@phdthesis{15094, abstract = {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. }, author = {Cultrera di Montesano, Sebastiano}, issn = {2663 - 337X}, pages = {108}, publisher = {Institute of Science and Technology Austria}, title = {{Persistence and Morse theory for discrete geometric structures}}, doi = {10.15479/at:ista:15094}, year = {2024}, } @inproceedings{15093, abstract = {We present a dynamic data structure for maintaining the persistent homology of a time series of real numbers. The data structure supports local operations, including the insertion and deletion of an item and the cutting and concatenating of lists, each in time O(log n + k), in which n counts the critical items and k the changes in the augmented persistence diagram. To achieve this, we design a tailor-made tree structure with an unconventional representation, referred to as banana tree, which may be useful in its own right.}, author = {Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Henzinger, Monika H and Ost, Lara}, booktitle = {Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)}, editor = {Woodruff, David P.}, location = {Alexandria, VA, USA}, pages = {243 -- 295}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Dynamically maintaining the persistent homology of time series}}, doi = {10.1137/1.9781611977912.11}, year = {2024}, } @unpublished{15091, abstract = {Motivated by applications in the medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective. Based on the persistent homology for images, kernels and cokernels, we design provably stable homological quantifiers that describe the geometric micro- and macro-structure of how the color classes mingle. These can be efficiently computed using chromatic variants of Delaunay and alpha complexes, and code that does these computations is provided.}, author = {Cultrera di Montesano, Sebastiano and Draganov, Ondrej and Edelsbrunner, Herbert and Saghafian, Morteza}, booktitle = {arXiv}, title = {{Chromatic alpha complexes}}, year = {2024}, } @article{13182, abstract = {We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs of theorems about the symmetry of persistence diagrams and the variation of such maps. In particular, we identify branching points and endpoints of networks as the sole source of asymmetry and relate the cycle basis in persistent homology with a version of the stable marriage problem. Our analysis provides the foundations of fast algorithms for maintaining a collection of sorted lists together with its persistence diagram.}, author = {Biswas, Ranita and Cultrera Di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza}, issn = {2367-1734}, journal = {Journal of Applied and Computational Topology}, publisher = {Springer Nature}, title = {{Geometric characterization of the persistence of 1D maps}}, doi = {10.1007/s41468-023-00126-9}, year = {2023}, } @article{10773, abstract = {The Voronoi tessellation in Rd is defined by locally minimizing the power distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined by locally maximizing the negative power distance to other such points. We prove that the average of the two piecewise quadratic functions is piecewise linear, and that all three functions have the same critical points and values. Discretizing the two piecewise quadratic functions, we get the alpha shapes as sublevel sets of the discrete function on the Delaunay mosaic, and analogous shapes as superlevel sets of the discrete function on the Voronoi tessellation. For the same non-critical value, the corresponding shapes are disjoint, separated by a narrow channel that contains no critical points but the entire level set of the piecewise linear function.}, author = {Biswas, Ranita and Cultrera Di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {811--842}, publisher = {Springer Nature}, title = {{Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics}}, doi = {10.1007/s00454-022-00371-2}, volume = {67}, year = {2022}, } @article{11660, abstract = {We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs of theorems about the symmetry of persistence diagrams and the variation of such maps. In particular, we identify branching points and endpoints of networks as the sole source of asymmetry and relate the cycle basis in persistent homology with a version of the stable marriage problem. Our analysis provides the foundations of fast algorithms for maintaining collections of interrelated sorted lists together with their persistence diagrams. }, author = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza}, journal = {LIPIcs}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs}}, year = {2022}, } @article{11658, abstract = {The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements.}, author = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza}, journal = {Leibniz International Proceedings on Mathematics}, publisher = {Schloss Dagstuhl - Leibniz Zentrum für Informatik}, title = {{Depth in arrangements: Dehn–Sommerville–Euler relations with applications}}, year = {2022}, } @unpublished{15090, abstract = {Given a locally finite set A⊆Rd and a coloring χ:A→{0,1,…,s}, we introduce the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in Rs+d that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with n=#A, and the coloring is random, then the chromatic Delaunay mosaic has O(n⌈d/2⌉) cells in expectation. In contrast, for Delone sets and Poisson point processes in Rd, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in R2 all colorings of a dense set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications.}, author = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Draganov, Ondrej and Edelsbrunner, Herbert and Saghafian, Morteza}, booktitle = {arXiv}, title = {{On the size of chromatic Delaunay mosaics}}, year = {2022}, } @inproceedings{9604, abstract = {Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in ℝ² to ℝ³, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of n ≥ 5 points in ℝ³, the number of regions in the order-k Voronoi tessellation is N_{k-1} - binom(k,2)n + n, for 1 ≤ k ≤ n-1, in which N_{k-1} is the sum of Euler characteristics of these function’s first k-1 sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-k Voronoi tessellation.}, author = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza}, booktitle = {Leibniz International Proceedings in Informatics}, isbn = {9783959771849}, issn = {18688969}, location = {Online}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Counting cells of order-k voronoi tessellations in ℝ3 with morse theory}}, doi = {10.4230/LIPIcs.SoCG.2021.16}, volume = {189}, year = {2021}, }