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 $f : X \rightarrow R$ to the reals, there is a persistent homology group for each interval $[r,s]$. If $X_r$ is the $r$-sublevel set of $f$ and $X_s$ the $s$-sublevel set of $f$, then the persistent homology group is image of the homomorphism $H_d(X_r) \rightarrow H_d(X_s)$ induced by the inclusion $X_r \subset X_s$. 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 $f : X \rightarrow M$, where $M$ is an oriented manifold. If $U$ is an open set of $M$, then the persistent homology over $U$ is a subgroup of $H_d(f^{-1}(U))$ that spreads out over $U$. The notion of persistent homology categorifies. Let $F: D \rightarrow C$ 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.