104 problems

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:

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

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 $\mathscr{L}_{G,F_1}$…

Two-component LSP compatibility conjecture. Then $T$ is not LSP and is compatible with the transfer system generated by adding the edge $e\to G$ to $T$.

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…

Let $L$ be a finite lattice, and let $A=KL$ be its incidence algebra. Assume that $A$ has a pure minimal injective coresolution. Distributivity conjecture. Then $L$ is distributive…

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…

Let $W$ be a finite Coxeter group and let $c$ be a Coxeter element. The $c$-noncrossing Bruhat order is the order on $W$ defined from the noncrossing Bruhat construction. Bruhat-or…

Let $W$ be a finite Coxeter group and let $c$ be a Coxeter element. Consider the $c$-noncrossing weak order and the $c$-noncrossing Bruhat order on $W$. Order-isomorphism conjectur…