18 problems

Takayama's conjecture. The holonomic dual of an $A$-hypergeometric system is itself a GKZ system.

Let $A$ be an integer matrix, let $\mathcal{M}_A(\beta)$ be the projectivized $A$-hypergeometric system with parameter $\beta$, and let $Y$ be a coordinate variety. For a filtratio…

Let $A$ be fixed, let $L$ be a filtration, let $\tau$ be the corresponding face or stratum, and let $\mu^{L,\tau}_{A,0}(\beta)$ denote the multiplicity of the associated component…

Hypergeometric pullback conjecture. For every constituent $\mathcal{C}$ of $Q$, there exist a torus $T_{\mathcal{C}}$, a morphism of tori

Let $A$ be an integer matrix, let $S_A$ be its semigroup ring, and let $\mathfrak m$ be its homogeneous maximal ideal. For each $i<\dim(S_A)$, let $H^i_{\mathfrak m}(S_A)$ denote t…

Let $A$ be the defining integer matrix and let ${\mathbb C}[{\mathbb N} A]$ be its semigroup ring. Let $H_A(\beta)$ denote the associated $A$-hypergeometric system, and let…

Let $L_0(s,\xi)$ be the greatest common divisor of the polynomials $\sigma(L_1)(s,\xi),\ldots,\sigma(L_k)(s,\xi)$, and let the residual resultant be the resultant of…

Consequences conjecture. For generic choices of $u\in\mathbb{C}^{n+1}$:

Integral representation conjecture. For generic choices of $u\in\mathbb{C}^{n+1}$, the morphism $\psi$ is an isomorphism of $D_Z$-modules.

Let $\mathcal{J}$ be the collection of subsets indexing the branes $\mathfrak{B}_{J}$, let $\hat{Z}_{\mathfrak{B}_{J}}(\mathbf{x})$ denote their A-periods, and let…

Let $A\in\mathbb{Z}^{m\times n}$ and $c\in\mathbb{C}^m$, and let $\operatorname{Horn}(A,c)$ be the hypergeometric system defined by … with…

Covering conjecture. On each member of this open covering, the system $E(n,m)$ is represented locally by a $GKZ$ system; equivalently, the hypergeometric $\mathcal{D}$-modules of…

Multiplicity conjecture. The multiplicity of $M_A(\beta_1,\beta_2)$ is $16$ for all $(\beta_1,\beta_2)$ except $(1,2)$.

Let … and … Define the Wronskian … and let $F=W\circ\lambda$, where … for $q$ in the punctured unit disc $\Delta^*=\{q\in\mathbb C:0<|q|<1\}$. Wronskian nonvanishing conjecture. Th…

Equality-region conjecture. The equality

Let $C$ be a rational polyhedral cone in a lattice $N$, let $C^\circ$ denote the associated dual data, and let $bbGKZ(C,0)$ and $bbGKZ(C^\circ,0)$ be the corresponding better-behav…

Let $C$ be a rational polyhedral cone in a lattice $N$, let $X=\operatorname{Spec}\mathbb{C}[C^{\vee}\cap N^{\vee}]$, and let $\Sigma_1$ and $\Sigma_2$ be adjacent projective simpl…

Sturmfels's conjecture. The systems in this family all have the same number of linearly independent solutions precisely when the corresponding affine semigroup ring is Cohen–Macaul…