thesis published Sebastiano Cultrera di Montesano author 34D2A09C-F248-11E8-B48F-1D18A9856A870000-0001-6249-0832 Herbert Edelsbrunner supervisor GradSch department HeEd department Alpha Shape Theory Extended project Mathematics, Computer Science project Persistent Homology, Algorithms and Stochastic Geometry project Persistence and stability of geometric complexes project Point sets, geometric networks, and arrangements of hyperplanes are fundamental objects in discrete geometry that have captivated mathematicians for centuries, if not millennia. This thesis seeks to cast new light on these structures by illustrating specific instances where a topological perspective, specifically through discrete Morse theory and persistent homology, provides valuable insights. At first glance, the topology of these geometric objects might seem uneventful: point sets essentially lack of topology, arrangements of hyperplanes are a decomposition of Rd, which is a contractible space, and the topology of a network primarily involves the enumeration of connected components and cycles within the network. However, beneath this apparent simplicity, there lies an array of intriguing structures, a small subset of which will be uncovered in this thesis. Focused on three case studies, each addressing one of the mentioned objects, this work will showcase connections that intertwine topology with diverse fields such as combinatorial geometry, algorithms and data structures, and emerging applications like spatial biology. https://research-explorer.ista.ac.at/download/15094/15112/Thesis Sebastiano.pdf application/pdfno https://research-explorer.ista.ac.at/download/15094/15113/Thesis (1).zip application/zip https://creativecommons.org/licenses/by-nc-sa/4.0/ Institute of Science and Technology Austria2024 eng 2663-337X10.15479/at:ista:15094 108 https://research-explorer.ista.ac.at/record/15091 https://research-explorer.ista.ac.at/record/15090 https://research-explorer.ista.ac.at/record/15093 https://research-explorer.ista.ac.at/record/13182 https://research-explorer.ista.ac.at/record/11658 https://research-explorer.ista.ac.at/record/11660 Cultrera di Montesano, Sebastiano. <i>Persistence and Morse Theory for Discrete Geometric Structures</i>. Institute of Science and Technology Austria, 2024, doi:<a href="https://doi.org/10.15479/at:ista:15094">10.15479/at:ista:15094</a>. Cultrera di Montesano, S. (2024). <i>Persistence and Morse theory for discrete geometric structures</i>. Institute of Science and Technology Austria. <a href="https://doi.org/10.15479/at:ista:15094">https://doi.org/10.15479/at:ista:15094</a> Cultrera di Montesano, Sebastiano. “Persistence and Morse Theory for Discrete Geometric Structures.” Institute of Science and Technology Austria, 2024. <a href="https://doi.org/10.15479/at:ista:15094">https://doi.org/10.15479/at:ista:15094</a>. S. Cultrera di Montesano, Persistence and Morse Theory for Discrete Geometric Structures, Institute of Science and Technology Austria, 2024. S. Cultrera di Montesano, “Persistence and Morse theory for discrete geometric structures,” Institute of Science and Technology Austria, 2024. Cultrera di Montesano S. Persistence and Morse theory for discrete geometric structures. 2024. doi:<a href="https://doi.org/10.15479/at:ista:15094">10.15479/at:ista:15094</a> Cultrera di Montesano S. 2024. Persistence and Morse theory for discrete geometric structures. Institute of Science and Technology Austria. 150942024-03-08T15:28:10Z2026-08-13T09:41:57Z