26 problems

Distortion ropelength minimizer conjecture. For each nontrivial knot type $K$, there exists a shortest curve $[?]\subset\mathbb R^3$ of distortion thickness $1$ in that knot type.…

Let $d\geq 2$, let $q>1$, and let $\gamma$ be a density matrix satisfying $0\leq\gamma\leq 1$. Write … for a rank-$N$ projection, and let $\rho_\gamma$ denote its density. The asso…

Strict-hierarchy conjecture. If $q<2$, then

Let $x=(x_1,\ldots,x_d)$ with $x_i\in{\mathbb R}^n$, signed masses $m(x_i)$ satisfying … and define … A configuration is stationary and stable when its first variation vanishes and…

For a tame knot class $K$, let $\mathcal{K}$ be the $W^{2,2}$-closure of the class of unit-length, arclength-parametrized closed curves in $K$: … A stable elastic knot for $K$ is a…

Let $N$ be a positive integer, and let a knot class have braid index and bridge index both equal to $N$. An $N$-fold circle is the corresponding circle traversed $N$ times, and an…

Let $q>0$ be the charge parameter, and consider the minimization problem defined by equation (minmin1), whose minimizers are configurations attaining its minimum energy. Nonexisten…

Let $(M,\zeta)$ be a closed contact $3$-manifold, where $\zeta$ is a contact distribution, and consider the Chern-Hamilton energy functional on the space of compatible metrics for…

Gravitational-field variational-shape conjecture. The preceding convergence in probability holds. The supplied context says that the solution of this variational problem is known i…

Variational solution threshold conjecture. There exists a constant $bbda_{b1,D}$ with $bbda_{b1,D}>bbda_{b1,D}$ such that, for every $bbda a bbda{b1,D}$, the variational problem ha…

Let $F_{\rm \max}$ be the maximal value function whose derivative appears as the limit shape in the paper's limit-shape corollary, and let $F_{\rm \max}'$ denote that limit shape. Co…

Let $q_{\mathrm{ic}}=\delta_{-\zeta}+\delta_{\zeta}$ for $\zeta>0$, set $\mathfrak{m}=1$ and $\xi_1=0$, and consider the variational problem in which $T$ and these data are fixed.…

Global optimizer conjecture. There is a number $\xi(j)\in(0,1)$ such that the globally least bent confined elastica is a $j$-$(j+1)$-clew if $r_L/R<j+\xi$, while the global optimiz…

Let $E$ be the elastic surface energy for topological discs, with positive parameters $\alpha$, $\beta$, $c_o$ and $b\neq 0$, and define … Assume that $\underline E\geq 0$. Axial s…

Stable-labeling variational asymptotic conjecture. If $n\to\infty$ and $p=p(n)$ satisfies $n^{-1/\Delta}\ll p\ll 1$, then

Clique-and-hub conjecture. The variational problem is asymptotically optimized by planting a clique and a hub.

Let $S=[0,1]^2$, and let $\phi^N$ be a stochastic interface model on $S_N$ with interaction potential $V=\mathsf{bead}$ or $V=\mathsf{\exp}$. Suppose its weight functions are … wher…

Uniqueness conjecture. The optimal measures $\mu$ are unique and satisfy

$L^\infty$ maximizer conjecture. For every $D\subset\Omega$, the unique maximizers of the problem are the odd function

Let $S^3$ be the three-sphere, let $E_1$ be the energy functional considered in the source, and let $\Sigma_{\frac{\sqrt{2}}{2}}\subset S^3$ denote the Clifford torus. A surface of…

Uniqueness conjecture. The shifted Heisenberg spheres above are the only closed minimizers for $E_1$ with zero energy in $\mathcal{H}_1$. The source establishes that these spheres…

Let $\Omega\subset\mathbb{R}^n$ be a ball, and consider the Trudinger–Moser variational problem … A maximizer is a function attaining this supremum. Uniqueness conjecture. Maximize…

Ignat–Merlet conjecture. Then $(\nabla\varphi_0)^\perp$ is a global minimizer of $\mathcal{I}_f$.

Let $\mu$ be a measure and, for $\lambda>0$ and $1\leq p<2$, consider the average-distance minimization problem whose global minimizers are parameterized curves. Non-injectivity co…

Freedman–He–Wang conjecture. The Möbius energy is minimized, among the class of all nontrivial links in $\mathbb R^3$, by the stereographic projection of the standard Hopf link in…