What is the difference between an empty glass and an open
empty bottle? Topologically there is none: they both are homeomorphic
to a disc. Geometrically, however, there is a difference: the glass is
flat (i.e. has zero Gaussian curvature everywhere), but the surface of
the bottle has points of positive, negative, and zero curvature.
Interestingly enough, it is possible to find this difference using
topological invariants: if we filter points by their curvature, and
then take the corresponding sublevel sets, some of them will be
topologically different: for example, the set of points of Gaussian
curvature not greater than zero is connected for the glass, but
disconnected for the bottle. This observation lies at the basis of
the idea of persistence in topological data analysis. I will
discuss this idea in more detail, and show other examples when this
approach gives us something reasonable.