Anastasios Stefanou

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  Partially Ordered Reeb Graphs, Tree Decompositions, and Phylogenetic Networks
  Anastasios Stefanou
  Inspired by the interval decomposition of persistence modules and the extended Newick format of phylogenetic networks, we show that, inside the larger category of partially ordered Reeb graphs, every Reeb graph with n leaves and first Betti number s, is equal to a coproduct of at most 2s trees with (n+s) leaves. An implication of this result, is that Reeb graphs are fixed parameter tractable when the parameter is the first Betti number. We propose partially ordered Reeb graphs as a natural framework for modeling time consistent phylogenetic networks. Furthermore, we define an interleaving distance on partially ordered Reeb graphs. Originated in Topological Data Analysis, the notion of interleaving distance can be defined on an arbitrary category with a notion of a flow. In contrast with other metrics for phylogenetic networks, the interleaving distance  gives us the ability to compare a pair of time consistent phylogenetic networks having, either different labelings on their leaves, or having some of their leaves labeled, or a combination of both, where the labelings are viewed as a partial order on the vertex set of the network. When the partial orders of a pair of Reeb graphs are trivial (no labeling) we obtain the interleaving distance for ordinary Reeb graphs.
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  Applied topology for phylogenetic networks
  Anastasios Stefanou

  In mathematical phylogenetics we are interested in developing mathematical methods that model the structure of evolutionary relationships among biological organisms known as phylogenetic networks. 

  
  

  Such models are often used to define metrics for comparison, that can be used for developing statistics and probability distributions on these structures. In practice we often encounter tree-like relationships called phylogenetic trees whose structure is already very well studied. 

  
  

  With the cophenetic map (Sokal and Rohlf, 1962) the isomorphism classes of phylogenetic trees with n leaves embed as vectors in the [n(n+1)/2]-dimensional Euclidean space. There we pull back l^p norms to define metrics on these trees. However quite few is known for the general case of phylogenetic networks. In this talk I will introduce some concepts from applied topology, namely categories, coproducts and Reeb graphs. I will discuss why Reeb graphs is the natural model for phylogenetic networks. 

  
  

  We show that any phylogenetic network with n-labelled leaves and s cycles has a canonical decomposition as a join of a set of phylogenetic trees with (n + s)-labelled leaves. By combining this tree-decomposition and the cophenetic map phylogenetic networks with n-labelled leave s cycles embed as point-clouds in the [(n+s)(n+s+1)/2]-dimensional Euclidean space. Thinking of phylogenetic networks as point-clouds we can study topological signatures such as  persistence diagrams.