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 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. In addition, the categorical viewpoint gives other insights such as providing convergence and approximation results for Mapper, a commonly used tool in TDA, as well as providing an understanding of these structures in higher dimensional settings. This work is joint with Vin de Silva, Amit Patel, Anastasios Stefanou, and Bei Wang.

Elizabeth Munch is an assistant professor in the Department of Mathematics and Statistics at the University at Albany. Her research specializes in Applied Topology and Topological Data Analysis, including work on persistent homology, statistics, Reeb graphs, and applied category theory. Liz received her PhD from the Department of Mathematics at Duke University in May 2013. She was a Postdoctoral Fellow at the Institute for Mathematics and its Applications at the University of Minnesota for the 2013-2014 thematic year on applications of topology. She also holds a Master of Arts in Mathematics from Duke University, a Bachelor of Science in Mathematics from the University of Rochester, and a Bachelor of Music in Harp Performance from the Eastman School of Music.