54 problems

Cohn–Elkies lifting conjecture. Every optimal solution $f$ can be lifted to such a function $h$ with the same value of $r$ and

Let … be the hexagonal lattice. A lattice is universally optimal if it minimizes every admissible energy among configurations of the same density. Cohn–Kumar's conjecture. The line…

Cohn–Miller rationality conjecture. The second Taylor coefficients of $f_8$, $\widehat{f}_8$, $f_{24}$, and $\widehat{f}_{24}$ are rational.

A 3-class association scheme is an association scheme with three nontrivial relations; here, consider a certain such scheme on $64$ points. Cohn's universal optimality conjecture.…

Let $Q$ be the region and let $P=(p_1,\ldots,p_n)$ be the generator locations. The multi-incenter objective is … where $V_i$ are the Voronoi cells of the generators. Sphere-packing…

Let $\phi$ be a completely monotonic potential of distance squared in dimension $2$, and consider point configurations at fixed density. The two-dimensional completely monotonic gr…

Three-way equivalence. The following conditions are equivalent:

Viazovska's uniqueness conjecture. This data, together with the nonzero value $f'\big(\sqrt{2}\big)$, should be enough to determine $f$ uniquely. The conjecture concerns the unique…

Large-q exponential-regime conjecture. For all $q$ sufficiently large, there exists an infinite sequence of $d$-regular graphs with $\lambda_2=\Theta(1/\sqrt{d})$ for some…

Let $D_3$ be the three-dimensional $D_3$ lattice, let $D_3^*$ be its dual lattice, and let $\Theta_\Lambda(q)$ denote the theta function of a lattice $\Lambda$. Dimension-three the…

Let $D_4$ denote the lattice in $\mathbb{R}^4$ generated by the $D_4$ root system, and consider sphere packings in $\mathbb{R}^4$ by congruent nonoverlapping spheres. $D_4$ sphere-…

Triangular-lattice universal optimality conjecture. The usual triangular lattice should minimize the interaction energy among infinite configurations of points with density $1$ for…

The Barnes–Wall lattices are lattices in dimensions $n=4$, $8$, $16$, and higher, with the lattices in dimensions $4$ and $8$ identified with $D_4$ and $E_8$, respectively. Their s…

Let $f\colon\mathbb{R}^d\to\mathbb{R}$ be an integrable, even function with integrable, real-valued Fourier transform, and let $\rho_1,\rho_2\geq 0$ satisfy $f(x)\geq 0$ for…

Viazovska's interpolation conjecture. The magic function is uniquely determined by its required roots; more generally, a radial Schwartz function on $\mathbb{R}^8$ is uniquely dete…

A lattice packing in $\mathbb{R}^4$ has density bounded above by the lattice three-point bound, obtained from a semidefinite-programming formulation of the three-point bound. Sharp…

$D_4$ packing optimality conjecture. The $D_4$ root lattice packing is conjectured to be optimal among sphere packings in dimension $4$. The four- and five-dimensional sphere packi…

Let $\omega_N\subset\mathbb{S}^2$ be a collection of $N$ antipodal points, meaning that $p\in\omega_N$ implies $-p\in\omega_N$. Let $\mathscr{E}_{\log}(\omega_N)$ denote its logari…

Sharpness conjecture. The upper bound is an equality:

Let $f:\mathbf{R}^2\to\mathbf{R}$ be a radial Schwartz function, and write $r=|x|$. Cohn–Elkies conjecture. There exists such an $f$ satisfying … … … For the hexagonal lattice, the…

Exact constant conjecture. The constant $\lambda$ satisfies

Asymptotic linear programming exponent conjecture. There exists a constant $\lambda$ with

For $d\geq 1$, let $\mathcal{A}_{LP}(d)$ be the class of continuous, even, real-valued functions $g\in L^1(\mathbb{R}^d)\setminus\{\mathbf{0}\}$ with…

Let $f(t)=\arccos|t|$, so that $f(\langle x,y\rangle)$ is the non-obtuse angle between the lines generated by $x$ and $y$. For a configuration of $N$ points, let $I_f$ denote the c…

PL-maximality conjecture. Every local maximum of $\delta$ on $\mathcal{P}_{n}$ is $PL$-maximal; equivalently, every locally maximal cluster of $n$ equal balls has this property, pr…