The work performed in this project was divided into four objectives, each targeting a particular direction needed for the broad development of the theory.
Objective I targeted the possibility to adapt automatic reconstruction methods to local properties of the data, such as more detailed shaping where the data provides the necessary information, while keeping the global integrity of the reconstruction in other areas. A particularly successful approach is the Wrap algorithm, and we have successfully generalized it to higher dimensions, non-Euclidean dissimilarities, k-fold covers, as well as the abstract context of monotonic functions on polyhedral complexes. The suitability of a version of the Wrap algorithm for the chromatic setting is subject of on-going research.
Objective II addressed the stochastic properties of data and our algorithms. The former targets the understanding of noise while the purpose of the latter is to understand and possibly improve the behaviour of the methods on the average. We had sweeping success in the study of Delaunay mosaics for Poisson point processes. This is the most fundamental geometric setting, and our topological approach to studying the geometry proved to be a fresh view on an old topic that allowed for major advances in our knowledge. In particular, we now have a complete understanding of the intervals and the critical structure of a broad class of Delaunay mosaics up to dimension 4. Importantly, we have complexity bounds for the chromatic setting, both for worst-case and random data as well as colourings.
Objective III focused on k-fold covers. Within the project, we were able to revive old subjects about order-k Voronoi diagrams, study new aspects, and extend their reach. For example, we now have an algorithm for computing persistence in depth, which means the characterization of the covering as the depth decreases. This kind of analysis is challenged by the absence of a consistent complex that represents the covers for different depths. There is also a connection between the order-k Voronoi diagram and the chromatic setting with k colors, which is subject of on-going research.
Objective IV extended the theory to periodic settings. Here we focused on the 3-dimensional case and questions that arise in the study of materials. We have made progress in the development of a stable invariant that can be used to search and organize periodic crystals. With the availability of millions of structures, this will be an important piece in the creation of new computational infra-structure supporting high-performance approaches to materials. Related to this work is the detailed analysis of Brillouin zones and tessellations. In particular, we proved the stability of Brillouin zones, derived bounds on the number of chambers, and showed the monotonicity of infimum and supremum angles with varying order, k.
