Amit Patel (University of Minnesota (IMA))
Persistent Sheaves for Stratified Maps
Given a stratified map f : X -> Y, we may study the homology (relative homology) of its fibers by studying its Leray cosheaf (sheaf), one for each dimension. The Leray cosheaf assigns to each sufficiently small open set U of Y the ordinary homology of the pre-image f^{-1}(U). The Leray sheaf assigns to each sufficiently small open set U of Y the relative homology of the pair of spaces (X, X - f^{-1}(U)). For a stratified map g : X -> R to the real line, the indecomposable summands of the Leray cosheaf (sheaf) are in one-to-one correspondence with the appropriate (level-set zigzag) persistence diagram of g. It is therefore tempting to define the persistence diagram of an arbitrary stratified map f : X -> Y as the list of indecomposable summands associated to the Leray sheaf or Leray cosheaf of f. However, the Leray cosheaf (sheaf) is in general not stable to perturbations to f. For example, an arbitrarily small perturbation to f may introduce a hole (a point with a trivial co-stalk) well in the interior of the support of the Leray cosheaf (sheaf).
In this talk, I will present the persistent sheaf for any stratified map f : X -> M to an oriented manifold M. Roughly speaking, the persistent sheaf is a 2-functor that assigns to each sufficiently small open set U of M, a subgroup of the homology of the pre-image f^{-1}(U). I will give a rough construction of the persistent sheaf and show how it is stable to sufficiently small perturbations to f.
This is joint work with Robert MacPherson (IAS).