7 problems

Let $T\in\widehat{\mathcal{RBT}}_n$ be a fixed phylogenetic tree, and let $T'$ be a random tree differing from $T$ by a single subtree-prune-and-regraft (SPR) move, chosen uniforml…

Tropicalization conjecture. The tropicalization $\operatorname{Trop}\mathcal{M}_{\mathrm{C}_{n-1}}$ is the space of ASPTs: the fan formed by the cones $C_{T,\varphi}$ with…

Let $T_1$ and $T_2$ be equidistant rooted phylogenetic trees with $n$ leaves in $X$, and let $\Gamma_{T_1,T_2}$ be the tropical line segment between them. A nearest neighbor interc…

Let ${\mathcal S}$ and ${\mathcal T}$ be balanced rooted binary phylogenetic $X$-trees, meaning that each has $2^m$ leaves for some non-negative integer $m$ and height $m$, and let…

The cluster-count criterion. One has

Let $T_1$ and $T_2$ be balanced, rooted, binary trees of height $m$. An agreement subtree is a common subtree of the two trees on a shared set of leaves. The square-root agreement-…

Weighted-tree tropical Grassmannian conjecture. For $m\geq 5$, one has