Abstract 

Periodic data is abundant in material science, for example, the atoms of a crystalline material repeat periodically. Additionally, periodic boundary conditions are used in many simulations, for example in molecular dynamics simulations of materials, to remove boundary effects. However, it is unclear how to deal with the periodicity of the data when computing topological descriptors, like the merge tree (which tracks how the connected components evolve along a growing scale parameter) or persistent homology (which tracks holes along the same scale parameter). A classical approach is to compute the respective descriptor on the torus. However, this does not give the information needed for many applications and is even unstable under certain types of noise. Therefore, we suggest decorating the merge tree gained from the torus with additional information, describing for each connected component on the torus how many components of the infinite periodic space map to it. As there are often infinitely many, we describe their density and growth rate inside a growing sphere. The resulting periodic merge tree and its induced periodic 0-th persistence barcode carry the desired information and fulfill the desired properties, in particular: stability and efficient computability (under mild assumptions, the running time is of order (n+m)log(m), where n and m are the number of vertices and edges per fundamental domain). This is joint work with Herbert Edelsbrunner.