieties whose geometric and analytic properties govern the Fourier structure of the associated quasicrystals.
The talk will introduce Fourier quasicrystals, Lee–Yang polynomials, and Lee–Yang varieties, and explain the geometric mechanism of generating summation formulas from algebraic varieties and how it is related to real-rootedness of trigonometric equations. No prior background will be assumed.
Joint work with Alex Cohen, Cynthia Vinzant, Mario Kummer, and Pavel Kurasov.
Fourier Quasicrystals and Lee–Yang Varieties