9 problems

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

Flag nested-set real-rootedness conjecture. If $\mathbf{N}(\mathcal{L},\mathcal{G})$ is flag, then its $h$-polynomial is real-rooted.

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…

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

Geometry extremality conjectures.

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…

Let $L$ be a geometric lattice of rank $r$. The Whitney numbers of the second kind are the integers $W_i(L)$ for $0\leq i\leq r$. Rota's unimodality conjecture. For some…

Let $L$ be a geometric lattice of rank $r$. The Whitney numbers of the first kind are the integers $w_i(L)$ for $0\leq i\leq r$. Rota's unimodality conjecture. For some…