@phdthesis{18979,
  abstract     = {Topological Data Analysis (TDA) is a discipline utilizing the mathematical field of topology to study data, most prominently collections of point sets. This thesis summarizes three projects related to computations in TDA.

The first one establishes a variant of TDA for chromatic point sets, where each point is given a color. For example, we are given positions of cells within a tumor microenvironment, and color the cancerous cells red, and the immune cells blue.

The aim is then to give a quantitative description of how the two or more sets of points spatially interact. Building on image, kernel and cokernel variants of persistent homology, we suggest six-packs of persistent diagrams as such a descriptor.

We describe a construction of a chromatic alpha complex, which enables  efficient computation of several variants of the six-packs. We give topological descriptions of natural subcomplexes of the chromatic alpha complex, and show that the radii of the simplices form a discrete Morse function. Finally, we provide an implementation of the presented chromatic TDA pipeline.

The second part aims to translate a powerful tool of sheaf theory to elementary terms using labeled matrices. The goal is to enable their use in computational settings. We show that derived categories of sheaves over finite posets have, up to isomorphism, unique objects---minimal injective resolutions---and give a concrete algorithm to compute them. We further describe simple algorithms to compute derived pushforwards and pullbacks for monotonic maps, and their proper variants for inclusions, and demonstrate their tractability by providing an implementation. Finally, we suggest a discrete definition of microsupport and show desirable properties inspired by discrete Morse theory.

In the last part, we present a collection of observations about collapses. We give a characterization of collapsibility in terms of unitriangular submatrices of the boundary matrix, a cotree-tree decomposition, and the optimal solution to a variant of the Procrustes problem. We establish relation between dual collapses and relative Morse theory and pose several open questions. Finally, focusing on complexes embedded in the three-dimensional Euclidean space, we describe a relation between the collapsibility and the triviality of a polygonal knot.},
  author       = {Draganov, Ondrej},
  issn         = {2663-337X},
  keywords     = {topological data analysis, chromatic point set, alpha complex, persistent homology, six pack, sheaf, microlocal discrete Morse, injective resolution, collapse, knot, discrete Morse theory},
  pages        = {140},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Structures and computations in topological data analysis}},
  doi          = {10.15479/at:ista:18979},
  year         = {2025},
}

@phdthesis{7460,
  abstract     = {Many methods for the reconstruction of shapes from sets of points produce ordered simplicial complexes, which are collections of vertices, edges, triangles, and their higher-dimensional analogues, called simplices, in which every simplex gets assigned a real value measuring its size. This thesis studies ordered simplicial complexes, with a focus on their topology, which reflects the connectedness of the represented shapes and the presence of holes. We are interested both in understanding better the structure of these complexes, as well as in developing algorithms for applications.

For the Delaunay triangulation, the most popular measure for a simplex is the radius of the smallest empty circumsphere. Based on it, we revisit Alpha and Wrap complexes and experimentally determine their probabilistic properties for random data. Also, we prove the existence of tri-partitions, propose algorithms to open and close holes, and extend the concepts from Euclidean to Bregman geometries.},
  author       = {Ölsböck, Katharina},
  issn         = {2663-337X},
  keywords     = {shape reconstruction, hole manipulation, ordered complexes, Alpha complex, Wrap complex, computational topology, Bregman geometry},
  pages        = {155},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{The hole system of triangulated shapes}},
  doi          = {10.15479/AT:ISTA:7460},
  year         = {2020},
}