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 sizeable perturbations. Specifically, we assign to each homology class of the intersection its robustness, the magnitude of a perturbation 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.