Quantifying Transversality by Measuring the Robustness of Intersections

By definition, transverse intersections are stable under infinitesimal perturbations. Using persistent homology, we extend this notion to a measure. Given a space of perturbations, we assign to each homology class of the intersection its robustness, the magnitude of a perturbation in this space necessary to kill it, and prove that robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for contours of smooth mappings. Communicated by Gunnar Carlsson AMS Subject Classification - 55N99 Keywords. Smooth mappings, transversality, fixed points, contours, homology, filtrations, zigzag modules, persistence, perturbations, stability.

• Type: Journal
• Year: 2011
• Where: FOCM
• Authors: Herbert Edelsbrunner, Dmitriy Morozov, Amit K. Patel
• Keywords: Continuous Mappings, FOCM 2011, Formal Methods, Gunnar Carlsson, Homology Class