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Reconstructing a parsimonious phylogenetic network that displays multiple phylogenetic trees is an important problem in theory of phylogenetics, where the complexity of the inferred networks is measured by reticulation numbers. The…
Understanding the population structure and patterns of gene flow within species is of fundamental importance to the study of evolution. In the fields of population and evolutionary genetics, measures of genetic differentiation are commonly…
Genetic sequences are known to possess non-trivial composition together with symmetries in the frequencies of their components. Recently, it has been shown that symmetry and structure are hierarchically intertwined in DNA, suggesting a…
Multiple genome alignment remains a challenging problem. Effects of recombination including rearrangement, segmental duplication, gain, and loss can create a mosaic pattern of homology even among closely related organisms. We describe a…
Phylogenetics begins with reconstructing biological family trees from genetic data. Since Nature is not limited to tree-like histories, we use networks to organize our data, and have discovered new polytopes, metric spaces, and simplicial…
Following genetic ancestry in eukaryote populations poses several open problems due to sexual reproduction and recombination. The history of extant genetic material is usually modeled backwards in time, but tracking chromosomes at a large…
DNA rearrangement processes recombine gene segments that are organized on the chromosome in a variety of ways. The segments can overlap, interleave or one may be a subsegment of another. We use directed graphs to represent segment…
Whether or not biodiversity dynamics tend toward stable equilibria remains an unsolved question in ecology and evolution with important implications for our understanding of diversity and its conservation. Phylo/population genetic models…
Grouping elements into families to analyse them separately is a standard analysis procedure in many areas of sciences. We propose herein a new algorithm based on the simple idea that members from a family look like each other, and don't…
A phylogenetic variety is an algebraic variety parameterized by a statistical model of the evolution of biological sequences along a tree. Understanding this variety is an important problem in the area of algebraic statistics with…
This thesis develops and expands upon known techniques of mathematical physics relevant to the analysis of the popular Markov model of phylogenetic trees required in biology to reconstruct the evolutionary relationships of taxonomic units…
We studied how to obtain a distribution for the number of ancestors in species of sexual reproduction. Present models concentrate on the estimation of distributions repetitions of ancestors in genealogical trees. It has been shown that is…
In a population with haploid reproduction any individual has a single parent in the previous generation. If all genealogical distances among pairs of individuals (generations from the closest common ancestor) are known it is possible to…
De novo assembly is the process of reconstructing the genome sequence of an organism from sequencing reads. Genome sequences are essential to biology, and assembly has been a central problem in bioinformatics for four decades. Until…
Rooted phylogenetic networks provide an explicit representation of the evolutionary history of a set $X$ of sampled species. In contrast to phylogenetic trees which show only speciation events, networks can also accommodate reticulate…
One way to understand the role history plays on evolutionary trajectories is by giving ancient life a second opportunity to evolve. Our ability to empirically perform such an experiment, however, is limited by current experimental designs.…
Phylogenetic networks are a generalization of phylogenetic trees to leaf-labeled directed acyclic graphs that represent ancestral relationships between species whose past includes non-tree-like events such as hybridization and horizontal…
The idea that all life on earth traces back to a common beginning dates back at least to Charles Darwin's {\em Origin of Species}. Ever since, biologists have tried to piece together parts of this `tree of life' based on what we can observe…
Genotype-to-phenotype mappings translate genotypic variations such as mutations into phenotypic changes. Neutrality is the observation that some mutations do not lead to phenotypic changes. Studying the search trajectories in genotypic and…
How do mammalian cells that share the same genome exist in notably distinct phenotypes, exhibiting differences in morphology, gene expression patterns, and epigenetic chromatin statuses? Furthermore how do cells of different phenotypes…