Persistent homology and microlocal sheaf theory

Understand

We interpret some results of persistent homology and barcodes (in any dimension) with the language of microlocal sheaf theory.

• For that purpose we study the derived category of sheaves on a real finite-dimensional vector space V.
• By using the operation of convolution, we introduce a pseudo-distance on this category and prove in particular a stability result for direct images.
• Then we assume that V is endowed with a closed convex proper cone $\gamma$ with non empty interior and study $\gamma$-sheaves, that is, constructible sheaves with microsupport contained in the antipodal to the polar cone (equivalently, constructible sheaves for the $\gamma$-topology).