• Investigations in phylogenetics: tree inference and model identifiability

      Yourdkhani, Samaneh; Rhodes, John A.; Allman, Elizabeth S.; McIntyre, Julie; Williams, Gordon (2020-05)
      This thesis presents two projects in mathematical phylogenetics. The first presents a new, statistically consistent, fast method for inferring species trees from topological gene trees under the multispecies coalescent model. The algorithm of this method takes a collection of unrooted topological gene trees, computes a novel intertaxon distance from them, and outputs a metric species tree. The second establishes that numerical and non-numerical parameters of a specic Prole Mixture Model of protein sequence evolution are generically identifiable. Algebraic techniques are used, especially a theorem of Kruskal on tensor decomposition.
    • On the Klein-Gordon equation originating on a curve and applications to the tsunami run-up problem

      Gaines, Jody; Rybkin, Alexei; Bueler, Ed; Nicolsky, Dmitry (2019-05)
      Our goal is to study the linear Klein-Gordon equation in matrix form, with initial conditions originating on a curve. This equation has applications to the Cross-Sectionally Averaged Shallow Water equations, i.e. a system of nonlinear partial differential equations used for modeling tsunami waves within narrow bays, because the general Carrier-Greenspan transform can turn the Cross-Sectionally Averaged Shallow Water equations (for shorelines of constant slope) into a particular form of the matrix Klein-Gordon equation. Thus the matrix Klein-Gordon equation governs the run-up of tsunami waves along shorelines of constant slope. If the narrow bay is U-shaped, the Cross-Sectionally Averaged Shallow Water equations have a known general solution via solving the transformed matrix Klein-Gordon equation. However, the initial conditions for our Klein-Gordon equation are given on a curve. Thus our goal is to solve the matrix Klein-Gordon equation with known conditions given along a curve. Therefore we present a method to extrapolate values on a line from conditions on a curve, via the Taylor formula. Finally, to apply our solution to the Cross-Sectionally Averaged Shallow Water equations, our numerical simulations demonstrate how Gaussian and N-wave profiles affect the run-up of tsunami waves within various U-shaped bays.
    • Species network inference under the multispecies coalescent model

      Baños Cervantes, Hector Daniel; Allman, Elizabeth S.; Rhodes, John A.; Barry, Ronald; Faudree, Jill (2019-05)
      Species network inference is a challenging problem in phylogenetics. In this work, we present two results on this. The first shows that many topological features of a level-1 network are identifable under the network multispecies coalescent model (NMSC). Specifcally, we show that one can identify from gene tree frequencies the unrooted semidirected species network, after suppressing all cycles of size less than 4. The second presents the theory behind a new, statistically consistent, practical method for the inference of level-1 networks under the NMSC. The input for this algorithm is a collection of unrooted topological gene trees, and the output is an unrooted semidirected species network.