18 problems

Let $p_a$, $p_b$, $p_c$, and $p_d$ be the edge-density quantities associated with the dimer model on an $n\times n$ torus in the thermodynamic limit. Consider an infinite sequence…

Let $\Delta$ be a convex lattice polygon containing the origin in its interior. A Laurent polynomial $W$ has Newton polygon $\Delta$, and a pair $(G,D)$ consists of a consistent di…

Let $(s,t)$ satisfy $|s|+|t| \leq 2$, and let $a,b,c,d$ be weights satisfying $ab=cd$ such that the average height function for weighted torus tilings has tilt $(s,t)$. Let…

In the dimer model on an $n \times n$ torus, let $p_a$, $p_b$, $p_c$, and $p_d$ denote the limiting densities of the four edge types in the thermodynamic limit as…

Gaussian free field–discrete Gaussian conjecture. The fluctuations of the dimer-model height function in the liquid region of a graph of scale $N$ are approximated in distribution…

Torus-knot dimer-model conjecture. Every $(p,q)$ torus knot admits a dimer model.

Positivity characterization conjecture. Traces of all $2$-webs in $H$ are positive if and only if, up to gauge equivalence, one of the following conditions holds:

Let $\mathbf{n}$ be a $3$-dimensional shape, and let $\mathcal{D}_3(\mathbb{Q}_{\mathbf{n}}^3)$ denote the graph of dimer configurations on $\mathbb{Q}_{\mathbf{n}}^3$ connected by…

Let $G$ be a planar bicoloured PBDTP graph, and let its edges carry the geometric signature. Form the associated sign matrix by assigning the corresponding geometric-signature sign…

Let $\mathcal G$ be a graph in the broader class considered by the cited work, with edge set $\mathcal E$, and let $\sigma:\mathcal E\mapsto\{\pm1\}$ be a signature. For every fini…

Let $\bar\Omega$ be the limiting region with red boundary segments corresponding to constrained boundary conditions and blue segments corresponding to free boundary conditions. Let…

Universal Edge Fluctuation Conjecture. Under suitable assumptions on the convergence of $G_n$, all scaling limits of the determinantal point processes at the frozen boundary are gi…

Let $Q_0$ be the vertex set of the dual quiver, let $\theta$ be a generic stability parameter, and let $\theta$ denote the corresponding moduli space with fan $\Sigma_\theta$. For a n…

For a sequence of bounded lecture hall tableaux with limiting height function $h(x,y)=\lim h_n(x,y)/n$, let $u(x,y)$ be the solution of the Burgers equation introduced for the asso…

Let … be the three-dimensional space of toric cluster variables described in the source. For self-intersecting contours, consider subgraphs with vertices of multiplicity one and tw…

Let $a_N$ denote the number of dimer packings of the odd lattice $(2k+1) \times (2k+1)$, and let $c_k$ be the integer in the factorization below. Factorization and squarefreeness c…

Let $G$ be a finite, non-abelian subgroup of $\operatorname{SL}(3,\mathbb{C})$, let $C$ be the chamber defining ${G}\operatorname{-Hilb}(\mathbb{C}^3)$, and let $S_\rho$, $L_\rho$,…

Dimer model realization conjecture. For a suitable choice of $W$ and distinguished basis, there is a bicolored graph $Y$ on $W^{-1}(0)$ such that an edge of $Y$ corresponds to an i…