13 problems

Let $n\geq 3$ and let $\vec{d}$ be a dimension vector. Write $P^{\mathrm{W}}_n(y)=P\left(\lvert\mathrm{W}_{n,\vec{d}}\rangle;y\right)$. Mainiero's generalized W Poincaré polynomial…

Let $n\geq 3$ and let $\vec{d}$ be a dimension vector. Write $P^{\mathrm{GHZ}}_n(y)=P\left(\lvert\mathrm{GHZ}_{n,\vec{d}}\rangle;y\right)$. Mainiero's generalized GHZ Poincaré poly…

Let $d$ be a positive integer and let $\hat\Omega_d^{\mathbb P^2}$ be the shifted Poincaré polynomial associated with the moduli space of degree-$d$ one-dimensional sheaves on…

Let $N_{r,d}$ be the moduli space of stable rank-$r$ bundles of degree $d$ on a smooth projective curve of genus $g$, and let $\Omega(N_{r,d},q,t)$ denote its refined Poincaré poly…

Let $N_{3,1}$ be the moduli space of stable rank-three bundles with fixed degree one on a smooth projective curve of genus $g$, and let $\Omega(N_{3,1},q,t)$ denote its refined Poi…

Let $K_n(t,q)$ denote the Poincaré polynomial of the Khovanov homology of the torus knot $T(n+1,n)$. Shumakovitch–Turner recurrence conjecture. The polynomials satisfy … and, for…

Let $K_n(t,q)$ denote the Poincaré polynomial of the Khovanov homology of $T(n+1,n)$, and let $L_n(t,q)$ denote that of $T(n,n)$. Let $C_n=\frac{1}{n+1}\binom{2n}{n}$ be the $n$th…

Brenti–Carnevale–Tenner's conjecture. Each odd diagram class is rank-symmetric in the Bruhat order.

Hilali conjecture. The homotopical Poincaré polynomial is bounded above by the Poincaré polynomial:

For each of the five types considered—type $A$, odd symplectic, even symplectic, odd orthogonal, and even orthogonal—let $P(q)$ denote the corresponding Poincaré polynomial. Unimod…

Let a $K$-node quiver have adjacency matrix $\alpha_{ab}$, dimension vector $\gamma=(N_1,\dots,N_K)$, stability parameters $\zeta=(\zeta_1,\dots,\zeta_K)$, and generic superpotenti…

Moment-graph formula conjecture. One has

Quiver–multiplicative Poincaré polynomial conjecture.