In order to understand the properties of a real-valued function on a
topological space, we can study the Reeb graph of that function. Since
it is efficient to compute and is a useful descriptor for the function,
it has found its place in many applications. However, as with many other
constructions in computational topology, we are interested in how to
deal with this construction in the context of noise. In this talk, we
will define the interleaving distance for Reeb graphs, discuss
computational complexity issues arising from this definition and
potential directions for approximation. The interleaving distance also
provides other insights such as convergence and approximation results
for Mapper, a commonly used tool in TDA, as well as an understanding of
these structures in higher dimensional settings.