15 problems

Let $M$ be a simple rank-4 matroid on the $n$-element set $S_n$, and let $W_i(M)$ denote the number of rank-$i$ flats. Points-lines-planes conjecture. For every such matroid, … The…

Let $L$ be a finite geometric lattice, and let its Whitney numbers of the second kind be the rank-number sequence counting elements of each rank. Stanley's conjecture. The Whitney…

Dominance conjecture. If $T\subseteq T\circ\beta$, then $T\circ\beta$ dominates $T$.

Let $\mathrm{M}$ be a matroid, let $\mathcal{L}(\mathrm{M})$ be its lattice of flats, and let $\mathcal{L}(\mathrm{M})^*$ denote its dual poset. A TN-poset is a lower rank-uniform…

Let $\mathrm{M}$ be a matroid. A TN-poset is a lower rank-uniform poset whose lower triangular matrix of lower Whitney numbers of the second kind is totally nonnegative, and UMEL-s…

Athanasiadis–Kalampogia-Evangelinou's interlacing conjecture. The polynomial $h_L(t)$ is real-rooted and is interlaced by the Eulerian polynomial $A_n(t)$ for every geometric latti…

Mayhew–Newman–Welsh conjecture. Almost all geometric lattices are paving lattices, in the precise sense that

Upper combinatorial uniformity conjecture. Every upper combinatorially uniform geometric lattice is a TN-poset.

Geometry extremality conjectures.

A hyperplane arrangement over $\mathbb{C}$ is a finite collection of hyperplanes in a complex vector space, and its intersection poset is a geometric lattice; an operad in geometri…

Let $L^{(n)}$ be a sequence of primitive geometric lattices, meaning geometric lattices that cannot be written as products of smaller geometric lattices, and suppose that … Let…

Real-rootedness and Eulerian interlacing conjecture. The polynomial $h_{\mathcal L}(x)$ has only real roots and is interlaced by the Eulerian polynomial $A_n(x)$ for every geometri…

Rota's conjecture. The Whitney numbers $w_0,w_1,\dots,w_n$ of the first kind for every geometric lattice are log-concave.

Rota's conjecture. The rank-level sizes form a unimodal sequence: for some $p$,

Let $L$ be a geometric lattice. The coefficients of its characteristic polynomial are defined analogously to those of a matroid characteristic polynomial. Geometric-lattice extensi…