Discrete Morse theory, persistent homology and Forman-Ricci curvature (Q2171716)

The author recalls a polyhedral Morse theory that is based on an index for critical points. On a 2-dimensional surface the index is +1 for minima and maxima, and it is \(-1\) for ordinary saddle points, \(-2\) for a monkey saddle, \(-3\) for a dog saddle and so on, as described by \textit{T. F. Banchoff} [Am. Math. Mon. 77, 475--485 (1970; Zbl 0191.52801)]. The barycentric subdivision of the underlying complex makes the function into a discrete Morse function in the sense of R. Forman, and critical cells in Forman's sense correspond to critical barycentres in Banchoff's sense. This correspondence is attributed to E. Bloch. While networks are considered as graphs, hypernetworks are considered as hypergraphs and, ultimately, as simplicial complexes. This article aims at a persistent homology scheme for networks and hypernetworks. Among other tools, Forman's discrete Ricci curvature is used. However, the real applications are only promised in a section ``Discussion and future work''. Concerning various versions of PL Morse theory, the reader may compare the forthcoming paper by \textit{R. Grunert} et al. [Adv. Geom. 23, No. 1, 135--150 (2023; Zbl 07655032)].