@article{8323, author = {Pach, János}, issn = {14320444}, journal = {Discrete and Computational Geometry}, pages = {571--574}, publisher = {Springer Nature}, title = {{A farewell to Ricky Pollack}}, doi = {10.1007/s00454-020-00237-5}, volume = {64}, year = {2020}, } @inproceedings{8580, abstract = {We evaluate the usefulness of persistent homology in the analysis of heart rate variability. In our approach we extract several topological descriptors characterising datasets of RR-intervals, which are later used in classical machine learning algorithms. By this method we are able to differentiate the group of patients with the history of transient ischemic attack and the group of hypertensive patients.}, author = {Graff, Grzegorz and Graff, Beata and Jablonski, Grzegorz and Narkiewicz, Krzysztof}, booktitle = {11th Conference of the European Study Group on Cardiovascular Oscillations: Computation and Modelling in Physiology: New Challenges and Opportunities, }, isbn = {9781728157511}, location = {Pisa, Italy}, publisher = {IEEE}, title = {{The application of persistent homology in the analysis of heart rate variability}}, doi = {10.1109/ESGCO49734.2020.9158054}, year = {2020}, } @article{10867, abstract = {In this paper we find a tight estimate for Gromov’s waist of the balls in spaces of constant curvature, deduce the estimates for the balls in Riemannian manifolds with upper bounds on the curvature (CAT(ϰ)-spaces), and establish similar result for normed spaces.}, author = {Akopyan, Arseniy and Karasev, Roman}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, keywords = {General Mathematics}, number = {3}, pages = {669--697}, publisher = {Oxford University Press}, title = {{Waist of balls in hyperbolic and spherical spaces}}, doi = {10.1093/imrn/rny037}, volume = {2020}, year = {2020}, } @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}, } @phdthesis{7944, abstract = {This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph. For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton. In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars.}, author = {Masárová, Zuzana}, isbn = {978-3-99078-005-3}, issn = {2663-337X}, keywords = {reconfiguration, reconfiguration graph, triangulations, flip, constrained triangulations, shellability, piecewise-linear balls, token swapping, trees, coloured weighted token swapping}, pages = {160}, publisher = {Institute of Science and Technology Austria}, title = {{Reconfiguration problems}}, doi = {10.15479/AT:ISTA:7944}, year = {2020}, } @inproceedings{8703, abstract = {Even though Delaunay originally introduced his famous triangulations in the case of infinite point sets with translational periodicity, a software that computes such triangulations in the general case is not yet available, to the best of our knowledge. Combining and generalizing previous work, we present a practical algorithm for computing such triangulations. The algorithm has been implemented and experiments show that its performance is as good as the one of the CGAL package, which is restricted to cubic periodicity. }, author = {Osang, Georg F and Rouxel-Labbé, Mael and Teillaud, Monique}, booktitle = {28th Annual European Symposium on Algorithms}, isbn = {9783959771627}, issn = {18688969}, location = {Virtual, Online; Pisa, Italy}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Generalizing CGAL periodic Delaunay triangulations}}, doi = {10.4230/LIPIcs.ESA.2020.75}, volume = {173}, year = {2020}, } @article{8163, abstract = {Fejes Tóth [3] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the square of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.}, author = {Vegter, Gert and Wintraecken, Mathijs}, issn = {1588-2896}, journal = {Studia Scientiarum Mathematicarum Hungarica}, number = {2}, pages = {193--199}, publisher = {Akadémiai Kiadó}, title = {{Refutation of a claim made by Fejes Tóth on the accuracy of surface meshes}}, doi = {10.1556/012.2020.57.2.1454}, volume = {57}, year = {2020}, } @article{9157, abstract = {Representing an atom by a solid sphere in 3-dimensional Euclidean space, we get the space-filling diagram of a molecule by taking the union. Molecular dynamics simulates its motion subject to bonds and other forces, including the solvation free energy. The morphometric approach [12, 17] writes the latter as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted mean curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [3], and the weighted Gaussian curvature [1], this yields the derivative of the morphometric expression of the solvation free energy.}, author = {Akopyan, Arseniy and Edelsbrunner, Herbert}, issn = {2544-7297}, journal = {Computational and Mathematical Biophysics}, number = {1}, pages = {51--67}, publisher = {De Gruyter}, title = {{The weighted mean curvature derivative of a space-filling diagram}}, doi = {10.1515/cmb-2020-0100}, volume = {8}, year = {2020}, } @article{9156, abstract = {The morphometric approach [11, 14] writes the solvation free energy as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted Gaussian curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [4], and the weighted mean curvature in [1], this yields the derivative of the morphometric expression of solvation free energy.}, author = {Akopyan, Arseniy and Edelsbrunner, Herbert}, issn = {2544-7297}, journal = {Computational and Mathematical Biophysics}, number = {1}, pages = {74--88}, publisher = {De Gruyter}, title = {{The weighted Gaussian curvature derivative of a space-filling diagram}}, doi = {10.1515/cmb-2020-0101}, volume = {8}, year = {2020}, } @article{15064, abstract = {We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior. We establish a sampling theorem to recover the eigenspaces of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for Čech and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive Čech complexes to the computationally efficient Delaunay triangulations. The fast chain map algorithm has applications beyond dynamical systems.}, author = {Bauer, U. and Edelsbrunner, Herbert and Jablonski, Grzegorz and Mrozek, M.}, issn = {2367-1734}, journal = {Journal of Applied and Computational Topology}, number = {4}, pages = {455--480}, publisher = {Springer Nature}, title = {{Čech-Delaunay gradient flow and homology inference for self-maps}}, doi = {10.1007/s41468-020-00058-8}, volume = {4}, year = {2020}, } @article{6515, abstract = {We give non-degeneracy criteria for Riemannian simplices based on simplices in spaces of constant sectional curvature. It extends previous work on Riemannian simplices, where we developed Riemannian simplices with respect to Euclidean reference simplices. The criteria we give in this article are in terms of quality measures for spaces of constant curvature that we develop here. We see that simplices in spaces that have nearly constant curvature, are already non-degenerate under very weak quality demands. This is of importance because it allows for sampling of Riemannian manifolds based on anisotropy of the manifold and not (absolute) curvature.}, author = {Dyer, Ramsay and Vegter, Gert and Wintraecken, Mathijs}, issn = {1920-180X}, journal = {Journal of Computational Geometry }, number = {1}, pages = {223–256}, publisher = {Carleton University}, title = {{Simplices modelled on spaces of constant curvature}}, doi = {10.20382/jocg.v10i1a9}, volume = {10}, year = {2019}, } @inproceedings{6628, abstract = {Fejes Tóth [5] and Schneider [9] studied approximations of smooth convex hypersurfaces in Euclidean space by piecewise flat triangular meshes with a given number of vertices on the hypersurface that are optimal with respect to Hausdorff distance. They proved that this Hausdorff distance decreases inversely proportional with m 2/(d−1), where m is the number of vertices and d is the dimension of Euclidean space. Moreover the pro-portionality constant can be expressed in terms of the Gaussian curvature, an intrinsic quantity. In this short note, we prove the extrinsic nature of this constant for manifolds of sufficiently high codimension. We do so by constructing an family of isometric embeddings of the flat torus in Euclidean space.}, author = {Vegter, Gert and Wintraecken, Mathijs}, booktitle = {The 31st Canadian Conference in Computational Geometry}, location = {Edmonton, Canada}, pages = {275--279}, title = {{The extrinsic nature of the Hausdorff distance of optimal triangulations of manifolds}}, year = {2019}, } @inproceedings{6648, abstract = {Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable the usage of existing computational topology software in this context.}, author = {Edelsbrunner, Herbert and Virk, Ziga and Wagner, Hubert}, booktitle = {35th International Symposium on Computational Geometry}, isbn = {9783959771047}, location = {Portland, OR, United States}, pages = {31:1--31:14}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Topological data analysis in information space}}, doi = {10.4230/LIPICS.SOCG.2019.31}, volume = {129}, year = {2019}, } @inproceedings{6989, abstract = {When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with hole(s) to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special simple holes guarantee foldability. }, author = {Aichholzer, Oswin and Akitaya, Hugo A and Cheung, Kenneth C and Demaine, Erik D and Demaine, Martin L and Fekete, Sandor P and Kleist, Linda and Kostitsyna, Irina and Löffler, Maarten and Masárová, Zuzana and Mundilova, Klara and Schmidt, Christiane}, booktitle = {Proceedings of the 31st Canadian Conference on Computational Geometry}, location = {Edmonton, Canada}, pages = {164--170}, publisher = {Canadian Conference on Computational Geometry}, title = {{Folding polyominoes with holes into a cube}}, year = {2019}, } @article{6671, abstract = {In this paper we discuss three results. The first two concern general sets of positive reach: we first characterize the reach of a closed set by means of a bound on the metric distortion between the distance measured in the ambient Euclidean space and the shortest path distance measured in the set. Secondly, we prove that the intersection of a ball with radius less than the reach with the set is geodesically convex, meaning that the shortest path between any two points in the intersection lies itself in the intersection. For our third result we focus on manifolds with positive reach and give a bound on the angle between tangent spaces at two different points in terms of the reach and the distance between the two points.}, author = {Boissonnat, Jean-Daniel and Lieutier, André and Wintraecken, Mathijs}, issn = {2367-1734}, journal = {Journal of Applied and Computational Topology}, number = {1-2}, pages = {29–58}, publisher = {Springer Nature}, title = {{The reach, metric distortion, geodesic convexity and the variation of tangent spaces}}, doi = {10.1007/s41468-019-00029-8}, volume = {3}, year = {2019}, } @article{6050, abstract = {We answer a question of David Hilbert: given two circles it is not possible in general to construct their centers using only a straightedge. On the other hand, we give infinitely many families of pairs of circles for which such construction is possible. }, author = {Akopyan, Arseniy and Fedorov, Roman}, journal = {Proceedings of the American Mathematical Society}, pages = {91--102}, publisher = {AMS}, title = {{Two circles and only a straightedge}}, doi = {10.1090/proc/14240}, volume = {147}, year = {2019}, } @article{6634, abstract = {In this paper we prove several new results around Gromov's waist theorem. We give a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk-Ulam-Crofton technique, consider waists of real and complex projective spaces, flat tori, convex bodies in Euclidean space; and establish waist-type results in terms of the Hausdorff measure.}, author = {Akopyan, Arseniy and Hubard, Alfredo and Karasev, Roman}, journal = {Topological Methods in Nonlinear Analysis}, number = {2}, pages = {457--490}, publisher = {Akademicka Platforma Czasopism}, title = {{Lower and upper bounds for the waists of different spaces}}, doi = {10.12775/TMNA.2019.008}, volume = {53}, year = {2019}, } @article{6756, abstract = {We study the topology generated by the temperature fluctuations of the cosmic microwave background (CMB) radiation, as quantified by the number of components and holes, formally given by the Betti numbers, in the growing excursion sets. We compare CMB maps observed by the Planck satellite with a thousand simulated maps generated according to the ΛCDM paradigm with Gaussian distributed fluctuations. The comparison is multi-scale, being performed on a sequence of degraded maps with mean pixel separation ranging from 0.05 to 7.33°. The survey of the CMB over 𝕊2 is incomplete due to obfuscation effects by bright point sources and other extended foreground objects like our own galaxy. To deal with such situations, where analysis in the presence of “masks” is of importance, we introduce the concept of relative homology. The parametric χ2-test shows differences between observations and simulations, yielding p-values at percent to less than permil levels roughly between 2 and 7°, with the difference in the number of components and holes peaking at more than 3σ sporadically at these scales. The highest observed deviation between the observations and simulations for b0 and b1 is approximately between 3σ and 4σ at scales of 3–7°. There are reports of mildly unusual behaviour of the Euler characteristic at 3.66° in the literature, computed from independent measurements of the CMB temperature fluctuations by Planck’s predecessor, the Wilkinson Microwave Anisotropy Probe (WMAP) satellite. The mildly anomalous behaviour of the Euler characteristic is phenomenologically related to the strongly anomalous behaviour of components and holes, or the zeroth and first Betti numbers, respectively. Further, since these topological descriptors show consistent anomalous behaviour over independent measurements of Planck and WMAP, instrumental and systematic errors may be an unlikely source. These are also the scales at which the observed maps exhibit low variance compared to the simulations, and approximately the range of scales at which the power spectrum exhibits a dip with respect to the theoretical model. Non-parametric tests show even stronger differences at almost all scales. Crucially, Gaussian simulations based on power-spectrum matching the characteristics of the observed dipped power spectrum are not able to resolve the anomaly. Understanding the origin of the anomalies in the CMB, whether cosmological in nature or arising due to late-time effects, is an extremely challenging task. Regardless, beyond the trivial possibility that this may still be a manifestation of an extreme Gaussian case, these observations, along with the super-horizon scales involved, may motivate the study of primordial non-Gaussianity. Alternative scenarios worth exploring may be models with non-trivial topology, including topological defect models.}, author = {Pranav, Pratyush and Adler, Robert J. and Buchert, Thomas and Edelsbrunner, Herbert and Jones, Bernard J.T. and Schwartzman, Armin and Wagner, Hubert and Van De Weygaert, Rien}, issn = {14320746}, journal = {Astronomy and Astrophysics}, publisher = {EDP Sciences}, title = {{Unexpected topology of the temperature fluctuations in the cosmic microwave background}}, doi = {10.1051/0004-6361/201834916}, volume = {627}, year = {2019}, } @article{6793, abstract = {The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here, we give a simple geometric proof of Regge symmetries in Euclidean, spherical, and hyperbolic geometry.}, author = {Akopyan, Arseniy and Izmestiev, Ivan}, issn = {14692120}, journal = {Bulletin of the London Mathematical Society}, number = {5}, pages = {765--775}, publisher = {London Mathematical Society}, title = {{The Regge symmetry, confocal conics, and the Schläfli formula}}, doi = {10.1112/blms.12276}, volume = {51}, year = {2019}, } @article{6828, abstract = {In this paper we construct a family of exact functors from the category of Whittaker modules of the simple complex Lie algebra of type to the category of finite-dimensional modules of the graded affine Hecke algebra of type . Using results of Backelin [2] and of Arakawa-Suzuki [1], we prove that these functors map standard modules to standard modules (or zero) and simple modules to simple modules (or zero). Moreover, we show that each simple module of the graded affine Hecke algebra appears as the image of a simple Whittaker module. Since the Whittaker category contains the BGG category as a full subcategory, our results generalize results of Arakawa-Suzuki [1], which in turn generalize Schur-Weyl duality between finite-dimensional representations of and representations of the symmetric group .}, author = {Brown, Adam}, issn = {0021-8693}, journal = {Journal of Algebra}, pages = {261--289}, publisher = {Elsevier}, title = {{Arakawa-Suzuki functors for Whittaker modules}}, doi = {10.1016/j.jalgebra.2019.07.027}, volume = {538}, year = {2019}, }