221 problems

Belfiore–Solé conjecture. For every unimodular lattice $L$, the function $\Xi_L(y)$ attains its global maximum on the positive imaginary axis at the symmetry point $y=1$.

For a Coxeter system $(W,S)$, the extended weak order is the partial order introduced by Dyer that generalizes the weak order to possibly infinite Coxeter groups. Dyer's lattice co…

Let $w$ be a permutation, and let $\mathsf{PD}(w)$ be the set of pipe dreams for $w$ equipped with the chute-move partial order $\mathrel{\leq_{\mathrm{chute}}}$, where…

Let $Q\colon\mathbb{Z}^d\to\mathbb{Z}$ be positive definite and unimodular, let $b$ be the corresponding even bilinear form, and let $\Theta_Q=\Theta_b$ be its theta function. Let…

Finite Coxeter Cambrian mutation conjecture. The Cambrian lattice $\operatorname{Camb}(Q)$ is mutable.

Shortest-path bisaturation conjecture. A shortest path between the trivial $G$-transfer system and the complete $G$-transfer system on a Hasse diagram will pass through as many bis…

Ceballos–Pons geometric realization conjecture. The Hasse diagram of the ${\boldsymbol{s}}$-weak order can be realized as an orientation of the skeleton of a polyhedral subdivision…

Formally unimodular secrecy-gain conjecture. The secrecy function of every formally unimodular lattice $\Lambda$ achieves its maximum at $\tau=1$:

Let $Y$ be an Enriques surface and let $H$ be any line bundle on $Y$. Borisov–Nuer's conjecture. There is a line bundle $N\in\operatorname{Pic}(Y)$ such that … This condition is eq…

Let $X$ be a singular K3 surface, let $p$ be a supersingular prime for $X$, and let $T_X$ be its positive-definite transcendental lattice. Define the supersingular reduction lattic…

Let $S$ be a lattice. An $S$-K3 surface is a K3 surface whose Picard lattice contains $S$, and let $S^\perp$ denote the orthogonal complement of $S$ in the relevant K3 lattice. Con…

A lattice has a basis of minimum vectors if its minimal vectors contain a lattice basis. Up to dimension $7$, every perfect lattice has a basis of minimum vectors. Martinet's conje…

Gaifman's conjecture. For every completion $T$ of $\mathsf{PA}$ and every finite distributive lattice $D$ with a unique atom, there is an end-extensional type $p(x)$ of $T$ that pr…

Mockingbird partial-order and lattice conjecture. The relation $\preccurlyeq$ on the terms over ${\operatorname{\mathbf{M}}}$ is a partial order relation, and each $\equiv$-equival…

Let $\Lambda$ be a lattice in three-dimensional Euclidean space, and for $s>3$ define its Epstein zeta function by … The FCC lattice denotes the face-centered cubic lattice. Sarnak…

For each of the 24 Niemeier lattices $N$, consider non-algebraic $K3$ surfaces $X$ and their Picard lattices $P(X)$. Nikulin's realization conjecture. There exists a non-algebraic…

Let $L$ be a lattice, and let $\mathcal{I}$ be the set of its irreducible ideals. For an ideal $A$, write $A^C$ for its complement in the underlying set of the lattice. Frankl's id…

Lattice strong locality conjecture. The vertex operator algebra $V_\Lambda$ is strongly local.

Thomas's trimness conjecture. The Cambrian semilattice $\mathcal{C}_{\gamma}$ is trim.

For each positive integer $k$, let $\Delta_k(n)$ denote the maximum upper density of a subset $S\subseteq\mathbb{Z}^k$ whose maximum degree is at most $n$, where degree counts neig…

Conjecture A. If $d(S)\leq 3$, then $S$ has upper density at most $1/2$ in $L$. This is presented as a stronger version of the still-life density problem: patterns with density nea…

Acketa's conjecture. The lattices $L_5$, $L_6$, and $L_7$ are Tr-lattices.

A perfect Delaunay polytope is a perfect Delaunay polytope in dimension $d$, and let $ED_d$ denote Dutour's polytope. Dutour's conjecture. The polytope $ED_d$ has the largest numbe…

A Cambrian lattice is a lattice associated with a Coxeter group and its Coxeter element; a lattice is trim when it is both left modular and extremal. The Cambrian lattices in types…

Let $\Lambda$ and $\Lambda_0$ be lattices associated with an $8$-dimensional perfection relation, and suppose that $[\Lambda:\Lambda_0]=5$. Two known examples have parameter pairs…