Abstract:  

"For a function f on a topological space X, Cohen–Steiner–Edelsbrunner–Harer introduced the notion of the extended barcode of f. It is a combinatorial construct that compactly stores information about,
in particular, the homology of any interlevel set f^{-1}([a,b]) (over a field F). When X is a closed manifold and f is a Morse function without multiple critical values, we propose an equivalent, geometrical definition of the extended barcode.
It relies on a Bruhat decomposition, which describes the relative position of two complete flags in a vector space. As an important by-product of this approach, we naturally associate a number (i.e. an element of F)
with each bar. It turns out that, in certain cases, the product of all these numbers does not depend on the function. In one such case, this product equals the Reidemeister torsion of X. Time permitting, we will also discuss
the behaviour of these objects in families of functions. Based on several papers, one of which is co-authored with P. Pushk"