Clemens Bannwart: Tagged barcodes for vector fields and their stability

Speaker
Clemens Bannwart, Università di Modena e Reggio Emilia

Abstract
We construct an invariant called ‘tagged barcode’ that can be assigned to a gradient-like Morse-Smale vector field on a manifold satisfying some conditions. We start by considering the Morse complex, which is a chain complex defined in terms of singular points and flow lines of the vector field and whose homology is isomorphic to the homology of the underlying manifold. We then identify a sequence of pairs of singular points along which we can simplify the Morse complex. Recording the distances between the pairs we simplified yields the tagged barcode. This procedure is locally stable in the sense that it is a continuous map if we consider the Whitney \(C^1\) topology for vector fields and the topology induced from the bottleneck distance for tagged barcodes. We sketch the proof of this result and explain why global stability does not hold.