12 problems

Dispersion conjecture. Localised smooth initial data satisfying the non-cavitation condition do not lead to stable structures localised in both spatial dimension. The solutions exi…

Let $F$ and $G$ be unimodal distributions in the sense that their quantile functions are differentiable almost everywhere, with derivatives $(F^{-1})'$ and $(G^{-1})'$ decreasing f…

For dimension $d$, consider the lattices associated with the rings of integers of the real number fields $\mathbb{Q}[2\cos(\pi/(2d+1))]$, where $2d+1$ is prime. Real cyclotomic lat…

Consider the three-dimensional lattice associated with the ring of integers of the splitting field $\mathbb{Q}[2\cos(\pi/7)]$ of $x^3-x^2-2x+1$. Its normalized dispersion is…

Two-dimensional dispersion optimality conjecture. The upper bound $1.89442\ldots$ is optimal, so

Let $P_n\subseteq[0,1]^s$ be a point set of size $n$, and let … be the minimum dispersion among such point sets. Bukh's order conjecture. The actual order of the minimum dispersion…

Planar dispersion optimality conjecture. The exact value of $\liminf_{N\to\infty}N\,\mathrm{disp}(N,2)$ is $\varphi^3/\sqrt{5}$. The lower and upper bounds leave the exact value un…

Modified Fibonacci lattice dispersion conjecture. We conjecture that this dispersion formula holds for all $m\geq 5$. Numerical computations for $m\in\{5,6,7,8,9,10\}$ support the…

Let $N(\varepsilon,d)$ denote the least number of points in the $d$-dimensional unit cube needed to obtain dispersion at most $\varepsilon$. Ullrich–Vybíral conjecture. The quantit…

Let ${{\mathcal P}_{n}}\subset [0,1]^2$ be a point set with $n$ points, and let $\operatorname{disp}({{\mathcal P}_{n}})$ denote the area of the largest rectangle in the two-dimens…

Let $E\ll -1$. A large-negative-energy behavior conjecture asserts that the KdV soliton is stable under NV dynamics and that localized initial data will be radiated away to infinit…

Combined-dispersion conjecture. Based on the model and analysis, the combined dispersion of the larvae and juveniles amplifies the goby population.