11 problems

Let $A_0\geq A_2\geq A_1$ be areas, and let a flat torus have area $A_0+A_1+A_2$. A non-tiling minimizing double bubble is a perimeter-minimizing double bubble enclosing areas…

Let $L=C\mathbb{Z}^n\subset\mathbb{R}^n$ be a lattice with Gram matrix $G=C^{\mathsf T}C$ and associated flat torus $T_L=\mathbb{R}^n/L$. Normalize to unit volume, so $\det G=1$. I…

Let $T$ be an $n$-dimensional flat torus of volume $1$, represented by a lattice in Euclidean space. Let the height of $T$ be $\zeta'(0)$, equivalently the negative logarithm of it…

For $\beta>0$, let … and, for $m\geq 2$, define … A configuration is $L^m_\beta$-universally optimal if it minimizes every admissible periodic energy among $2m$-point configuration…

Let $K\in\mathbb{R}$, $n\in\mathbb{N}$, and $v,D>0$. For every $\epsilon>0$, consider Riemannian $n$-tori $M$ satisfying $\operatorname{ric}_M\geq K$,…

Bourgain–Rudnick conjecture. The lower bound for the number of nodal intersections is of order $\lambda$; equivalently,

Let $M$ be a symplectic surface and let $S_{cl}$ be the classical functional for maps from $M$ to a flat torus $(S^1)^n$. For each $N$, let $\Phi_1,\dots,\Phi_n\in U(N)$ and let…

Minimal-energy equality conjecture. The three inequalities in Proposition 1 are equalities.

Let ${\mathcal L}_0$ be a Diophantine irrational lattice. Write $n_j$ for the distinct Laplacian eigenvalues, let $\delta_j=n_j-n_{j-1}>0$, let … and define the mean-normalized spa…

Nine-point packing conjecture. There is one unique, up to isometry, optimal arrangement of 9 points in ${\mathbb T}^2$, shown in the paper's Figure 9, with

Kitaoka's conjecture. If the flat tori $\mathbb{R}^n/L_1$ and $\mathbb{R}^n/L_2$ are spectrally commensurable, then, up to an isometry of $\mathbb{R}^n$, the lattices $L_1$ and…