Institute for Advanced Study | Higher Dimensional Fourier Quasicrystals from Lee-Yang Varieties - Pavel Kurasov @videosfromIAS | Uploaded 2 days ago | Updated 2 hours ago
Analysis and Mathematical Physics
Topic: Higher Dimensional Fourier Quasicrystals from Lee-Yang Varieties
Speaker: Pavel Kurasov
Affiliation: Stockholm University
Date: October 08, 2024
Fourier Quasicrystals (FQ) are defined as crystalline measures
μ=∑λ∈Λaλδλ,μ̂ =∑s∈Sbsδs, so that not only μ (and hence μ̂ ) are tempered distributions, but also |μ|:=∑λ∈Λ|aλ|δλand|μ̂ |:=∑s∈S|bs|δs,
are tempered.
One-dimensional FQs with positive integer weights (that is aλ∈ℕ) can be described using stable Lee-Yang polynomials, as was proven in a joint work with Peter Sarnak. Multidimensional Fourier quasicrystals are discussed in the current talk. It is shown that a rather general family of FQs in ℝd
with positive integer weights can be constructed using co-dimension d Lee-Yang varieties in ℂn,n greater than d.
These complex algebraic varieties are symmetric and avoid certain regions in ℂn, thus generalising zero sets of Lee-Yang polynomials.
It is shown that such FQs can be supported by Delaunay almost periodic sets and are genuinely multidimensional in the sense that their restriction to any one-dimensional subspace is not given by a one-dimensional FQ. Connections to alternative recent approaches by Yves Meyer, Lawton-Tsikh and de Courcy-Ireland-K. are clarified. Is it possible that our construction gives all multidimensional FQs with positive integer masses?
This is joint work with L. Alon, M. Kummer, and C. Vinzant (arXiv:2407.11184).
Analysis and Mathematical Physics
Topic: Higher Dimensional Fourier Quasicrystals from Lee-Yang Varieties
Speaker: Pavel Kurasov
Affiliation: Stockholm University
Date: October 08, 2024
Fourier Quasicrystals (FQ) are defined as crystalline measures
μ=∑λ∈Λaλδλ,μ̂ =∑s∈Sbsδs, so that not only μ (and hence μ̂ ) are tempered distributions, but also |μ|:=∑λ∈Λ|aλ|δλand|μ̂ |:=∑s∈S|bs|δs,
are tempered.
One-dimensional FQs with positive integer weights (that is aλ∈ℕ) can be described using stable Lee-Yang polynomials, as was proven in a joint work with Peter Sarnak. Multidimensional Fourier quasicrystals are discussed in the current talk. It is shown that a rather general family of FQs in ℝd
with positive integer weights can be constructed using co-dimension d Lee-Yang varieties in ℂn,n greater than d.
These complex algebraic varieties are symmetric and avoid certain regions in ℂn, thus generalising zero sets of Lee-Yang polynomials.
It is shown that such FQs can be supported by Delaunay almost periodic sets and are genuinely multidimensional in the sense that their restriction to any one-dimensional subspace is not given by a one-dimensional FQ. Connections to alternative recent approaches by Yves Meyer, Lawton-Tsikh and de Courcy-Ireland-K. are clarified. Is it possible that our construction gives all multidimensional FQs with positive integer masses?
This is joint work with L. Alon, M. Kummer, and C. Vinzant (arXiv:2407.11184).
