127 problems

Let $xi_{\ell,r,c}$ denote the integer in the reduced $Gr_2$-simplex in layer $\ell$, row $r$, and NE-SW diagonal $c$, and set $\xi_{\ell,r,c}=0$ when there is no entry in that pos…

Arrange the entries in each layer of the $Gr_2$-simplex as coefficients of a polynomial in $a,b,c$ according to the arrangement described in the paper. Divisibility conjecture. Eac…

Let $T\in\operatorname{SSYT}(k,[n])$ be a tableau whose distinct entries are $a_1<\ldots<a_r$. Let $n'\ge n$ and let $f:\{a_1,\ldots,a_r\}\to[n']$ satisfy $f(a_1)<\ldots<f(a_r)$; w…

For positive integers $k,n,r$, let $N_{k,n,r}$ denote the number of cluster variables of rank $r$ in $\mathbb{C}[\operatorname{Gr}(k,n)]$. Cluster-variable counting conjecture. The…

Let $\mathbb G(r,n)$ denote the Grassmannian of $r$-planes in projective $n$-space, and let $h$ be a positive integer. A projective variety is called $h$-defective when its $h$-sec…

Let $m,p>1$, let $\operatorname{Grass}(p,m+p)$ be the Grassmannian, and let $\alpha^1,\ldots,\alpha^n$ be Schubert data. A real upper-triangular matrix with diagonal entries $1$ is…

Mirror-family conjecture. The pair $M_y+W_y$ is the mirror family of $X_A+Y_A$, with $M_y$ and $W_y$ forming different parts of the B-model moduli space and $X_A$ and $Y_A$ forming…

Let ${\mathcal G}_k$ be the Grassmannian of $k$-dimensional singular subspaces of a building of type $\textsf{X}_{n}=\textsf{C}_{n},\textsf{D}_{n}$, and let ${\mathfrak G}_k$ be it…

Let $\mathbf{G}$ be the Grassmannian in its Plücker embedding, let $\mathbf{Pf}$ be its projective-dual Pfaffian variety, and write $\mathbf{P}^*$ for the dual projective space. A…

The Schubert calculus conjecture. For generic $z=(z_1,\ldots,z_n)$, the intersection of Schubert varieties $\mathcal{I}_{\{{\bf w}\}}(z)$ is transversal. Here generic means that…

Chow quotient and log canonical model conjecture. The inclusion $X(r,n)\subset \overline{X}(r,n)$ is the log canonical model precisely in the cases

Let $[1\dots n]$ be labeled cyclically, and let a non-crossing family be a family of pairwise non-crossing $k$-subsets, where non-crossing means that no chord with endpoints in…

The $e_m$-overkiller conjecture. For $3\leq m\leq k$,

The exact $e_m$-killer conjecture. For $1\leq m\leq k$,

The filtration-saturation conjecture. For $1\leq m\leq k$,

The filtered Hilbert-series conjecture. For $m\geq 0$,

For integers $k$ and $n$ in the range where the polynomial $\hat{E}_{k,n}(q)$ is defined, let $[q]\hat{E}_{k,n}(q)$ denote the coefficient of $q$ in $\hat{E}_{k,n}(q)$. Linear-coef…

Let $d\geq 1$ be an integer, and let $\lambda$, $\mu$, and $\nu$ be Young diagrams indexing Schubert classes in the Grassmannian. Denote by $\langle\lambda,\mu,\nu\rangle_d$ the co…

Let $X$ be the Grassmannian, and let $\sigma_{\lambda}$ and $\sigma_{\mu}$ be Schubert classes indexed by Young diagrams. Write $d_{\min}$ and $d_{\max}$ for the smallest and large…

Let $\operatorname{W}(k,n)$ be the set of maximal collections of pairwise weakly separated $k$-subsets of $[1\dots n]$, and let a $(2,4)$-move replace one of the two indicated memb…

Let $X_0$ be a conifold degeneration of a generic Calabi–Yau complete intersection $X\subset G(k,n)$, let $p$ be the number of nodes of $X_0$, and set $\alpha=(k-1)(n-k-1)$. Let…

Let $G(r,s)$ be the Grassmannian and let $X_{[a,b]}$ be the variables indexed by $1\leq a\leq s-r$ and $1\leq b\leq r$, with $X_{a,b}=0$ when $a>s-r$ or $b>r$. Let $q$ be the quant…

Let $G(k,n)$ be the Grassmannian, let $P(k,n)$ be its Gorenstein toric degeneration, and let $A_{G(k,n)}(q)$ denote the special solution to the quantum ${\cal D}$-module determined…

Compatible-splitting conjecture. For any diagram $D$ and any Weyl group elements $u_1,\ldots,u_r$, the pairs

Let $\mathcal{S}_{ij}$ be a Schubert variety in ${\rm Gr}(2,n)$. GD-degree conjecture. If $1<j-i<n-1$, then the Schubert variety $\mathcal{S}_{ij}$ has GD degree $2$. The paper rep…