Abstract:  

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