@article{10071, author = {Adams, Henry and Kourimska, Hana and Heiss, Teresa and Percival, Sarah and Ziegelmeier, Lori}, issn = {1088-9477}, journal = {Notices of the American Mathematical Society}, number = {9}, pages = {1511--1514}, publisher = {American Mathematical Society}, title = {{How to tutorial-a-thon}}, doi = {10.1090/noti2349}, volume = {68}, year = {2021}, } @inproceedings{10367, abstract = {How information is created, shared and consumed has changed rapidly in recent decades, in part thanks to new social platforms and technologies on the web. With ever-larger amounts of unstructured and limited labels, organizing and reconciling information from different sources and modalities is a central challenge in machine learning. This cutting-edge tutorial aims to introduce the multimodal entailment task, which can be useful for detecting semantic alignments when a single modality alone does not suffice for a whole content understanding. Starting with a brief overview of natural language processing, computer vision, structured data and neural graph learning, we lay the foundations for the multimodal sections to follow. We then discuss recent multimodal learning literature covering visual, audio and language streams, and explore case studies focusing on tasks which require fine-grained understanding of visual and linguistic semantics question answering, veracity and hatred classification. Finally, we introduce a new dataset for recognizing multimodal entailment, exploring it in a hands-on collaborative section. Overall, this tutorial gives an overview of multimodal learning, introduces a multimodal entailment dataset, and encourages future research in the topic.}, author = {Ilharco, Cesar and Shirazi, Afsaneh and Gopalan, Arjun and Nagrani, Arsha and Bratanič, Blaž and Bregler, Chris and Liu, Christina and Ferreira, Felipe and Barcik, Gabriek and Ilharco, Gabriel and Osang, Georg F and Bulian, Jannis and Frank, Jared and Smaira, Lucas and Cao, Qin and Marino, Ricardo and Patel, Roma and Leung, Thomas and Imbrasaite, Vaiva}, booktitle = {59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing, Tutorial Abstracts}, isbn = {9-781-9540-8557-2}, location = {Bangkok, Thailand}, pages = {29--30}, publisher = {Association for Computational Linguistics}, title = {{Recognizing multimodal entailment}}, doi = {10.18653/v1/2021.acl-tutorials.6}, year = {2021}, } @article{10608, abstract = {We consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.}, author = {Weighill, Thomas and Yamauchi, Takamitsu and Zava, Nicolò}, issn = {2199-6768}, journal = {European Journal of Mathematics}, publisher = {Springer Nature}, title = {{Coarse infinite-dimensionality of hyperspaces of finite subsets}}, doi = {10.1007/s40879-021-00515-3}, year = {2021}, } @inproceedings{9296, abstract = { matching is compatible to two or more labeled point sets of size n with labels {1,…,n} if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of n points there exists a compatible matching with ⌊2n−−√⌋ edges. More generally, for any ℓ labeled point sets we construct compatible matchings of size Ω(n1/ℓ) . As a corresponding upper bound, we use probabilistic arguments to show that for any ℓ given sets of n points there exists a labeling of each set such that the largest compatible matching has O(n2/(ℓ+1)) edges. Finally, we show that Θ(logn) copies of any set of n points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge.}, author = {Aichholzer, Oswin and Arroyo Guevara, Alan M and Masárová, Zuzana and Parada, Irene and Perz, Daniel and Pilz, Alexander and Tkadlec, Josef and Vogtenhuber, Birgit}, booktitle = {15th International Conference on Algorithms and Computation}, isbn = {9783030682101}, issn = {16113349}, location = {Yangon, Myanmar}, pages = {221--233}, publisher = {Springer Nature}, title = {{On compatible matchings}}, doi = {10.1007/978-3-030-68211-8_18}, volume = {12635}, year = {2021}, } @article{9465, abstract = {Given a locally finite set 𝑋⊆ℝ𝑑 and an integer 𝑘≥0, we consider the function 𝐰𝑘:Del𝑘(𝑋)→ℝ on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space.}, author = {Edelsbrunner, Herbert and Nikitenko, Anton and Osang, Georg F}, issn = {14208997}, journal = {Journal of Geometry}, number = {1}, publisher = {Springer Nature}, title = {{A step in the Delaunay mosaic of order k}}, doi = {10.1007/s00022-021-00577-4}, volume = {112}, year = {2021}, } @inproceedings{9345, abstract = {Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functionsthat facilitates the efficient search for new materials and material properties. We prove invarianceunder isometries, continuity, and completeness in the generic case, which are necessary featuresfor the reliable comparison of crystals. The proof of continuity integrates methods from discretegeometry and lattice theory, while the proof of generic completeness combines techniques fromgeometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and relatedinclusion-exclusion formulae. We have implemented the algorithm and describe its application tocrystal structure prediction.}, author = {Edelsbrunner, Herbert and Heiss, Teresa and Kurlin , Vitaliy and Smith, Philip and Wintraecken, Mathijs}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, issn = {1868-8969}, location = {Virtual}, pages = {32:1--32:16}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{The density fingerprint of a periodic point set}}, doi = {10.4230/LIPIcs.SoCG.2021.32}, volume = {189}, year = {2021}, } @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}, } @inproceedings{9824, abstract = {We define a new compact coordinate system in which each integer triplet addresses a voxel in the BCC grid, and we investigate some of its properties. We propose a characterization of 3D discrete analytical planes with their topological features (in the Cartesian and in the new coordinate system) such as the interrelation between the thickness of the plane and the separability constraint we aim to obtain.}, author = {Čomić, Lidija and Zrour, Rita and Largeteau-Skapin, Gaëlle and Biswas, Ranita and Andres, Eric}, booktitle = {Discrete Geometry and Mathematical Morphology}, isbn = {9783030766566}, issn = {16113349}, location = {Uppsala, Sweden}, pages = {152--163}, publisher = {Springer Nature}, title = {{Body centered cubic grid - coordinate system and discrete analytical plane definition}}, doi = {10.1007/978-3-030-76657-3_10}, volume = {12708}, year = {2021}, } @article{8317, 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 one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special “basic” holes guarantee foldability.}, author = {Aichholzer, Oswin and Akitaya, Hugo A. and Cheung, Kenneth C. and Demaine, Erik D. and Demaine, Martin L. and Fekete, Sándor P. and Kleist, Linda and Kostitsyna, Irina and Löffler, Maarten and Masárová, Zuzana and Mundilova, Klara and Schmidt, Christiane}, issn = {09257721}, journal = {Computational Geometry: Theory and Applications}, publisher = {Elsevier}, title = {{Folding polyominoes with holes into a cube}}, doi = {10.1016/j.comgeo.2020.101700}, volume = {93}, year = {2021}, } @article{8773, abstract = {Let g be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g-modules Y(χ,η) introduced by Kostant. We prove that the set of all contravariant forms on Y(χ,η) forms a vector space whose dimension is given by the cardinality of the Weyl group of g. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M(χ,η) introduced by McDowell.}, author = {Brown, Adam and Romanov, Anna}, issn = {1088-6826}, journal = {Proceedings of the American Mathematical Society}, keywords = {Applied Mathematics, General Mathematics}, number = {1}, pages = {37--52}, publisher = {American Mathematical Society}, title = {{Contravariant forms on Whittaker modules}}, doi = {10.1090/proc/15205}, volume = {149}, year = {2021}, } @inproceedings{9253, abstract = {In March 2020, the Austrian government introduced a widespread lock-down in response to the COVID-19 pandemic. Based on subjective impressions and anecdotal evidence, Austrian public and private life came to a sudden halt. Here we assess the effect of the lock-down quantitatively for all regions in Austria and present an analysis of daily changes of human mobility throughout Austria using near-real-time anonymized mobile phone data. We describe an efficient data aggregation pipeline and analyze the mobility by quantifying mobile-phone traffic at specific point of interests (POIs), analyzing individual trajectories and investigating the cluster structure of the origin-destination graph. We found a reduction of commuters at Viennese metro stations of over 80% and the number of devices with a radius of gyration of less than 500 m almost doubled. The results of studying crowd-movement behavior highlight considerable changes in the structure of mobility networks, revealed by a higher modularity and an increase from 12 to 20 detected communities. We demonstrate the relevance of mobility data for epidemiological studies by showing a significant correlation of the outflow from the town of Ischgl (an early COVID-19 hotspot) and the reported COVID-19 cases with an 8-day time lag. This research indicates that mobile phone usage data permits the moment-by-moment quantification of mobility behavior for a whole country. We emphasize the need to improve the availability of such data in anonymized form to empower rapid response to combat COVID-19 and future pandemics.}, author = {Heiler, Georg and Reisch, Tobias and Hurt, Jan and Forghani, Mohammad and Omani, Aida and Hanbury, Allan and Karimipour, Farid}, booktitle = {2020 IEEE International Conference on Big Data}, isbn = {9781728162515}, location = {Atlanta, GA, United States}, pages = {3123--3132}, publisher = {IEEE}, title = {{Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic}}, doi = {10.1109/bigdata50022.2020.9378374}, year = {2021}, } @article{9317, abstract = {Given a locally finite X⊆Rd and a radius r≥0, the k-fold cover of X and r consists of all points in Rd that have k or more points of X within distance r. We consider two filtrations—one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k—and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in Rd+1 whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module of Delaunay mosaics that is isomorphic to the persistence module of the multi-covers.}, author = {Edelsbrunner, Herbert and Osang, Georg F}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {1296–1313}, publisher = {Springer Nature}, title = {{The multi-cover persistence of Euclidean balls}}, doi = {10.1007/s00454-021-00281-9}, volume = {65}, year = {2021}, } @article{9602, abstract = {An ordered graph is a graph with a linear ordering on its vertex set. We prove that for every positive integer k, there exists a constant ck > 0 such that any ordered graph G on n vertices with the property that neither G nor its complement contains an induced monotone path of size k, has either a clique or an independent set of size at least n^ck . This strengthens a result of Bousquet, Lagoutte, and Thomassé, who proved the analogous result for unordered graphs. A key idea of the above paper was to show that any unordered graph on n vertices that does not contain an induced path of size k, and whose maximum degree is at most c(k)n for some small c(k) > 0, contains two disjoint linear size subsets with no edge between them. This approach fails for ordered graphs, because the analogous statement is false for k ≥ 3, by a construction of Fox. We provide some further examples showing that this statement also fails for ordered graphs avoiding other ordered trees.}, author = {Pach, János and Tomon, István}, issn = {0095-8956}, journal = {Journal of Combinatorial Theory. Series B}, pages = {21--37}, publisher = {Elsevier}, title = {{Erdős-Hajnal-type results for monotone paths}}, doi = {10.1016/j.jctb.2021.05.004}, volume = {151}, year = {2021}, } @article{9821, abstract = {Heart rate variability (hrv) is a physiological phenomenon of the variation in the length of the time interval between consecutive heartbeats. In many cases it could be an indicator of the development of pathological states. The classical approach to the analysis of hrv includes time domain methods and frequency domain methods. However, attempts are still being made to define new and more effective hrv assessment tools. Persistent homology is a novel data analysis tool developed in the recent decades that is rooted at algebraic topology. The Topological Data Analysis (TDA) approach focuses on examining the shape of the data in terms of connectedness and holes, and has recently proved to be very effective in various fields of research. In this paper we propose the use of persistent homology to the hrv analysis. We recall selected topological descriptors used in the literature and we introduce some new topological descriptors that reflect the specificity of hrv, and we discuss their relation to the standard hrv measures. In particular, we show that this novel approach provides a collection of indices that might be at least as useful as the classical parameters in differentiating between series of beat-to-beat intervals (RR-intervals) in healthy subjects and patients suffering from a stroke episode.}, author = {Graff, Grzegorz and Graff, Beata and Pilarczyk, Pawel and Jablonski, Grzegorz and Gąsecki, Dariusz and Narkiewicz, Krzysztof}, issn = {19326203}, journal = {PLoS ONE}, number = {7}, publisher = {Public Library of Science}, title = {{Persistent homology as a new method of the assessment of heart rate variability}}, doi = {10.1371/journal.pone.0253851}, volume = {16}, year = {2021}, } @article{10222, abstract = {Consider a random set of points on the unit sphere in ℝd, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case d = 3, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density.}, author = {Akopyan, Arseniy and Edelsbrunner, Herbert and Nikitenko, Anton}, issn = {1944-950X}, journal = {Experimental Mathematics}, pages = {1--15}, publisher = {Taylor and Francis}, title = {{The beauty of random polytopes inscribed in the 2-sphere}}, doi = {10.1080/10586458.2021.1980459}, year = {2021}, } @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}, }