Foundations of Computational Mathematics

, Volume 11, Issue 3, pp 345–361 | Cite as

Quantifying Transversality by Measuring the Robustness of Intersections

  • Herbert Edelsbrunner
  • Dmitriy Morozov
  • Amit Patel
Article

Abstract

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 then we prove that the 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.

Keywords

Smooth mappings Transversality Fixed points Contours Homology Filtrations Zigzag modules Persistence Perturbations Stability 

Mathematics Subject Classification (2000)

55N99 

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Copyright information

© SFoCM 2011

Authors and Affiliations

  • Herbert Edelsbrunner
    • 1
    • 2
    • 3
  • Dmitriy Morozov
    • 4
  • Amit Patel
    • 1
    • 3
  1. 1.Departments of Computer Science and of MathematicsDuke UniversityDurhamUSA
  2. 2.GeomagicResearch Triangle ParkUSA
  3. 3.Institute of Science and Technology AustriaKlosterneuburgAustria
  4. 4.Departments of Computer Science and of MathematicsStanford UniversityStanfordUSA

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