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Computational topology, the new frontier.Geometry/Topology
|Speaker: ||Pawel Dlotko, University of Pennsylvania|
|Location: ||3106 MSB|
|Start time: ||Thu, Jan 8 2015, 4:10PM|
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.