Related papers: Point configurations, phylogenetic trees, and diss…
Let T be a weighted tree with n numbered leaves and let D be its distance matrix, so D(i,j) is the distance between the leaves i and j. If m is an integer between 2 and n, we prove a tropical formula to compute the m-dissimilarity map of T…
We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all \ ultrametrics. The ${\rm CAT}(0)$-metric of Billera-Holmes-Vogtman arises from the…
Phylogenetic trees are the fundamental mathematical representation of evolutionary processes in biology. They are also objects of interest in pure mathematics, such as algebraic geometry and combinatorics, due to their discrete geometry.…
While the positive Grassmannian is deeply understood through the rich combinatorics of plabic graphs and positroid cells, its tropical counterpart, the positive tropical Grassmannian Trop$_{>0}G(k,n)$, has lacked a comparable structural…
We introduce and study three different notions of tropical rank for symmetric and dissimilarity matrices in terms of minimal decompositions into rank 1 symmetric matrices, star tree matrices, and tree matrices. Our results provide a close…
Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical…
Ecological studies have now gone beyond measures of species turnover towards measures of phylogenetic and functional dissimilarity with a main objective: disentangling the processes that drive species distributions from local to broad…
The metric space of phylogenetic trees defined by Billera, Holmes, and Vogtmann, which we refer to as BHV space, provides a natural geometric setting for describing collections of trees on the same set of taxa. However, it is sometimes…
Let $\mathbf{v}_1,\ldots,\mathbf{v}_m$ be points in a metric space with distance $d$, and let $w_1,\ldots,w_m$ be positive real weights. The weighted Fermat-Weber points are those points $\mathbf{x}$ which minimize $\sum w_i d(\mathbf{v}_i,…
In a metric space, the Fermat-Weber points of a sample are statistics to measure the central tendency of the sample and it is well-known that the Fermat-Weber point of a sample is not necessarily unique in the metric space. We investigate…
Ducros, Hrushovski, and Loeser gave maps from families of archimedean diffrential forms to non-archiemedean (or tropical) ones, which are compatible with integrals on algebraic varieties. In this paper, we introduce slight modifications of…
We develop a combinatorial framework to study certain polyhedral maps which are higher-dimensional analogues of tropical covers between metric graphs. Under a mild combinatorial assumption, we show that a map satisfies the so-called…
We investigate a canonical extension of a conventional combinatorial notion of reduced divisors to a notion of tropical projections, which can be defined as the unique minimizers of the so-called $B$-pseudonorms with respect to compact…
In this paper we introduce and study three new measures for efficient discriminative comparison of phylogenetic trees. The NNI navigation dissimilarity $d_{nav}$ counts the steps along a "combing" of the Nearest Neighbor Interchange (NNI)…
We study tropicalisations of quasi-automorphisms of cluster algebras and show that their induced action on the g-vectors can be realized by tropicalising their action on the homogeneous $\hat{y}$ (or $\mathcal{X}$) variables of a chosen…
We show that the number of combinatorial types of clusters of type $D_4$ modulo reflection-rotation is exactly equal to the number of combinatorial types of tropical planes in $\mathbb{TP}^5$. This follows from a result of Sturmfels and…
We study the conditions under which the isometry of spaces with metrics generated by weights given on the edges of finite trees is equivalent to the isomorphism of these trees. Similar questions are studied for ultrametric spaces generated…
We study a parametric version of the Fermat-Weber problem with respect to an asymmetric distance function, which occurs naturally in tropical geometry. Our results yield a method for constructing phylogenetic supertrees.
We introduce tree dimension and its leveled variant in order to measure the complexity of leaf sets in binary trees. We then provide a tight upper bound on the size of such sets using leveled tree dimension. This, in turn, implies both the…
Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and…