27 problems

Matroid-base quadraticity conjecture. The toric ideal $I_{P_B(\mathcal{M})}$ is quadratic.

Let $M$ be a dyadic matroid, meaning a matroid representable over the dyadic partial field. Let $P(M)$ be its matroid base polytope. Dyadic matroid triangulation conjecture. The ma…

Let $Q$ be an MV polytope of type $A$, and let $Q(\theta_I)$ denote the Schubert matroid polytope associated with $I\subseteq [n]$. For collections…

Let $M$ be a connected matroid, and let $P_M$ be its base polytope. Say that $P_M$ admits a series-parallel subdivision if it can be subdivided into base polytopes of series-parall…

A positroid is a matroid arising from a cell of the totally positive Grassmannian. A matroid base polytope is Ehrhart positive when all coefficients of its Ehrhart polynomial are n…

Let $a,b$ be positive integers. Let ${\texttt{Pan}}_{b+1,a+b,a+b+2}$ be the panhandle matroid, let $i(M,t)$ denote the Ehrhart polynomial of the matroid polytope of $M$, and let…

Geometry extremality conjectures.

Complete-graph graphic-matroid conjecture. For every $n\geq 1$, the graphic matroid $\mathsf{K}_n$ of the complete graph on $n$ vertices is an extremal matroid.

Direct-sum extremality conjecture. If $\mathsf{M}_1,\ldots,\mathsf{M}_s$ are extremal matroids, then $\mathsf{M}_1\oplus\cdots\oplus\mathsf{M}_s$ is extremal.

Let $\mathrm{U}_{r,n}$ be the uniform matroid of rank $r$ on $[n]$, and let $\Delta(\mathrm{U}_{r,n})$ be its matroid base polytope. A matroidal decomposition is a decomposition of…

Connected series-parallel subdivision conjecture. The polynomial $\wwtilde{g}_{\mathsf{M}}(t-1)$ has non-negative coefficients. This is a stronger positivity statement beyond Schub…

Let $\mathsf{M}$ be a matroid, and let $g_{\mathsf{M}}(t)\in\mathbb{Z}[t]$ denote its $g$-polynomial. Speyer's conjecture. For every matroid $\mathsf{M}$, the polynomial…

Let $\mathcal{S}$ be a subdivision of $\Delta_{k,n}$ into smaller matroid polytopes. For each $1\leq i\leq n$, let $f_i$ be the number of cells of $\mathcal{S}$ of dimension $n-i$…

For each positive integer $n$, let $\mathsf{S}_n$ be the sparse paving matroid on $[n]$ of rank $\lfloor n/2\rfloor$ with the maximal possible number of circuit-hyperplanes and lex…

Positroid Ehrhart-positivity conjecture. Every positroid is Ehrhart positive.

Eight-face conjecture. The facet of $\mathbf{N}_{3,n}$ minimizing $\gamma_{j_1,j_2,j_3}$ has exactly eight codimension-$3$ faces combinatorially isomorphic to

Let $\mathsf{M}$ be a matroid, and let $\mathscr{P}(\mathsf{M})$ be its matroid base polytope. Denote by $\operatorname{ehr}(\mathscr{P}(\mathsf{M}),t)$ its Ehrhart polynomial. Ehr…

De Loera–Haws–Köppe conjecture. The basis polytope $\mathscr{P}(M)$ of every matroid $M$ is Ehrhart positive.

A hypersimplex is the matroid polytope of a uniform matroid, and its dual polytope is the polytope dual to that hypersimplex. A shelling is extendable if every partial shelling can…

Let $M$ be a connected matroid of rank $k$ on $n$ elements. Write $i(M,t)$ for its Ehrhart polynomial, and let $T_{k,n}$ and $U_{k,n}$ denote respectively the minimal and uniform m…

Let $M$ be a matroid, and let $\mathscr{P}(M)$ denote its matroid polytope. The $h^*$-polynomial of $\mathscr{P}(M)$ is the polynomial whose coefficient vector is being considered.…

Let $M$ be a rank-$k$ matroid on $[n]:=\{1,\dots,n\}$. A tropical linear space is a $(k-1)$-dimensional balanced polyhedral complex in tropical projective space, dual to a coherent…

Let $M$ be a matroid, and let $\mathscr{P}(M)$ denote its basis polytope. Its Ehrhart polynomial is the polynomial counting lattice points in integer dilations of $\mathscr{P}(M)$.…

Mutation connectivity conjecture for uniform matroid polytopes. Every pair of uniform matroid polytopes of rank $4$ on the same ground set can be transformed into each other by a f…

Deloera–Haws–Köppe conjecture. For any matroid $M$, the matroid polytope $P(M)$ is both Ehrhart positive and Ehrhart unimodal.