C04
Persistence and Stability of Geometric Complexes

The project is concerned with multiscale families of complexes constructed from geometric or abstract data. Commonly, the data is a sample of some shape (a subspace of some metric space), and the complexes serve, in a precise sense, as a discretization of the topology of the shape at a given scale. The complexes in the family are linked by maps, and the resulting diagram contains global information that reveals more structure than just the individual complexes appearing in the diagram.  

We will investigate stable topological features of these complexes and develop computational methods to leverage this topological information for applications in dynamics and data analysis. One particular motivating problem arises from the study of invariant sets of discrete dynamical systems. 

Utilizing tools of discrete Morse theory, we aim to provide an efficient computational link between different construction of geometric complexes, and between different complexes approximating the same shape. An application of these methods lies in the problem of computing the homology of a self-map from a sample, a relevant problem in the study of discrete dynamical systems using Conley index theory.  

We have new results on the expected number of critical simplices and intervals of the radius functions of the Poisson--Delaunay mosaic in n-dimensional space.  Specifically, we have the expected numbers in dimensions n ≤ 4, and we their distribution dependent on the radius.  The expected number of Delaunay simplices in dimensions n ≤ 4 follow. 

The stability of persistence diagrams has recently been used to obtain integral geometric formulas for the first intrinsic volume (in R3 the total mean curvature) that converge for cubical approximations of shapes with smooth boundary.  The extension to most other intrinsic volumes is still open.  We propose to study possible extensions, in particular to the second intrinsic volume (in R3 the area).