Categorification of Reeb Graphs

Tuesday, October 22, 2013 - 1:30pm - 2:30pm
Lind 305
Elizabeth Munch (University of Minnesota, Twin Cities)
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. As with many other constructions in
computational topology, we are interested in how to deal with this
construction in the context of noise. In particular, we would like a
method to smooth out the topology to get rid of, for example, small loops
in the Reeb graph.

In this talk, we will define a generalization of a Reeb graph as a
functor. Using the added structure given by category theory, we can define
interleavings on Reeb graphs. This also gives an immediate method for
topological smoothing and we will discuss an algorithm for computing this
smoothed Reeb graph.