Discrete Morse theory has recently lead to new developments in the theory of random geometric complexes. This article surveys the methods and results obtained with this new approach, and discusses some of its shortcomings. It uses simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics.
Topological Data Analysis
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements No 78818 Alpha and No 638176). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF).
Edelsbrunner H, Nikitenko A, Ölsböck K, Synak P. Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. In: Topological Data Analysis. Vol 15. Springer Nature; 2020:181-218. doi:10.1007/978-3-030-43408-3_8
Edelsbrunner, H., Nikitenko, A., Ölsböck, K., & Synak, P. (2020). Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. In Topological Data Analysis (Vol. 15, pp. 181–218). Springer Nature. https://doi.org/10.1007/978-3-030-43408-3_8
Edelsbrunner, Herbert, Anton Nikitenko, Katharina Ölsböck, and Peter Synak. “Radius Functions on Poisson–Delaunay Mosaics and Related Complexes Experimentally.” In Topological Data Analysis, 15:181–218. Springer Nature, 2020. https://doi.org/10.1007/978-3-030-43408-3_8.
H. Edelsbrunner, A. Nikitenko, K. Ölsböck, and P. Synak, “Radius functions on Poisson–Delaunay mosaics and related complexes experimentally,” in Topological Data Analysis, 2020, vol. 15, pp. 181–218.
Edelsbrunner H, Nikitenko A, Ölsböck K, Synak P. 2020. Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. Topological Data Analysis. , Abel Symposia, vol. 15. 181–218.
Edelsbrunner, Herbert, et al. “Radius Functions on Poisson–Delaunay Mosaics and Related Complexes Experimentally.” Topological Data Analysis, vol. 15, Springer Nature, 2020, pp. 181–218, doi:10.1007/978-3-030-43408-3_8.
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