Geometric data analysis for phylogenetic trees and non-contractible manifolds

A phylogenetic tree is an acyclic graph with distinctly labeled leaves, whose internal edges have a positive weight. Given a set {1,2,...,n} of n leaves, the collection of all phylogenetic trees with this leaf set can be assembled into a metric cube complex known as phylogenetic tree space, or Billera-Holmes-Vogtmann tree space, after [9]. In Chapter 2, we show that the isometry group of this space is the symmetric group S [subscript n]. This fact is relevant to the analysis of some statistical tests of phylogenetic trees, such as those introduced in [11]. In Chapter 3, co-authored with Megan Owen, we give a rigorous framework for comparing trees in different moduli spaces of phylogenetic trees, and apply this to define extension spaces of trees, a conservative split-based supertree construction method, and two measures of compatibility between tree fragments. In Chapter 4, we discuss some techniques in manifold learning, and outline a new topologically-constrained nonlinear dimensionality reduction algorithm, which quickly reduces a nerve complex build on local tangent space approximations to produce a small number of manifold charts, visualized by a collection of least squares alignments of contractible components. We also give a method to optimize tangent space alignment on a sphere, and a template for using local tensor decomposition of higher-order moments to extend this technique to intersecting and stratified manifolds.