14 problems

Let an inflation tiling be one-dimensional, with inflation factor a Pisot–Vijayaraghavan number and inflation matrix $M$ having an irreducible characteristic polynomial. Here, a Pi…

Let a dynamical system satisfy reasonable assumptions on its type, and let its factors be dynamical systems of a similar kind or simpler. The dynamical spectrum is the spectrum ari…

Absolutely continuous diffraction conjecture. The absolutely continuous component of the diffraction measure is nonzero:

Let $\mathcal{E}(x,t)$ solve the diffraction-amplification equation on a $d$-dimensional torus $\Lambda\subset\mathbb{R}^d$, with diffusion constant $\mathcal{D}\geq 0$ replaced by…

Let $d\geq 2$ and let $\Lambda=\Lambda(\mathcal{W},\mathcal{L})\subset\mathbb{R}^d$ be an irreducible cut-and-project set. Let $\mathcal{A}\subset\mathbb{R}^d$ be a van Hove sequen…

Let $u_0$ be the limit temperature and let $H$ denote the mean curvature of the boundary of the domain. Curvature–temperature dependence conjecture. The design of the shape of the…

Singularity conjecture. The continuous component $\widehat{\gamma}^{\ }_{\textsf{cont}}$ must be singular, namely singular continuous rather than absolutely continuous.

Pisot inflation conjecture. The system $cS$ has pure point powder diffraction, and consequently the main theorem applies to its associated point process.

Geometric resonance conjecture. A zero determinant should imply the presence of homogeneous (Rayleigh–Bloch) waves.

Let $\boldsymbol{\Gamma}$ and $\boldsymbol{\Gamma}'$ be surfaces related by a deformation, and let $\sigma_j'$ be the relevant singularity set. Their bridge symbol with respect to…

Physical conjecture. The field has an exponential decay due to the presence of the imaginary part of $k$ corresponding to absorption in the medium.

Let $G$ and $\hat G$ be the matrix functions in the two vector Riemann–Hilbert problems, and define their indices by … The compatibility conditions relating the solutions of the tw…

Let $\{\mu_n\}\subseteq \mathfrak{S}(\mathbb{R}^d)^{\prime}$ be a sequence of uniformly translation bounded measures, each possessing a natural autocorrelation, such that the weak-…

General propagator expansion.