分野別セミナー

Persistent Betti numbers and persistence diagrams are expressions of a persistent homology, which are very important tools to understand topological features of data. In this talk, we introduce filtered complexes built over finite marked point data (configurations) on Euclidean spaces. Marks mean additional information of data. Given a marked point data, we define a filtration (increasing sequence) of simplicial complexes with the vertex sets in the data by assigning the birth time for each simplex. Examples of our filtered complexes include filtered $\check{C}$ech complexes (nerves) of a family of sets with various sizes, growths, and shapes. We discuss laws of large numbers for persistent Betti numbers and persistence diagrams of random filtered complexes built over marked point processes (randomly distributed marked point configurations).
This talk is based on a joint work with Tomoyuki Shirai (Kyushu University).