Toric topology and persistent homology

Based on the results of Baskakov, Buchstaber, Panov, and Franz, one possible way to apply toric topology in topological data analysis would be to define and study the properties of bigraded persistent (co)homology and bigraded barcode of a data set using (co)homology of (real) moment-angle-complexes over the corresponding Vietoris-Rips complexes.
However, as we will show in the talk, this version of a bigraded barcode satisfies only a weaker version of the stability theorem. Applying the so called double cohomology of a moment-angle-complex instead of the ordinary one, we will fix this problem and introduce such a construction of the bigraded persistent (co)homology that the corresponding bigraded barcode satisfies the stability theorem.
Based on the ongoing research project joint with A.Bahri, T.E.Panov, J.Song, and D.Stanley.