@article{8940, abstract = {We quantise Whitney’s construction to prove the existence of a triangulation for any C^2 manifold, so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is completely geometric.}, author = {Boissonnat, Jean-Daniel and Kachanovich, Siargey and Wintraecken, Mathijs}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, keywords = {Theoretical Computer Science, Computational Theory and Mathematics, Geometry and Topology, Discrete Mathematics and Combinatorics}, number = {1}, pages = {386--434}, publisher = {Springer Nature}, title = {{Triangulating submanifolds: An elementary and quantified version of Whitney’s method}}, doi = {10.1007/s00454-020-00250-8}, volume = {66}, year = {2021}, } @article{9111, abstract = {We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space X equipped with a continuous function f:X→R. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line R. We then introduce a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium on point-based graphics, 2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of (X,f) when it is applied to points randomly sampled from a probability density function concentrated on (X,f). Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (In: 32nd international symposium on computational geometry, volume 51 of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany, pp 53:1–53:16, 2016), we first show that the mapper graph of (X,f), a constructible R-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of (X,f) to the mapper of a super-level set of a probability density function concentrated on (X,f). Finally, building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data.}, author = {Brown, Adam and Bobrowski, Omer and Munch, Elizabeth and Wang, Bei}, issn = {2367-1734}, journal = {Journal of Applied and Computational Topology}, number = {1}, pages = {99--140}, publisher = {Springer Nature}, title = {{Probabilistic convergence and stability of random mapper graphs}}, doi = {10.1007/s41468-020-00063-x}, volume = {5}, year = {2021}, } @phdthesis{9056, abstract = {In this thesis we study persistence of multi-covers of Euclidean balls and the geometric structures underlying their computation, in particular Delaunay mosaics and Voronoi tessellations. The k-fold cover for some discrete input point set consists of the space where at least k balls of radius r around the input points overlap. Persistence is a notion that captures, in some sense, the topology of the shape underlying the input. While persistence is usually computed for the union of balls, the k-fold cover is of interest as it captures local density, and thus might approximate the shape of the input better if the input data is noisy. To compute persistence of these k-fold covers, we need a discretization that is provided by higher-order Delaunay mosaics. We present and implement a simple and efficient algorithm for the computation of higher-order Delaunay mosaics, and use it to give experimental results for their combinatorial properties. The algorithm makes use of a new geometric structure, the rhomboid tiling. It contains the higher-order Delaunay mosaics as slices, and by introducing a filtration function on the tiling, we also obtain higher-order α-shapes as slices. These allow us to compute persistence of the multi-covers for varying radius r; the computation for varying k is less straight-foward and involves the rhomboid tiling directly. We apply our algorithms to experimental sphere packings to shed light on their structural properties. Finally, inspired by periodic structures in packings and materials, we propose and implement an algorithm for periodic Delaunay triangulations to be integrated into the Computational Geometry Algorithms Library (CGAL), and discuss the implications on persistence for periodic data sets.}, author = {Osang, Georg F}, issn = {2663-337X}, pages = {134}, publisher = {Institute of Science and Technology Austria}, title = {{Multi-cover persistence and Delaunay mosaics}}, doi = {10.15479/AT:ISTA:9056}, year = {2021}, } @article{10204, abstract = {Two common representations of close packings of identical spheres consisting of hexagonal layers, called Barlow stackings, appear abundantly in minerals and metals. These motifs, however, occupy an identical portion of space and bear identical first-order topological signatures as measured by persistent homology. Here we present a novel method based on k-fold covers that unambiguously distinguishes between these patterns. Moreover, our approach provides topological evidence that the FCC motif is the more stable of the two in the context of evolving experimental sphere packings during the transition from disordered to an ordered state. We conclude that our approach can be generalised to distinguish between various Barlow stackings manifested in minerals and metals.}, author = {Osang, Georg F and Edelsbrunner, Herbert and Saadatfar, Mohammad}, issn = {1744-6848}, journal = {Soft Matter}, number = {40}, pages = {9107--9115}, publisher = {Royal Society of Chemistry }, title = {{Topological signatures and stability of hexagonal close packing and Barlow stackings}}, doi = {10.1039/d1sm00774b}, volume = {17}, year = {2021}, } @inproceedings{9605, abstract = {Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness. }, author = {Corbet, René and Kerber, Michael and Lesnick, Michael and Osang, Georg F}, booktitle = {Leibniz International Proceedings in Informatics}, isbn = {9783959771849}, issn = {18688969}, location = {Online}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Computing the multicover bifiltration}}, doi = {10.4230/LIPIcs.SoCG.2021.27}, volume = {189}, year = {2021}, } @inproceedings{9441, abstract = {Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. submanifolds of ℝ^d defined as the zero set of some multivariate multivalued smooth function f: ℝ^d → ℝ^{d-n}, where n is the intrinsic dimension of the manifold. A natural way to approximate a smooth isomanifold M is to consider its Piecewise-Linear (PL) approximation M̂ based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we describe a simple algorithm to trace isomanifolds from a given starting point. The algorithm works for arbitrary dimensions n and d, and any precision D. Our main result is that, when f (or M) has bounded complexity, the complexity of the algorithm is polynomial in d and δ = 1/D (and unavoidably exponential in n). Since it is known that for δ = Ω (d^{2.5}), M̂ is O(D²)-close and isotopic to M, our algorithm produces a faithful PL-approximation of isomanifolds of bounded complexity in time polynomial in d. Combining this algorithm with dimensionality reduction techniques, the dependency on d in the size of M̂ can be completely removed with high probability. We also show that the algorithm can handle isomanifolds with boundary and, more generally, isostratifolds. The algorithm for isomanifolds with boundary has been implemented and experimental results are reported, showing that it is practical and can handle cases that are far ahead of the state-of-the-art. }, author = {Boissonnat, Jean-Daniel and Kachanovich, Siargey and Wintraecken, Mathijs}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, isbn = {978-3-95977-184-9}, issn = {1868-8969}, location = {Virtual}, pages = {17:1--17:16}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Tracing isomanifolds in Rd in time polynomial in d using Coxeter-Freudenthal-Kuhn triangulations}}, doi = {10.4230/LIPIcs.SoCG.2021.17}, volume = {189}, year = {2021}, } @article{8338, abstract = {Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics that are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines that are diagonally related to lines of curvature is proved theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory.}, author = {Akopyan, Arseniy and Bobenko, Alexander I. and Schief, Wolfgang K. and Techter, Jan}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {938--976}, publisher = {Springer Nature}, title = {{On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs}}, doi = {10.1007/s00454-020-00240-w}, volume = {66}, year = {2021}, } @article{8248, abstract = {We consider the following setting: suppose that we are given a manifold M in Rd with positive reach. Moreover assume that we have an embedded simplical complex A without boundary, whose vertex set lies on the manifold, is sufficiently dense and such that all simplices in A have sufficient quality. We prove that if, locally, interiors of the projection of the simplices onto the tangent space do not intersect, then A is a triangulation of the manifold, that is, they are homeomorphic.}, author = {Boissonnat, Jean-Daniel and Dyer, Ramsay and Ghosh, Arijit and Lieutier, Andre and Wintraecken, Mathijs}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {666--686}, publisher = {Springer Nature}, title = {{Local conditions for triangulating submanifolds of Euclidean space}}, doi = {10.1007/s00454-020-00233-9}, volume = {66}, year = {2021}, } @article{7905, abstract = {We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1355–1370, ACM, New York, 2012), and the cohomology stratification algorithm given in Nanda (Found. Comput. Math. 20(2), 195–222, 2020). Additionally, we give examples of stratifications based on the geometric techniques of Breiding et al. (Rev. Mat. Complut. 31(3), 545–593, 2018), illustrating how the sheaf-theoretic approach can be used to study stratifications from both topological and geometric perspectives. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory in generalizing existing algorithms.}, author = {Brown, Adam and Wang, Bei}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {1166--1198}, publisher = {Springer Nature}, title = {{Sheaf-theoretic stratification learning from geometric and topological perspectives}}, doi = {10.1007/s00454-020-00206-y}, volume = {65}, year = {2021}, } @article{7567, abstract = {Coxeter triangulations are triangulations of Euclidean space based on a single simplex. By this we mean that given an individual simplex we can recover the entire triangulation of Euclidean space by inductively reflecting in the faces of the simplex. In this paper we establish that the quality of the simplices in all Coxeter triangulations is O(1/d−−√) of the quality of regular simplex. We further investigate the Delaunay property for these triangulations. Moreover, we consider an extension of the Delaunay property, namely protection, which is a measure of non-degeneracy of a Delaunay triangulation. In particular, one family of Coxeter triangulations achieves the protection O(1/d2). We conjecture that both bounds are optimal for triangulations in Euclidean space.}, author = {Choudhary, Aruni and Kachanovich, Siargey and Wintraecken, Mathijs}, issn = {1661-8289}, journal = {Mathematics in Computer Science}, pages = {141--176}, publisher = {Springer Nature}, title = {{Coxeter triangulations have good quality}}, doi = {10.1007/s11786-020-00461-5}, volume = {14}, year = {2020}, }