Colby Long

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  Rank Conditions for Phylogenetic Inference
  Colby Long

  Algebraic relations among DNA site-pattern frequencies have proven useful for inferring phylogenetic relationships between species. Often, these algebraic relations can be expressed as rank conditions on certain matrices constructed from phylogenetic data. In this talk, we will discuss how certain rank conditions for phylogenetic sequence data can be interpreted as conditional independence statements that give evidence of the evolutionary tree that produced the sequences. We will also present some new results that generalize the classic phylogenetic rank conditions to coalescent models of evolution. We show that rank-based reconstruction methods based on these results are robust to violations of model assumptions and perform well even in the presence of gene flow.

  
  

   
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  Algebraic Geometry of Phylogenetic Models
  Colby Long

  Phylogenetic statistical models can be used to determine evolutionary relationships between species. These models have the structure of algebraic varieties and understanding their geometry is important for proving many of their properties. For various biological reasons, phylogenetic mixture models are often a better model for species evolution. The variety of probability distributions associated to an r-tree mixture model is in general a join variety. In this talk, we examine some of the algebraic geometry underlying phylogenetic mixture models. In particular, we will show some computational tools for determining their generating sets.