Related papers: Markov invariants and the isotropy subgroup of a q…

Using the theory of representations of the symmetric group, we propose an algorithm to compute the invariant ring of a permutation group. Our approach have the goal to reduce the amount of linear algebra computations and exploit a thinner…

The aim of this review is to present and analyze the probabilistic models of mathematical phylogenetics which have been intensively used in recent years in biology as the cornerstone of attempts to infer and reconstruct the ancestral…

Inference of species networks from genomic data under the Network Multispecies Coalescent Model is currently severely limited by heavy computational demands. It also remains unclear how complicated networks can be for consistent inference…

The classical Fourier transform is, in essence, a way to take data and extract components (in the form of complex exponentials) which are invariant under cyclic shifts. We consider a case in which the components must instead be invariant…

Evolutionary models used for describing molecular sequence variation suppose that at a non-recombining genomic segment, sequences share ancestry that can be represented as a genealogy--a rooted, binary, timed tree, with tips corresponding…

Phylogenetic networks generalise phylogenetic trees and allow for the accurate representation of the evolutionary history of a set of present-day species whose past includes reticulate events such as hybridisation and lateral gene transfer.…

Less rigid than phylogenetic trees, phylogenetic networks allow the description of a wider range of evolutionary events. In this note, we explain how to extend the rank invariants from phylogenetic trees to phylogenetic networks evolving…

This dissertation presents a multifaceted look into the structural decomposition of permutation classes. The theory of permutation patterns is a rich and varied field, and is a prime example of how an accessible and intuitive definition…

Graph invariants are a useful tool in graph theory. Not only do they encode useful information about the graphs to which they are associated, but complete invariants can be used to distinguish between non-isomorphic graphs. Polynomial…

Phylogenetic tree shapes capture fundamental signatures of evolution. We consider ``ranked'' tree shapes, which are equipped with a total order on the internal nodes compatible with the tree graph. Recent work has established an elegant…

Tree Containment is a fundamental problem in phylogenetics useful for verifying a proposed phylogenetic network, representing the evolutionary history of certain species. Tree Containment asks whether the given phylogenetic tree (for…

An important problem in phylogenetics is the construction of phylogenetic trees. One way to approach this problem, known as the supertree method, involves inferring a phylogenetic tree with leaves consisting of a set $X$ of species from a…

Phylogenetic (evolutionary) trees and networks are leaf-labeled graphs that are widely used to represent the evolutionary relationships between entities such as species, languages, cancer cells, and viruses. To reconstruct and analyze…

How can we effectively find the best structures in tree models? Tree models have been favored over complex black box models in domains where interpretability is crucial for making irreversible decisions. However, searching for a tree…

We apply the theory of markov random fields on trees to derive a phase transition in the number of samples needed in order to reconstruct phylogenies. We consider the Cavender-Farris-Neyman model of evolution on trees, where all the inner…

The goal of these lectures is to survey some of the recent progress on the description of large-scale structure of random trees. We use the framework of Markov-Branching sequences of trees and discuss several applications.

Unrooted phylogenetic networks are graphs used to represent evolutionary relationships. Accurately reconstructing such networks is of great relevance for evolutionary biology. It has recently been conjectured that all phylogenetic networks…

As an alternative to parsimony analyses, stochastic models have been proposed (Lewis, 2001), (Nylander, et al., 2004) for morphological characters, so that maximum likelihood or Bayesian analyses may be used for phylogenetic inference. A…

We compare the phylogenetic tensors for various trees and networks for two, three and four taxa. If the probability spaces between one tree or network and another are not identical then there will be phylogenetic tensors that could have…

In this paper we study group-based Markov models of evolution and their mixtures. In the algebreo-geometric setting, group-based phylogenetic tree models correspond to toric varieties, while their mixtures correspond to secant and join…