On Interleaving Distance between Reeb Trees as 𝑹-Pospaces

Abstract: A Reeb graph is one of the mathematical tools to summarize topology of real-valued functions. Silva, Munch and Patel proposed the interleaving distance between two Reeb graphs as 𝑹-spaces. Although this metric is useful to show the stability, the estimation is rough in terms of data resolution. In this talk we will introduce an interleaving distance between tree pospaces, which can be also used to prove the stability of Reeb trees. A pospace, a partially ordered space, is a poset endowed with a compatible topology. By considering the compatibility of both topology and order, we can achieve sharp evaluation by incorporating data resolutions as discrete order structures, while maintaining the comparison of “soft” topological structures of trees. Furthermore, the metric can compare discrete and/or continuous structured 𝑹-pospaces, which is expected to be useful for evaluating the convergence.

computational geometryalgebraic topologycombinatoricsgeometric topologyprobability

Audience: advanced learners

Asia Pacific Seminar on Applied Topology and Geometry