27 problems

Let a simply-connected, bounded Jordan domain be approximated by a sequence of electrical networks, for instance networks arising from a sequence of disk packings. Choose conductan…

Let $G_{m,n}$ be the degree-$m$ part of the grove algebra, and let $c$ generate the cyclic action … The basis elements are to be indexed by height $m$ staircase plane partitions of…

Kenyon–Wilson's conjecture. The matrix $M$ is the response matrix of a circular electrical network belonging to a cell of co-dimension $r$ in the poset $EP_n$ if and only if the fi…

Let $\mathcal{E}$ be a well-connected network with an odd number of boundary nodes and response matrix $M_R(\mathcal{E})$. Let $A(\mathcal{E})$ be the matrix constructed from this…

Let $\mathcal{X}_n$ be the space of electrical networks, let $\mathcal{X}_\tau$ be the electroid variety indexed by a matching $\tau$ on $2n$ points, and let…

Let $G_{d,n}$ be the degree-$d$ homogeneous piece of the electrical network coordinate ring, let $\mathcal{C}^{(d)}_n$ index the proposed basis, let $E_{\mathbf{P}}$ denote its ele…

Let $L_s^{(i)}$, for $i\in\{1,2\}$, be the edge-resistance quantities defined in the paper, and let $\operatorname{num}(x)$ denote the numerator of the maximally reduced fractional…

Let $T_{n,k}$ denote the triangular grid obtained after $k$ transformations from the $n$-grid, and let $T_{n,n-s}^s$ denote an $s$-subgrid of $T_{n,n-s}$. For an edge, its edge val…

Edge-label ratio conjecture. For the $T_1(k,c)$ grid,

Let $({\mathbb{Z}}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\,p)$ be a biased disordered random network, with $d\geq 3$ and $0<\lambda_1\leq 1<\lambda_2$. A network is…

Let $p_c$ denote the critical probability for Bernoulli bond percolation on ${\mathbb{Z}}^d$, and let $({\mathbb{Z}}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\,p)$ den…

Conditional local-move existence conjecture. For every integer $n\geq 3$, there exists a conditional local move on $\Pi_n$ such that:

Hexagon conditional local-move conjecture. There is a hexagon conditional local move of the displayed form, with the conjectural conductance formulas described in the paper, that p…

Local-move completeness conjecture. Any two electrically equivalent graphs associated to rnpds are connected by the local moves stated in this section.

Let $N$ be the network from the hexwheel example, let $m\geq 3$, and write $\operatorname{cp}_m(\lambda)=\operatorname{rcp}(M(N))$. Let $\omega_\pm=\lambda-2\pm\sqrt{\lambda^2+4}$.…

Resistance exponent conjecture. The exponent $\mathfrak{r}$ exists and satisfies

A semicontiguous minor is a minor $M_A^B$ in which only one of $A$ or $B$ is contiguous. Domino-tiling conjecture. Every semicontiguous minor has an associated domino-tiling region…

Let $L$ be the response matrix of a minimal circular planar network with $m$ edges, and call a minor of $L$ noninterlaced when its row and column index sets are noninterlaced. Base…

Let $L$ be the response matrix of a minimal circular planar network with $m$ edges, and let a noninterlaced minor of $L$ mean a minor whose row and column index sets are noninterla…

Rank-unimodality conjecture. $EP_n$ is rank-unimodal; equivalently, the sequence $(|EP_{n,r}|)_r$ is unimodal. The conjecture concerns the distribution of electrical networks by ra…

Let $EP_n$ be the graded poset of electrical networks on $n$ boundary vertices, with rank given by the number of edges in a critical representative. Lexicographic shellability conj…

Let $G$ be a distance-regular graph of diameter $D$, and let $\phi_0,\ldots,\phi_{D-1}$ denote its Biggs potentials. For an integer $m$ with $0\leq m\leq D-2$, the universal bounde…

Root-antiroot conjecture. The root and the antiroot maximize the resistance in $\mathcal G(d,m,n)$.

Subpolynomial resistance conjecture. The maximal resistance in $\mathcal G(2,m,n)$ grows slower than any power of $n$:

Transience conjecture. The graph $\mathcal G(3,2,\infty)$ is transient.