63 problems

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…

Sullivant's conjecture. For any finite abelian group $G$, $\phi(G)\leq |G|$.

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 $T$ be a binary tree, let ${\mathcal A}_T$ be the configuration whose columns are the exponent vectors of the toric parametrization, and let $P_T$ be the convex hull of the col…

Let $T$ be a binary tree with $n$ nodes, and let $I_T$ be the toric ideal of phylogenetic invariants associated to the homogeneous binary Markov model. A generating set of $I_T$ sh…

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…

Maximally probable topology conjecture with simultaneity. The unlabeled topology whose labelings have the largest number of tie-permitting labeled histories has the form in Theorem…

Maximal-topology conjecture. The topology in Theorem Hammersley is unique in having the maximal total number of tie-permitting labeled histories. Moreover, for every

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-…

STEAC inferiority conjecture. In the presence of substitution uncertainty, STEAC is uniformly inferior to METAL for all $$ .

Let $\cal{E}^\mathcal{M}_n$ be the linear subspace of $\otimes^n \mathbb{C}^4$ cut out by all linear homogeneous model invariants, and let $\cal{Z}^\mathcal{M}_n$ be the linear sub…

The paper considers binary phylogenetic trees $T$, the alignment $A_k(T)$ of binary characters of parsimony score $k$ on $T$, and unique maximum-parsimony recovery within the NNI n…

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 $\mathcal{S}$ be a set of taxa on characters $\mathcal{C}$. A perfect transfer network (PTN) for $\mathcal{S}$ is a network with an $\mathcal{S}$-map $sigma$ that explains…

Let $T$ be a tree with Fitch-labeling $ambda$, and let $G$ and $l$ be the network and labeling produced by Algorithm. For each transfer edge of $G$, remove the edge and any resulti…

Let $T$ be a gene tree, $S$ a species tree, and $sigma$ a leaf-association map. For a prescribed set of reconciliation edges $H\subseteq E(T)$, consider whether there exists a reco…

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…

Let $T$ be a binary phylogenetic tree with $n$ leaves, and let $A_k(T)$ denote the set of binary characters requiring precisely $k$ nucleotide substitutions on $T$. A tree is a max…

Let $G$ be a graph with treewidth at least $4$, and suppose that $G$ has a ladder $L$ of length $2$ or longer. The preceding discussion gives examples showing that reducing such a…

Outlier-spectrum conjecture. A tree with outliers has more outlier $J_D$-eigenvalues.