20 problems

Equal matroids conjecture. The two join varieties have the same algebraic matroid:

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 $\tau$ be a binary tree on binary random variables, and let $I_f$ be the ideal of phylogenetic invariants of its general Markov model. Each edge of $\tau$ induces a split of th…

Let $T$ be a rooted tree, let ${\mathcal A}_T$ be the configuration of exponent vectors from the toric parametrization, and let $P_T$ be the convex hull of the columns of…

Let $T$ be a path of length $n$, and let $I_T$ be its toric ideal of phylogenetic invariants. The ideal $I_T$ should be generated in degree $3$, with $2n-4$ generators of degree…

Let $T$ be a rooted tree and let $I_T$ be the toric ideal associated to the homogeneous binary Markov model. Suppose either that $T$ is binary, or more generally that every non-lea…

Let mutation matrices be governed by a single order parameter $\theta$, and let $\theta(e)$ denote the parameter of the mutation matrix $M^e$ on edge $e$. Consider the Markov rando…

Let $d_r(n)$ be the maximum, over all binary phylogenetic trees with $n$ leaves, of the minimum number of $r$-state characters required to define the tree. Let $n_r$ be the least i…

Let $k$, $\delta$, and $h$ be the parameters of the Map-$\varepsilon$-PDD problem, where $h$ is the value of $H$ in a phylogenetic tree. Map-$\varepsilon$-PDD conjecture. Map-…

Let $T$ be a binary phylogenetic $X$-tree with $|X|=n$. For a character alignment $A_k(T)$ consisting of the binary characters of parsimony score $k$ on $T$, let…

Let $T$ be a binary phylogenetic $X$-tree with $|X|=n$. For a character alignment $A_k(T)$ consisting of the binary characters of parsimony score $k$ on $T$, let $k<\frac{n}{4}$. M…

Let $k\in\mathbb{N}_{\geq 3}$, let $X$ be a taxon set, and let $T$ and $\widetilde{T}$ be binary phylogenetic trees on $X$. For a binary phylogenetic tree $T$, let $A_k(T)$ denote…

Finite-character-subset conjecture. For each positive integer $r$, there exists an integer $f(r)$ such that, for any finite set $X$ and any set $C$ of $r$-state characters on $X$,…

Charleston and Steel's conjecture. For $T_k$ and $r\geq 2$,

Let $V^{(n)}_{a}(\alpha,0)$ and $V^{(n)}_{c}(\alpha)$ denote the anagenetic and cladogenetic contributions to the variance, and let $C^{(n)}_{a}(\alpha,0)$ and…

Let two groves be strictly informative if each satisfies the strict informativeness criterion for determining supertrees. Let the groves be collections of taxon sets, with intersec…

Let $\mathcal{S}$ be a set of taxon sets, and let a grove be a collection of taxon sets with the potential to construct informative supertrees. A grove is maximal in $\mathcal{S}$…

Let $S:=f_1f_2\ldots f_n$ be a sequence of characters (an alignment) on a set $X$ of taxa, with $|X|=m$, and let $\mathcal{T}$ be a maximum parsimony tree for $S$. For a subset…

Let $\mathcal{T}_n$ be the space of trees on $n$ leaves. For $T,T'\in\mathcal{T}_n$, let $d_{TBR}(T,T')$ denote the minimum number of tree-bisection-regrafting moves needed to tran…

For $x\in\{C,A\}$, let $(x,x)(t)$ and $x,x$ denote the corresponding pair-frequency functions in the RN+YpR model, and let $(x)_*$ be the stationary frequency of $x$. Diffeomorphis…