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http://math.colorado.edu/seminars/ics/frag.ics
as the source.
Thunderbird
The Algebraic Geometry Seminar, AKA FRAGMENT, meets at Boulder on Thursdays 11:15 AM -- 12:15 PM in MATH 350 and at CSU on Thursdays 3 -- 5 PM in Weber 201. For more information, or to provide a speaker, please contact Renzo Cavalieri, Jonathan Wise, or Sebastian (Yano) Casalaina.
Thu, Feb. 26 11:15am (MATH …
Lior Alon (MIT)
X
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