Peter Bubenik

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  Topological spaces of persistence modules and their properties
  Peter Bubenik

  Persistence modules are a central algebraic object arising in topological data analysis. Interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules and the relationships between them. We also study the resulting topological spaces and their basic topological properties.

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  Towards statistical topology: homology, persistent homology and persistence landscapes
  Peter Bubenik
  One of the principal uses of topology is to patch together local quantitative data to obtain global qualitative information not readily accessible to other methods. While the early development of topology was largely driven by applications, many later advances were motivated by strictly mathematical concerns. Now the field of applied topology is returning topology to its roots, adapting some of the later advances in topological methods to current questions in applications. I will survey some of the central constructions in topological data analysis, introducing homology and persistent homology.
  
  There is a clear need to combine these tools with statistical analysis. However there are difficulties in doing so, as the space of the usual topological descriptor is not a manifold. I define a new topological descriptor, the persistence landscape, whose definition allows for the calculation of means and standard deviations, laws of large numbers, central limit theorems and hypothesis testing.
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  Higher Interpolation and Extension for Persistence Modules
  Peter Bubenik
  Persistence modules are the central algebraic object in topological data analysis. This
  motivates the study of the geometry of the space of persistence modules. We isolate an elegant
  coherence condition that guarantees the interpolation and extension of sets of persistence
  modules. This "higher interpolation" is a consequence of the existence of certain universal
  constructions. As an application, it allows one to compare Vietoris-Rips and Cech complexes
  built within the space of persistence modules. This is joint work with Vin de Silva and Vidit
  Nanda.
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  Topological analysis of biological data using persistence landscapes
  Peter Bubenik
  One approach to combining geometry, topology and statistics in the analysis of data consists of the following steps: (1) use the data to construct a geometric object; (2) apply topology to obtain a summary; and (3) apply statistics to the resulting summaries. From a statistical viewpoint, it is fruitful to replace the standard topological summary, the persistence diagram, with a vector (or better yet, a point in a Hilbert space). One such construction with particularly nice properties (e.g. reversability) is the persistence landscape. I will give an overview of this pipeline and apply it to analyze protein data and brain imaging data.