12 problems

Let $G$ be a plabic graph, and let $rG$ denote any rotation of $G$. Its associated Newton–Okounkov body is denoted by $\Delta_G$. Rotation invariance conjecture. All rotations of a…

Let $\X=\G/\P$ be a homogeneous space with a dominant weight $\varpi$ yielding a projective embedding $\X\hookrightarrow\bP(V_\varpi)$, and let $\bs$ be a reduced expression for…

Let $\X=\G/\P$ be a homogeneous space with a projective embedding $\X\hookrightarrow\bP(V_\varpi)$, and let $\bs$ be a reduced expression for $\wP$. The Chevalley polytope Newton–O…

Let $X$ be a smooth projective variety of dimension $d$ over an algebraically closed field $\kappa$, and let $L$ be a big line bundle on $X$. For each valuation $\nu$ in the higher…

Let $X$ be a projective variety of dimension $d$, let $L$ be a big line bundle on $X$, and let $D$ be a simple normal crossing divisor whose dual cone complex $\Sigma(D)$ is pure o…

A conservative reaction network is a chemical reaction network in which each species occurs in at least one nonnegative conservation law. Let the mixed-volume bound be the upper bo…

Sharpness conjecture. If $X$ is $\mathbb P^2$, $\mathbb P^1\times\mathbb P^1$, or a Hirzebruch surface, then

Let $S$ be a smooth projective algebraic surface, let $D$ be a big divisor on $S$, and write $\rho=\rho(S)$. A smooth projective surface $\tilde S$ with a proper projective morphis…

The proposed bijection. The resulting point is the corresponding point in $NO_G^r$.

Let $X$ be a Mori dream space, and let $Y_\bullet$ be a flag for which the global semigroup $\Gamma_{Y_\bullet}(X)$ is defined. The Cox ring of $X$ is denoted by…

Let $G$ be a complex reductive group, let $B$ be a Borel subgroup, let $w_0$ be the longest Weyl-group element, and let $\mathbf{i}$ be a reduced expression of $w_0$. For a dominan…

Minimal valuation conjecture. If $C$ is sufficiently general and $s\geqslant 8+\frac{1}{36}$, then