Persistence and stability of geometric complexes

The topic belongs to the general area of Computational Topology, and more specifically to the Persistent Homology sub-area. Building on recent results by the two authors, the project suggests several extensions and improvements of past work. A. We propose a stochastic analysis of the discrete Morse theory of the Delaunay triangulation of a Poisson point process. B. We aim at extending the proof of convergence of the modified Crofton Formula for intrinsic volumes. C. We to develop topological data analysis for Bregman divergences and analyze its stability. D. We study sparse complexes that preserve or approximate the persistence of standard distance functions. E. We apply the discrete Morse theory of Cech and Delaunay complexes to find solutions to sampled dynamical systems. Each problem will need its own methods, and solutions to one problem will feed into approaches to others. As an overarching method, we mention that discrete Morse theory -- as developed by Robin Forman about 20 years ago -- applies to all five topics and ties the work together to a coherent project. If successful, the advances will create a discrete theory that bridges several up to now disjoint mathematical topics. The completion of this theory is driven by the applications of the results and the corresponding computational tools to data analysis questions.

There are two significant results we obtained during this project. The first is a proof-of-concept of the topologically adaptive reconstruction of shapes. The second is a sweeping analysis of the expected behavior of Poisson--Delaunay mosaics. Topological adaptivity in reconstruction means that the algorithm is guided by the hole systems that have the potential to arise. For the non-adaptive reconstruction, we get a default hole system from local considerations of size. But there can be competing interests, such as the functionality of a protein in channeling ions through a cell membrane. Two different hole systems are however incompatible, or only partially compatible, and the organized presentation of the dependences between them is at the core of the adaptive algorithm. A Poisson point process is an unbiased way to randomly pick points in a space, for example in the Euclidean plane. A Poisson--Delaunay mosaic is the Delaunay mosaic of the randomly sampled points. A curious fact is that we can expect that half the triangles are acute and the other half is obtuse. This insight is not new, but it is representative for the many analytic results on Poisson--Delaunay mosaics obtained in this project. In the stepwise construction of the mosaic guided by the radius function, the acute triangles are critical while the obtuse triangles are non-critical. It follows that the fact about acute triangles informs us about the expected number of times the homotopy type of the mosaic changes during the stepwise construction.