FoCM 2014 conference
Workshop B2 - Computational Topology and Geometry
December 16, 15:30 ~ 15:55 - Room C11
Persistent Objects
Amit Patel
Institute for Advanced Study, USA - amit@ias.edu
For a continuous map f:X→R to the reals, there is a persistent homology group for each interval [r,s]. If Xr is the r-sublevel set of f and Xs the s-sublevel set of f, then the persistent homology group is image of the homomorphism Hd(Xr)→Hd(Xs) induced by the inclusion Xr⊂Xs. This is the homology that spreads out over the interval [r,s]. Recently, it has been shown that the persistent homology group can be defined for any map f:X→M, where M is an oriented manifold. If U is an open set of M, then the persistent homology over U is a subgroup of Hd(f−1(U)) that spreads out over U. The notion of persistent homology categorifies. Let F:D→C be a diagram in a category C. Under some mild assumptions on C, there is a notion of a persistent object for F. This is the object in C that spreads out over then entire diagram. In this talk, I will present the persistent homology group of maps to the reals, the persistent homology group for maps to any oriented manifold, and the persistent object.