Peter Bubenik

Persistence Diagrams of Cortical Surface Data

We present a novel framework for characterizing signals in images using techniques from computational algebraic topology. This technique is general enough for dealing with noisy multivariate data including geometric noise. The main tool is persistent homology which can be encoded in persistence diagrams. These diagrams visually show how the number of connected components of the sublevel sets of the signal changes. The use of local critical values of a function differs from the usual statistical parametric mapping framework, which mainly uses the mean signal in quantifying imaging data. Our proposed method uses all the local critical values in characterizing the signal and by doing so offers a completely new data reduction and analysis framework for quantifying the signal. As an illustration, we apply this method to a 1D simulated signal and 2D cortical thickness data. In case of the latter, extra homological structures are evident in an control group over the autistic group.

Categorification of Persistent Homology

We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are

Metrics for Generalized Persistence Modules

\mathbf {(\mathbb {R},\leq)}

A persistence landscapes toolbox for topological statistics

-indexed diagrams in some target category. A set of such diagrams has an interleaving distance, which we show generalizes the previously studied bottleneck distance. To illustrate the utility of this approach, we generalize previous stability results for persistence, extended persistence, and kernel, image, and cokernel persistence. We give a natural construction of a category of ε-interleavings of

Min-Type Morse Theory for Configuration Spaces of Hard Spheres

\mathbf {(\mathbb {R},\leq)}

A Model Category for Local Po-spaces

-indexed diagrams in some target category and show that if the target category is abelian, so is this category of interleavings.

Context for Models of Concurrency

We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of examples, inverse-image persistence modules, which occur whenever a topological space is mapped to a metric space. Several standard theories of persistence and their stability can be described in this framework. This includes the classical case of sublevelset persistent homology. We introduce a distinction between ‘soft’ and ‘hard’ stability theorems. While our treatment is direct and elementary, the approach can be explained abstractly in terms of monoidal functors.

Free and semi-inert cell attachments

Topological data analysis provides a multiscale description of the geometry and topology of quantitative data. The persistence landscape is a topological summary that can be easily combined with tools from statistics and machine learning. We give efficient algorithms for calculating persistence landscapes, their averages, and distances between such averages. We discuss an implementation of these algorithms and some related procedures. These are intended to facilitate the combination of statistics and machine learning with topological data analysis. We present an experiment showing that the low-dimensional persistence landscapes of points sampled from spheres (and boxes) of varying dimensions differ.

Higher Interpolation and Extension for Persistence Modules

We study configuration spaces of hard spheres in a bounded region. We develop a general Morse-theoretic framework, and show that mechanically balanced configurations play the role of critical points. As an application, we find the precise threshold radius for a configuration space to be homotopy equivalent to the configuration space of points.

Separated Lie Models and the Homotopy Lie Algebra

Locally partial-ordered spaces (local po-spaces) have been used to model concurrent systems. We provide equivalences for these spaces by constructing a model category containing the category of local po-spaces. We show the category of simplicial presheaves on local po-spaces can be given Jardines model structure, in which we identify the weak equivalences between local po-spaces. In the process we give an equivalence between the category of sheaves on a local po-space and the category ofbundles over a local po- space. Finally we describe a localization that should provide a good framework for studying concurrent systems.