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Description

Computational topology has recently became an important tool in the applied science. Until now the state of the art methods allowed to construct a filtered simplicial or cubical complexes and to compute persistence diagrams or Betti numbers or a representative cycles of those complexes by using matrix reduction methods. All the computations were made on a single computer with only very basic parallel methods. There was no way to do a statistical analysis of larger families of results. Still, this basic scheme has already provided very good results and therefore further development of the filed is currently taking place. In this talk I will present my idea of a new frontier in computational topology. First I will address the problem of scaling up the computations we are able to perform by using a distributed divide and conquer scheme to compute homology and persistent homology of very large complexes. Unlike the computations so far, this approach allows computations on clusters of computers for much larger data sets. Later I will address the problem of rigorous computation of homology and persistent homology of level sets of a continuous scalar function. This approach allows mathematically accurate transition from the continuous to discrete setting and open new perspectives in numerical analysis and beyond. Finally I will present a Persistence Landscapes Toolbox - a set of ready to use programs based on the idea of Persistence Landscapes allowing to do basic statistical operations on persistence modules. I will show how to compute averages, standard deviations, distances between persistence modules, and even how to build a simple classifier that bases on persistence.