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Multiparameter Persistent Homology for Shape Comparison: Continuous versus Discrete

Wednesday, March 19, 2014 - 1:30pm - 2:30pm
Lind 305
Tomasz Kaczynski (University of Sherbrooke)
The theory of multiparameter persistent homology was initially
developed in the discrete setting of filtered simplicial complexes.
Stability of persistence was proved for topological spaces filtered by
continuous vector-valued functions. Our aim is to provide a formal
bridge between the continuous setting, where stability properties hold,
and the discrete setting, where actual computations are carried out. The
existence of this bridge is not obvious for two reasons. One is related
to the Sard’s lemma, namely, we do not have the isolation of critical
values for generic vector functions. The other one is due to the
phenomenon of structural gap between the two settings which appears in
the multi-parameter case when using the standard piecewise linear
interpolation of the discrete model.