Qingsong Wang (06/19/24): Measure Theoretic Reeb Graphs
Abstract: A Reeb graph is a graphical representation of a scalar function on a topological space that encodes the topology of the level sets. In this talk, we introduce a novel construction of Reeb graphs incorporating a measure. Specifically, we introduce measure-theoretic Reeb graphs when the domain is modeled as a metric measure space (i.e.,~a metric space equipped with a measure). Our main goal is to enhance the robustness of the Reeb graph in representing the topological features of a scalar field while accounting for the distribution of the measure. We first introduce a Reeb graph with local smoothing and prove its stability with respect to the interleaving distance. We then prove the stability of a Reeb graph of a metric measure space with respect to the measure, defined using the distance to a measure or the kernel distance to a measure, respectively.