20 problems

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…

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

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 $T$ 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…

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 $\mathbb{T}^d$ 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…

The Germain–Myerson conjecture. If $\delta>\lambda^{-1+\kappa}$ for some fixed $\kappa\in(0,1]$, then there is a constant $C=C(n,p,\kappa,L^*)$ such that

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…

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

Odd-prime-power torus packing conjecture. There exists $n_0$ such that, for all $n \ge n_0$, $C_k^n$ admits a perfect induced $H$-packing.

Four-source nonexistence conjecture. The equation has no solution for any $\tau\in i\mathbb{R}_{>0}$ if and only if $(n_0,n_1,n_2,n_3)$ satisfies neither of these two conditions.

Exact-solution-count conjecture. The equation possesses exactly $n$ solutions.

Let $F$ be a global field, let $X_F$ be a hyperbolic affine curve contained in $\mathbf P^1_F$, and let $j:X_F\rightarrow T_F$ be a closed immersion into a torus $T_F$. For a finit…

Let $G$ be an absolutely almost simple simply connected algebraic group over a finitely generated field $K$ of characteristic zero, or over a field of good characteristic relative…

Uniqueness of rectifiers. For $T$ as above, $T$ admits a unique rectifier up to equivalence.

The conjecture on intersections of subvarieties with tori. For every $n\in\mathbb{N}$ and every irreducible subvariety $W\subseteq\mathbb{G}_{\mathrm{m}}^n$, the set…