From quasicrystals to Kepler's packing problem
A talk in the Mathematisches Kolloquium & Mathematisches Kolloquium (SFB 1283) series by
Yves F. Meyer from ENS Paris-Saclay
| Abstract: | An atomic measure $\mu$ on $R^n$ is a crystalline measure if (a) $\mu$ is supported by a
locally finite set and if (b) its distributional Fourier transform $\widehat{\mu}$ is also an atomic
measure supported by a locally finite set. Crystalline measures were introduced
in 1959 by Andrew Guinand, Jean-Pierre Kahane, and Szolem Mandelbrojt. To
any crystalline measure $\mu$ a generalized zeta function $\zeta_\mu(s)$ can be associated as
it is proved by Guinand et al. If $\mu$ is a Dirac comb $\zeta_\mu(s)$ is the Riemann zeta
function. The discovery of quasicrystals by D. Shechtman et al. renewed the in-
terest in Guinand’s work. This led to the following time-frequency interpolation
problem: We are given two locally finite sets $E\subset R^n$ and $F\subset R^n$. We say that
$E$ and $F$ complement each other if a Schwartz function $f$ is uniquely defined by
its restriction to $E$ together with the restriction of its Fourier transform to $F$.
A spectacular example is given by Maryna Viazovska. Finally it applies to the
solution by Viazovska of the packing Kepler problem in dimensions 8 and 24.
Yves F. Meyer. $\textit{Measures with locally finite support and spectrum}$, PNAS (2016) 113 3152-3158. D. Radchenko and M. Viazovska, $\textit{Interpolation on the real line}$, Publ. Math. IHES 129, (2019) 5181. $\href{https://geotop-bielefeld.de/file/486/download?token=6Dsy8XgK}{\textbf{Präsentation (Datei zum Herunterladen)}}$ |