381 problems

Let $\Sigma(r,n)$ be the hypersimplex, and consider a subdivision of $\Sigma(r,n)$ into matroid base polytopes. An interior face is a face of the subdivision not contained in the b…

Let $G$ be a multigraph, and let $\Gamma$ be the metric multigraph obtained from $G$ by assigning length $1$ to every edge. Let the Baker–Norine rank of a divisor be computed eithe…

Let $QH=\mathbb{C}P^1\times\mathbb{C}P^1$ and let $\Sigma_2$ be the second Hirzebruch surface. For integers $a,b\geq 0$, let $\square_{a,b}$ and $\triangle_{a,b}$ denote the homolo…

Consider the Gross–Siebert construction of a family of Calabi–Yau manifolds from an integral affine manifold with singularities and its associated tropical geometry; let $B$ denote…

Passare–Rullgård's solid amoeba conjecture. The amoeba of a maximally sparse polynomial should be solid, meaning that

Dimension-equality conjecture. For every $g\geq 0$ and $d\geq 1$,

Tropical Brill–Noether cycle conjecture. Assume $\rho\geq 0$. Then there exists a canonical tropical subvariety $Z^r_d\subseteq W^r_d$ of pure dimension $\rho$ such that

Non-crossing chord diagram conjecture. The regions contributing to $A_n^{phi^4}$ are in bijection with the $C_{n/2-1}$ possible non-crossing chord diagrams. The region associated w…

Let $X$ admit a Lagrangian torus fibration … and let the semi-flat SYZ mirror carry Maurer–Cartan solutions whose Fourier modes are taken along its Lagrangian torus fibers. Near a…

Let $V\subseteq\mathbb{C}^n\times(\mathbb{C}^{\times})^n$ be an algebraic variety, and let $\pi_1$ and $\pi_2$ be the projections to the additive and multiplicative factors. Call…

Let $\mathcal{A}_{n,k,2}$ be the $m=2$ amplituhedron, and let $\operatorname{Trop}^+{\rm Gr}(k+1,n)$ denote the positive tropical Grassmannian. A regular positroidal subdivision is…

Mikhalkin–Zharkov conjecture. The map above is an isomorphism for all $p\geq q$.

Let $(X,\mathcal E)$ be a compact non-singular real tropical hypersurface. Let ${\mathcal K}_1,\dots,{\mathcal K}_n$ and ${\mathcal S}_{\mathcal E}$ be the coefficient systems appe…

Ardila–Develin's correspondence conjecture. Tropical oriented matroids are in bijection with subdivisions of

Let $V$ be a subvariety of the torus $(\mathbb{C}^\times)^n$ of codimension $k{+}1$, and let its amoeba be the image under the coordinatewise map $z\mapsto \log|z|$. A subset…

Let $\tau$ be a binary tree on binary random variables, let $f$ be the associated multilinear map, and let $I_f$ be its prime ideal. For a $2 \times \cdots \times 2$ table, flatten…

Let $\mathbb R^n$ carry coordinate-wise tropical addition, and let an additive tropical set-variety mean an additive tropical set-variety in the sense of the source. A tropical set…

Let ${\mathbb T}=\mathbb R\cup\{-\infty\}$ be the tropical semifield, with tropical addition and multiplication given by $a\oplus b=\max\{a,b\}$ and $a\odot b=a+b$. A tropical poly…

Tropical Torelli conjecture. The map from the moduli space of tropical curves to $\mathcal{J}$ is tropical of degree $1$.

Let $L$ be an $(n-k)$-dimensional linear subspace of $K^n$ whose tropical Plücker coordinates are all finite, and let $M\in K^{m\times n}$, with $m\geq k$, have nonzero rows in the…

Tropical halfspace characterization conjecture. A tropical polytope $P$ is pure and full dimensional if and only if it has a halfspace description whose apices are in general posit…

Tropical halfspace representation conjecture. If $P$ is pure, then the halfspaces from a generic lift $\overline{P}$ of $P$ map to tropical halfspaces whose intersection is $P$ its…

The f-vector conjecture. The subdivision $\mathcal{D}_P$ has at most

Let $C$ be a curve of support $I$ and Newton polygon $Delta$, and let $\Gamma$ be the refinement of the subdivision of $Delta$. Let $q_1,\ldots,q_{\delta-1}$ be points in $C$. The…

Maximal-stratum local tropical zero-set conjecture. The auxiliary polynomial is unnecessary on the maximal stratum, in the sense that