105 problems

For integers $a,b\geq 0$, let $\mathsf{Shuf}(a,b)$ be the shuffle poset, with rank function $\mathsf{rk}$, and define … For $n>0$, consider the case $a=n-1$ and $b=1$. The interval…

For every finite group $G$, there exists a finite lattice $L$ such that $\operatorname{Aut}(L)\cong G$ and the action of $\operatorname{Aut}(L)$ on the elements of $L$ has at most…

Kapranov–Voevodsky conjecture. The map $f$ is a quotient by a weak order congruence. This conjecture asks whether the higher-dimensional analogue of the map from weak Bruhat order…

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…

Greene–Owens cubiquity conjecture. $\Sigma(L)$ bounds a rational homology $4$-ball if and only if $\Lambda(L)$ is cubiquitous.

Let $k\geq 3$ and let $(n_1,\ldots,n_k)\neq(1,\ldots,1)$. Let $L_k(n_1,\ldots,n_k)$ be the stretched finite lattice with chains … for $i=1,\ldots,k$, and let…

Ribbon cobordism converse. If $\Lambda(L_2)$ admits a cubiquitous embedding into a stabilization of $\Lambda(L_1)$, then there exists a ribbon cobordism from $\Sigma(L_1)$ to…

Let $L_k(n_1,n_2,\ldots,n_k)$ be the lattice considered in the paper, and let $I_{L_k(n_1,n_2,\ldots,n_k)}$ denote its join-meet ideal in the associated polynomial ring. Crystal co…

Let $h_{105}(x)$ be the coordinator polynomial of the lattice $\mathbb Z[\zeta_{105}]$ with respect to the set of all $105$th roots of unity. A polynomial is palindromic if its coe…

Let $\zeta_{15}=e^{2\pi i/15}$, and let $h_{15}(x)$ be the coordinator polynomial of the lattice $\mathbb Z[\zeta_{15}]$ with respect to the set of all $15$th roots of unity. Parke…

Let $p$ be an odd prime, let $\zeta_{2p}=e^{2\pi i/(2p)}$, and let $h_{2p}(x)$ be the coordinator polynomial of $\mathbb Z[\zeta_{2p}]$ with respect to the set of all $2p$th roots…

Let $m$ be a positive integer. Write $\sqrt m$ for the squarefree part of $m$, let $\zeta_m=e^{2\pi i/m}$, and let $h_m(x)$ be the coordinator polynomial of the lattice…

Let $W$ be a finite reflection group containing $-1$, let $G$ be its Dynkin diagram, and let $\bar G$ be an orientation. Form the Coxeter element $c$ from the oriented diagram, ord…

Let $C(\bar G)$ be a Cambrian lattice associated with an oriented Dynkin diagram $\bar G$ of a finite reflection group. A lattice is trim if it has a left modular maximal chain and…

Let $d$ be fixed. For a finitely supported function $\xi:{\mathbb Z}^d\to{\mathbb R}$, let $W_{2n}({\bf v},\xi)$ denote the associated lattice random-walk quantity at ${\bf v}$ and…

Let $B$ be the associated braid group, let $R$ be its set of braid reflections, and let $B_+$ be the submonoid of $B$ generated by $R$. Define a relation $\preccurlyeq$ on $B_+$ by…

Let $L$ be a nonmodular lattice, and let $I_L$ denote its join–meet ideal. An ideal is linearly related when its linear syzygies generate all of its syzygies. Nonmodular-lattice jo…

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…

Orbit-structure conjecture. The orbit structure of the alt $\nu$-Tamari lattice $\mathsf{Tam}_{\delta}(\nu)$ is independent of the increment vector $\delta$.

Ordovician lattice structure conjecture. The following hold:

N-regularity conjecture. Every vertex of the underlying undirected graph of $\operatorname{Hasse}(L)$ has degree $N$; equivalently, the graph is $N$-regular.

Bounded-arity symmetric-part conjecture. For any lattice variety axiomatized by identities of arity at most $k$, one has

Symmetric-part conjecture. Each lattice variety $\mathcal{V}$ equals its symmetric part:

Let $G$ be a graph, and let $F_1$ and $F_2$ be framings of $G$. For each $k=0,1,2,\ldots$, consider the linear intervals of length $k$ in the framing lattices $\mathscr{L}_{G,F_1}$…