35 problems

Let $\mathbb{T}^d:=\mathbb{R}^d/(2\pi\mathbb{Z})^d$ be the $d$-dimensional torus, and let $\phi_\lambda$ be an $L^2$-normalized Laplace eigenfunction with eigenvalue parameter…

Sormani's conjecture. If $g$ satisfies the min-A condition, then $(T^n,g)$ should be close to a flat torus in the intrinsic flat topology.

Duke's torus-realization conjecture. Every triangulated torus can be realized as a polyhedron in $\mathbb{R}^3$.

Let $T$ be the torus, and let $\Lambda_2(T)$ denote its second topological eigenvalue. Kao–Lai–Osting conjecture. … This conjecture concerns the sharp value of the second eigenvalu…

Lin–Wang's nonexistence conjecture. For every $n\geq2$, this equation has no solutions when $\tau\in i\mathbb{R}_{>0}$.

Let $X\subset \mathbf{G}_m^N(\overline{\mathbf{Q}})$ be a variety, and let $V_X$ be the smallest translate of a subtorus containing $X$. Write $\widehat{\mu}^{\mathrm{ess}}(X)$ for…

Let $X\subset \mathbf{G}_m^N(\overline{\mathbf{Q}})$ be a variety, and let $U_X^{\mathbf{Q}}$ be the smallest union of torsion varieties containing $X$ that is defined over…

Kühn­el–Lassmann torus conjecture. The Kühn­el–Lassmann series of combinatorial $d$-tori $T^d$ with $2^{d+1}-1$ vertices is vertex-minimal.

Let $d\geq 1$, let $\mathbb{T}^d$ be the $d$-dimensional torus, and let $V\in L^\infty_x(\mathbb{T}^d)$. Let $\mathcal{H}_V=-\Delta+V$, and let $f$ be any eigenfunction of…

Let $d\geq 1$, let $\mathbb{T}^d=\mathbb{R}^d/(2\pi\mathbb{Z})^d$, and let $V\in L^\infty_x(\mathbb{T}^d)$ be a bounded potential. For $u_0\in L^2_x(\mathbb{T}^d)$, write…

Let $n$ be an integer with $n\geq 5$, and let $\Gamma^\prime$ be the $(n+1)$-regular colored graph constructed from the colored triangulation described in the paper. It has…

Let an elliptic curve mean a complex torus, and let $S^1$ denote the unit circle. An embedding of $S^1\times S^1$ into Euclidean space induces a complex structure on the torus thro…

Let $\mathbb{T}^d$ be the $d$-dimensional torus with $d>1$, equipped with a smooth Riemannian metric, and let $W_t(P)$ denote the wave front at time $t$ from a point…

Let $m_d(r)$ be the optimal observability constant for Laplacian eigenfunctions on the $d$-dimensional torus observed on a ball of radius $r$. For $d\geq 4$, let $C_d$ and $C'_d$ b…

Let $d\geq 2$. Let $\Gamma_d(\lambda)$ denote the maximal order of vanishing of a Laplacian eigenfunction on the $d$-dimensional torus with frequency parameter $\lambda$, and let…

Gutiérrez–Maldonado's conjecture. For an arbitrary torus,

Bounded-translate conjecture. For each dimension $d$ there is a constant $c$ such that, whenever

Let $c_q(\mathbb{Z}_m^n)$ denote the number of proper $q$-colourings of the torus $\mathbb{Z}_m^n$. Even-colouring free-energy conjecture. For even $q$, along even values of $m$, ……

Free-energy asymptotic conjecture. Along even values of $m$,

Engbers and Galvin's conjecture. There is such a decomposition satisfying all the conclusions of their decomposition theorem and, for some polynomial $p(n)$ whose degree depends on…

Let $S^n$ be the standard $n$-sphere and let $T^k$ be the $k$-torus, with $n\geq2$ and $k$ a positive integer. Let ${\rm Ein}(M)$ denote the Einstein invariant of a compact manifol…

Let $B$ be a bounded region in $\mathbb{R}^r$, let $r\geq 3$, and let $(\cdot,\cdot)$ be the positive-definite integral quadratic form used to define the frequency sphere. For coef…

Let $p$ be a prime integer and let $k$ be a $p$-closed field of characteristic different from $p$. Let $G$ be an affine algebraic group defined over $k$ whose connected component…

Let $G$ be a graph that triangulates the torus, and let a packing on a flat torus have contact graph $G$. Smooth edge-flip flow conjecture. One can flip an edge of $G$ and smoothly…

Optimal resolvent-region conjecture. For every $\varepsilon > 0$, there is a constant $C_{\varepsilon}$ such that, whenever $z \in \mathcal{R}_{\mathrm{opt}}$,