CIMMS Lunchtime Series:
Elevation and Extended Persistence

Dr. Herbert Edelsbrunner, Duke University,
Arts and Sciences Professor of Computer Science and Mathematics

Given a smoothly embedded 2-manifold M in 3-dimensional Euclidean space, the elevation is an almost everywhere smooth function E: M --> R.  It is based on the 2-sphere of height functions and the idea of persistence used to pair up critical points.  There are different rules for pairing depending on whether a critical point is responsible for transient or permanent homology of M. For 2-manifolds, both rules are fairly elementary and can be explained in terms of branch-points and end-points of the Reeb graph.  There is an algebraic framework in which conventional persistence and extended persistence generalize the two rules to manifolds of dimension beyond two.