🍩 Database of Original & Non-Theoretical Uses of Topology

(found 2 matches in 0.001039s)

  1. Persistent Homology Analysis of Brain Artery Trees (2016)

    Paul Bendich, J. S. Marron, Ezra Miller, Alex Pieloch, Sean Skwerer
    Abstract New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number of representations of each data tree arise from persistence diagrams that quantify branching and looping of vessels at multiple scales. Novel approaches to the statistical analysis, through various summaries of the persistence diagrams, lead to heightened correlations with covariates such as age and sex, relative to earlier analyses of this data set. The correlation with age continues to be significant even after controlling for correlations from earlier significant summaries.

  2. Fruit Flies and Moduli: Interactions Between Biology and Mathematics (2015)

    Ezra Miller
    Abstract Possibilities for using geometry and topology to analyze statistical problems in biology raise a host of novel questions in geometry, probability, algebra, and combinatorics that demonstrate the power of biology to influence the future of pure mathematics. This expository article is a tour through some biological explorations and their mathematical ramifications. The article starts with evolution of novel topological features in wing veins of fruit flies, which are quantified using the algebraic structure of multiparameter persistent homology. The statistical issues involved highlight mathematical implications of sampling from moduli spaces. These lead to geometric probability on stratified spaces, including the sticky phenomenon for Frechet means and the origin of this mathematical area in the reconstruction of phylogenetic trees.