Connecting Persistent Homology Groups

Thursday, October 10, 2013 - 2:00pm - 2:50pm
Keller 3-180
Amit Patel (Institute for Advanced Study)
This talk is about the categorical structure of the persistent homology group. This is part of an upcoming paper with Robert MacPherson.

Given a function f: X -> R, persistence examines the evolution of the homology of the sublevel sets. There is a persistent homology group for each pair of values r M to manifolds. For now, we are calling the generalized persistent homology group the well group. To each open set of M, there is a well group. For a pair of nested open sets V subset U, we construct a morphism from the well group over U to the well group over V. For a triple of nested open sets W subset V subset U, the triangle consisting of the three well groups and three morphisms does not commute. However, it does commute up to a 2-morphism. We will use pictures to illustrate these ideas.