Categorification of Reeb Graphs

Thursday, March 6, 2014 - 10:15am - 11:05am
Keller 3-180
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. The Reeb graph is a construction which summarizes the connectivity of the level sets. 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 which can be used to compare them. This also gives an immediate method for topological smoothing and we will discuss an algorithm for computing this smoothed Reeb graph.

This is joint work with Vin de Silva and Amit Patel.
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