I will survey of a wide variety of software packages for applied and computational topology.
We learn software best by running it on examples, and I will describe some of my favorite exercises for students, including sampling points from a torus, Klein bottle, or projective plane.
Two real-world datasets that can be analyzed with applied topology software include the three-circle model for optical image patches, and the conformation space of the cyclo-octane molecule (which is a Klein bottle glued to a sphere along two circles of singularities).
I’ll conclude by describing two applied topology online research seminars and YouTube channels, which allow one to remain connected to the community.
Data exhibiting complicated spatial structures are common in many areas of science (e.g. cosmology, biology), but can be difficult to analyze. Persistent homology is a popular approach within the area of Topological Data Analysis (TDA) that offers a way to represent, visualize, and interpret complex data by extracting topological features, which can be used to infer properties of the underlying structures. For example, TDA may be useful for analyzing the large-scale structure (LSS) of the Universe, which is an intricate and spatially complex web of matter.
The output from persistent homology, called persistence diagrams, summarizes the different ordered holes in the data (e.g. connected components, loops, voids). I will discuss how persistent homology can be used for inference for spatially complex data.
This talk gives an introduction to the R package TDA, which provides some tools for Topological Data Analysis. The R package TDA provides functions to sample on various geometric objects. It also provides functions that, given some data, provide topological information about the underlying space, such as distance functions and density functions. The salient topological features of data can be quantified with persistent homology.
The R package TDA provides an R interface for the efficient algorithms of the C++ libraries GUDHI, Dionysus, and PHAT for computing the persistent homology. Specifically, The R package TDA includes functions for computing the persistent homology of Rips complex, alpha complex, alpha shape complex, and a function for the persistent homology of sublevel sets (or superlevel sets) of arbitrary functions evaluated over a grid of points or on data points.
The R package TDA also provides functions for functional summaries of the persistent homology, such as the landscape function and the silhouette function. The R package TDA also provides a function for computing the confidence band that determines the significance of the features in the resulting persistence diagrams.
Topological data analysis can have widespread applications in fields whose practitioners are unfamiliar with algebraic topology. TDAstats is an R package that aims to lower the barrier for entry to the field of topological data analysis by permitting, even encouraging, introduction of persistent homology computation to non-experts. By wrapping the fast Ripser library in R, incorporating ggplot2-compatible visualizations, and providing basic tools for statistical inference of persistent homology, TDAstats provides an introductory, yet efficient and powerful, library for application of topological data analysis.