21 problems

Ideal-exponent factorization conjecture. This new polynomial should factor according to the ideal exponents $m_1,m_2,\dots,m_k$, in the same way that the Poincaré polynomial of the…

Let ${{\cal A}_n(m)}$ and ${{\cal C}_{n+1}(m,l)}$ be the deformed configurations with multiplicities $m$ and $(m,l)$, respectively, and let $P_{stable}^{{{\cal A}_n(m)}}$ and…

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

Polynomiality conjecture. For every such satellite $H_I$, the ratio $R_I(t^{-1})$ is a polynomial $R_I$ in $t^{-1}$ with integral coefficients.

Let $\Phi$ be a reduced root system with positive roots $\Phi^+$, let $\mathcal I$ be an ideal in $\Phi^+$, and let $\mathcal I(t)$ be its Poincaré polynomial. Let…

Feigin–Veselov factorization conjecture. All arrangements of vee-systems are free, so the corresponding Poincare polynomials are always factorizable in such a form.

Hausel–Villegas conjecture. The functions $H_n(w,z)$ are polynomials in $w,z$ with integer coefficients, and

Let $W$ be a finite simply-laced Weyl group with $n$ generators. An element $w\in W$ is $k$-palindromic if, writing its Poincaré polynomial as $P_w(q)=\sum a_i q^i$, one has…

Quiver–multiplicative Poincaré polynomial conjecture.

Emery's conjecture. If the rank of $X$ is greater than $1$, then $-1$ is a multiple root of $P(t;X)$.