33 problems

Let $\mathcal M$ denote the parameter space of conformal classes represented by the flat metrics $g_{a,b}$, and let $g_{a,b}^{1,1,0}$ be the rotationally symmetric critical metric…

Let $T(a,b)=\mathbb{R}^2/\Gamma(a,b)$ be the flat torus associated with the lattice $\Gamma(a,b)=\mathbb{Z}(1,0)\oplus\mathbb{Z}(a,b)$, where $(a,b)$ lies in the moduli region … Wr…

For $\lambda>0$, consider lattice points on the circle $|x|=\lambda$ and arcs on that circle of length $\lambda^\delta$. Cilleruelo–Córdoba conjecture. For every $\delta<1$, there…

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…

Flat tori conjecture. Then $\Gamma\backslash G$ is diffeomorphic to a flat torus.

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…

Germain's conjecture. One should have

Choir-number conjecture.

Transition-threshold conjecture. For odd $k$, the threshold $b_k$ is given by the upper bound

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 $\operatorname{ric}_M\geq K$,…

Let $M$ be a closed Riemannian manifold as in part (2) of the characterization theorem: it is $\delta$-Gromov-Hausdorff close to a flat $n$-torus $T$ with…

Let ${\mathbb S}^1_A \times {\mathbb S}^1_B$ be a flat product torus with $A>B$, and let $\psi^{\lambda\,\textrm{ near }(k/B)^2}$ denote an approximately selective embedding into…

Solution-count conjecture. Except for finitely many $\tau$ modulo the $SL(2,\mathbb{Z})$ action, the Toda system has exactly $N(n_1,n_2)$ solutions. The conjecture extends the esta…

Let $D_4$, $E_8$ and $\Lambda_{24}$ denote the indicated lattices, and consider the flat tori obtained by quotienting Euclidean space by these lattices. Lattice-torus unicorn conje…

Let $N(A_2)$ denote the set of sizes of lattice codes in the triangular torus $\mathbb{R}^2/A_2$. Triangular-torus deletion conjecture. For every $n\in N(A_2)$ with $n>1$ and…

Hauswirth's three-phase conjecture. The isoperimetric profile of $T^2\times\mathbb R$ is composed of three parts: for small volumes, isoperimetric regions are spheres $B^3_R$; for…

Transition-volume equality conjecture. In fact,

Three-phase profile conjecture. The precise isoperimetric profile of $(T^2\times {\mathbb R},h_2+dt^2)$ should be generated by regions such as spheres $B^3_R$, cylinders…

Let $d\ge3$, let $\Sigma\subset\mathbb{T}^d$ be a smooth submanifold of dimension $k$ with $1\le k\le d-2$, and let $e_\lambda$ be an eigenfunction of $\mathbb{T}^d$. Higher-dimens…

Let $({\alpha}_{ij})$ be a symmetric matrix and set … Let $E$ be the set of symmetric matrices satisfying $-2\leq\alpha_{ij}\leq 2$ and whose smallest absolute eigenvalue satisfies…

Let $d\geq 1$, let $({\alpha}_{ij})$ be a diagonal matrix, and define … Call $({\alpha}_{ij})$ generic when it lies outside an exceptional set $E$ of diagonal matrices having $d$-d…

Let $M$) be a flat torus, with $M=\mathbb{R}/\theta_i\mathbb{Z}$ in the rectangular case. Call the spectrum of $\Delta_M$ rational when all ratios of its eigenvalues are rational n…

Let $\mathcal C\subset \mathbf{T}^2$ be a smooth curve with non-zero curvature, and let $F$ be a toral eigenfunction with eigenvalue parameter $\lambda$. Write $\mathcal Z(F)$ for…