TY - CONF AB - 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. AU - Edelsbrunner, Herbert AU - Nikitenko, Anton AU - Ölsböck, Katharina AU - Synak, Peter ID - 8135 SN - 21932808 T2 - Topological Data Analysis TI - Radius functions on Poisson–Delaunay mosaics and related complexes experimentally VL - 15 ER - TY - JOUR AB - Rhombic dodecahedron is a space filling polyhedron which represents the close packing of spheres in 3D space and the Voronoi structures of the face centered cubic (FCC) lattice. In this paper, we describe a new coordinate system where every 3-integer coordinates grid point corresponds to a rhombic dodecahedron centroid. In order to illustrate the interest of the new coordinate system, we propose the characterization of 3D digital plane with its topological features, such as the interrelation between the thickness of the digital plane and the separability constraint we aim to obtain. We also present the characterization of 3D digital lines and study it as the intersection of multiple digital planes. Characterization of 3D digital sphere with relevant topological features is proposed as well along with the 48-symmetry appearing in the new coordinate system. AU - Biswas, Ranita AU - Largeteau-Skapin, Gaëlle AU - Zrour, Rita AU - Andres, Eric ID - 9249 IS - 1 JF - Mathematical Morphology - Theory and Applications SN - 2353-3390 TI - Digital objects in rhombic dodecahedron grid VL - 4 ER - TY - CONF AB - We call a multigraph non-homotopic if it can be drawn in the plane in such a way that no two edges connecting the same pair of vertices can be continuously transformed into each other without passing through a vertex, and no loop can be shrunk to its end-vertex in the same way. It is easy to see that a non-homotopic multigraph on n>1 vertices can have arbitrarily many edges. We prove that the number of crossings between the edges of a non-homotopic multigraph with n vertices and m>4n edges is larger than cm2n for some constant c>0 , and that this bound is tight up to a polylogarithmic factor. We also show that the lower bound is not asymptotically sharp as n is fixed and m⟶∞ . AU - Pach, János AU - Tardos, Gábor AU - Tóth, Géza ID - 9299 SN - 0302-9743 T2 - 28th International Symposium on Graph Drawing and Network Visualization TI - Crossings between non-homotopic edges VL - 12590 ER - TY - JOUR AB - 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. AU - Edelsbrunner, Herbert AU - Virk, Ziga AU - Wagner, Hubert ID - 9630 IS - 2 JF - Journal of Computational Geometry TI - Topological data analysis in information space VL - 11 ER - TY - JOUR AB - We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods. AU - Akopyan, Arseniy AU - Schwartz, Richard AU - Tabachnikov, Serge ID - 8538 JF - European Journal of Mathematics SN - 2199-675X TI - Billiards in ellipses revisited ER - TY - CONF AB - Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f: ℝ^d → ℝ^(d-n). A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine triangulation 𝒯. This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary. AU - Boissonnat, Jean-Daniel AU - Wintraecken, Mathijs ID - 7952 SN - 1868-8969 T2 - 36th International Symposium on Computational Geometry TI - The topological correctness of PL-approximations of isomanifolds VL - 164 ER - TY - CHAP AB - We study the Gromov waist in the sense of t-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov’s original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no t-neighborhood waist property, including a rather wide class of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2. We use a simpler form of Gromov’s pancake argument to produce some estimates of t-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. In the appendix of this paper we provide for reader’s convenience a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian measures. AU - Akopyan, Arseniy AU - Karasev, Roman ED - Klartag, Bo'az ED - Milman, Emanuel ID - 74 SN - 00758434 T2 - Geometric Aspects of Functional Analysis TI - Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures VL - 2256 ER - TY - JOUR AB - Slicing a Voronoi tessellation in ${R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as a power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane gives a generalized discrete Morse function. Assuming the Voronoi tessellation is generated by a Poisson point process in ${R}^n$, we study the expected number of simplices in the $k$-dimensional weighted Delaunay mosaic as well as the expected number of intervals of the Morse function, both as functions of a radius threshold. As a by-product, we obtain a new proof for the expected number of connected components (clumps) in a line section of a circular Boolean model in ${R}^n$. AU - Edelsbrunner, Herbert AU - Nikitenko, Anton ID - 7554 IS - 4 JF - Theory of Probability and its Applications SN - 0040585X TI - Weighted Poisson–Delaunay mosaics VL - 64 ER - TY - JOUR AB - Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups. AU - Edelsbrunner, Herbert AU - Ölsböck, Katharina ID - 7666 JF - Discrete and Computational Geometry SN - 01795376 TI - Tri-partitions and bases of an ordered complex VL - 64 ER - TY - JOUR AB - A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that some pair of them is not connected by any edge (n→∞). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that almost all string graphs on n vertices are intersection graphs of plane convex sets. AU - Pach, János AU - Reed, Bruce AU - Yuditsky, Yelena ID - 7962 IS - 4 JF - Discrete and Computational Geometry SN - 01795376 TI - Almost all string graphs are intersection graphs of plane convex sets VL - 63 ER - TY - JOUR AU - Pach, János ID - 8323 JF - Discrete and Computational Geometry SN - 01795376 TI - A farewell to Ricky Pollack VL - 64 ER - TY - CONF AB - 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. AU - Graff, Grzegorz AU - Graff, Beata AU - Jablonski, Grzegorz AU - Narkiewicz, Krzysztof ID - 8580 SN - 9781728157511 T2 - 11th Conference of the European Study Group on Cardiovascular Oscillations: Computation and Modelling in Physiology: New Challenges and Opportunities, TI - The application of persistent homology in the analysis of heart rate variability ER - TY - JOUR AB - 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. AU - Akopyan, Arseniy AU - Karasev, Roman ID - 10867 IS - 3 JF - International Mathematics Research Notices KW - General Mathematics SN - 1073-7928 TI - Waist of balls in hyperbolic and spherical spaces VL - 2020 ER - TY - THES AB - 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. AU - Ölsböck, Katharina ID - 7460 KW - shape reconstruction KW - hole manipulation KW - ordered complexes KW - Alpha complex KW - Wrap complex KW - computational topology KW - Bregman geometry SN - 2663-337X TI - The hole system of triangulated shapes ER - TY - THES AB - 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. AU - Masárová, Zuzana ID - 7944 KW - reconfiguration KW - reconfiguration graph KW - triangulations KW - flip KW - constrained triangulations KW - shellability KW - piecewise-linear balls KW - token swapping KW - trees KW - coloured weighted token swapping SN - 2663-337X TI - Reconfiguration problems ER - TY - CONF AB - 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. AU - Osang, Georg F AU - Rouxel-Labbé, Mael AU - Teillaud, Monique ID - 8703 SN - 18688969 T2 - 28th Annual European Symposium on Algorithms TI - Generalizing CGAL periodic Delaunay triangulations VL - 173 ER - TY - JOUR AB - 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. AU - Vegter, Gert AU - Wintraecken, Mathijs ID - 8163 IS - 2 JF - Studia Scientiarum Mathematicarum Hungarica SN - 0081-6906 TI - Refutation of a claim made by Fejes Tóth on the accuracy of surface meshes VL - 57 ER - TY - JOUR AB - 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. AU - Akopyan, Arseniy AU - Edelsbrunner, Herbert ID - 9157 IS - 1 JF - Computational and Mathematical Biophysics SN - 2544-7297 TI - The weighted mean curvature derivative of a space-filling diagram VL - 8 ER - TY - JOUR AB - 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. AU - Akopyan, Arseniy AU - Edelsbrunner, Herbert ID - 9156 IS - 1 JF - Computational and Mathematical Biophysics SN - 2544-7297 TI - The weighted Gaussian curvature derivative of a space-filling diagram VL - 8 ER - TY - JOUR AB - 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. AU - Bauer, U. AU - Edelsbrunner, Herbert AU - Jablonski, Grzegorz AU - Mrozek, M. ID - 15064 IS - 4 JF - Journal of Applied and Computational Topology SN - 2367-1726 TI - Čech-Delaunay gradient flow and homology inference for self-maps VL - 4 ER - TY - JOUR AB - 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. AU - Dyer, Ramsay AU - Vegter, Gert AU - Wintraecken, Mathijs ID - 6515 IS - 1 JF - Journal of Computational Geometry SN - 1920-180X TI - Simplices modelled on spaces of constant curvature VL - 10 ER - TY - CONF AB - 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. AU - Vegter, Gert AU - Wintraecken, Mathijs ID - 6628 T2 - The 31st Canadian Conference in Computational Geometry TI - The extrinsic nature of the Hausdorff distance of optimal triangulations of manifolds ER - TY - CONF AB - 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. AU - Edelsbrunner, Herbert AU - Virk, Ziga AU - Wagner, Hubert ID - 6648 SN - 9783959771047 T2 - 35th International Symposium on Computational Geometry TI - Topological data analysis in information space VL - 129 ER - TY - CONF AB - 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. AU - Aichholzer, Oswin AU - Akitaya, Hugo A AU - Cheung, Kenneth C AU - Demaine, Erik D AU - Demaine, Martin L AU - Fekete, Sandor P AU - Kleist, Linda AU - Kostitsyna, Irina AU - Löffler, Maarten AU - Masárová, Zuzana AU - Mundilova, Klara AU - Schmidt, Christiane ID - 6989 T2 - Proceedings of the 31st Canadian Conference on Computational Geometry TI - Folding polyominoes with holes into a cube ER - TY - JOUR AB - 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. AU - Boissonnat, Jean-Daniel AU - Lieutier, André AU - Wintraecken, Mathijs ID - 6671 IS - 1-2 JF - Journal of Applied and Computational Topology SN - 2367-1726 TI - The reach, metric distortion, geodesic convexity and the variation of tangent spaces VL - 3 ER - TY - JOUR AB - 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. AU - Akopyan, Arseniy AU - Fedorov, Roman ID - 6050 JF - Proceedings of the American Mathematical Society TI - Two circles and only a straightedge VL - 147 ER - TY - JOUR AB - 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. AU - Akopyan, Arseniy AU - Hubard, Alfredo AU - Karasev, Roman ID - 6634 IS - 2 JF - Topological Methods in Nonlinear Analysis TI - Lower and upper bounds for the waists of different spaces VL - 53 ER - TY - JOUR AB - 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. AU - Pranav, Pratyush AU - Adler, Robert J. AU - Buchert, Thomas AU - Edelsbrunner, Herbert AU - Jones, Bernard J.T. AU - Schwartzman, Armin AU - Wagner, Hubert AU - Van De Weygaert, Rien ID - 6756 JF - Astronomy and Astrophysics SN - 00046361 TI - Unexpected topology of the temperature fluctuations in the cosmic microwave background VL - 627 ER - TY - JOUR AB - 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. AU - Akopyan, Arseniy AU - Izmestiev, Ivan ID - 6793 IS - 5 JF - Bulletin of the London Mathematical Society SN - 00246093 TI - The Regge symmetry, confocal conics, and the Schläfli formula VL - 51 ER - TY - JOUR AB - 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 . AU - Brown, Adam ID - 6828 JF - Journal of Algebra SN - 0021-8693 TI - Arakawa-Suzuki functors for Whittaker modules VL - 538 ER - TY - CONF AB - We present LiveTraVeL (Live Transit Vehicle Labeling), a real-time system to label a stream of noisy observations of transit vehicle trajectories with the transit routes they are serving (e.g., northbound bus #5). In order to scale efficiently to large transit networks, our system first retrieves a small set of candidate routes from a geometrically indexed data structure, then applies a fine-grained scoring step to choose the best match. Given that real-time data remains unavailable for the majority of the world’s transit agencies, these inferences can help feed a real-time map of a transit system’s trips, infer transit trip delays in real time, or measure and correct noisy transit tracking data. This system can run on vehicle observations from a variety of sources that don’t attach route information to vehicle observations, such as public imagery streams or user-contributed transit vehicle sightings.We abstract away the specifics of the sensing system and demonstrate the effectiveness of our system on a "semisynthetic" dataset of all New York City buses, where we simulate sensed trajectories by starting with fully labeled vehicle trajectories reported via the GTFS-Realtime protocol, removing the transit route IDs, and perturbing locations with synthetic noise. Using just the geometric shapes of the trajectories, we demonstrate that our system converges on the correct route ID within a few minutes, even after a vehicle switches from serving one trip to the next. AU - Osang, Georg F AU - Cook, James AU - Fabrikant, Alex AU - Gruteser, Marco ID - 7216 SN - 9781538670248 T2 - 2019 IEEE Intelligent Transportation Systems Conference TI - LiveTraVeL: Real-time matching of transit vehicle trajectories to transit routes at scale ER - TY - JOUR AB - The order-k Voronoi tessellation of a locally finite set 𝑋⊆ℝ𝑛 decomposes ℝ𝑛 into convex domains whose points have the same k nearest neighbors in X. Assuming X is a stationary Poisson point process, we give explicit formulas for the expected number and total area of faces of a given dimension per unit volume of space. We also develop a relaxed version of discrete Morse theory and generalize by counting only faces, for which the k nearest points in X are within a given distance threshold. AU - Edelsbrunner, Herbert AU - Nikitenko, Anton ID - 5678 IS - 4 JF - Discrete and Computational Geometry SN - 01795376 TI - Poisson–Delaunay Mosaics of Order k VL - 62 ER - TY - JOUR AB - We use the canonical bases produced by the tri-partition algorithm in (Edelsbrunner and Ölsböck, 2018) to open and close holes in a polyhedral complex, K. In a concrete application, we consider the Delaunay mosaic of a finite set, we let K be an Alpha complex, and we use the persistence diagram of the distance function to guide the hole opening and closing operations. The dependences between the holes define a partial order on the cells in K that characterizes what can and what cannot be constructed using the operations. The relations in this partial order reveal structural information about the underlying filtration of complexes beyond what is expressed by the persistence diagram. AU - Edelsbrunner, Herbert AU - Ölsböck, Katharina ID - 6608 JF - Computer Aided Geometric Design TI - Holes and dependences in an ordered complex VL - 73 ER - TY - GEN AB - The input to the token swapping problem is a graph with vertices v1, v2, . . . , vn, and n tokens with labels 1,2, . . . , n, one on each vertex. The goal is to get token i to vertex vi for all i= 1, . . . , n using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge.Token swapping on a tree, also known as “sorting with a transposition tree,” is not known to be in P nor NP-complete. We present some partial results: 1. An optimum swap sequence may need to perform a swap on a leaf vertex that has the correct token (a “happy leaf”), disproving a conjecture of Vaughan. 2. Any algorithm that fixes happy leaves—as all known approximation algorithms for the problem do—has approximation factor at least 4/3. Furthermore, the two best-known 2-approximation algorithms have approximation factor exactly 2. 3. A generalized problem—weighted coloured token swapping—is NP-complete on trees, but solvable in polynomial time on paths and stars. In this version, tokens and vertices have colours, and colours have weights. The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved. AU - Biniaz, Ahmad AU - Jain, Kshitij AU - Lubiw, Anna AU - Masárová, Zuzana AU - Miltzow, Tillmann AU - Mondal, Debajyoti AU - Naredla, Anurag Murty AU - Tkadlec, Josef AU - Turcotte, Alexi ID - 7950 T2 - arXiv TI - Token swapping on trees ER - TY - CONF AB - Smallest enclosing spheres of finite point sets are central to methods in topological data analysis. Focusing on Bregman divergences to measure dissimilarity, we prove bounds on the location of the center of a smallest enclosing sphere. These bounds depend on the range of radii for which Bregman balls are convex. AU - Edelsbrunner, Herbert AU - Virk, Ziga AU - Wagner, Hubert ID - 188 TI - Smallest enclosing spheres and Chernoff points in Bregman geometry VL - 99 ER - TY - THES AB - We describe arrangements of three-dimensional spheres from a geometrical and topological point of view. Real data (fitting this setup) often consist of soft spheres which show certain degree of deformation while strongly packing against each other. In this context, we answer the following questions: If we model a soft packing of spheres by hard spheres that are allowed to overlap, can we measure the volume in the overlapped areas? Can we be more specific about the overlap volume, i.e. quantify how much volume is there covered exactly twice, three times, or k times? What would be a good optimization criteria that rule the arrangement of soft spheres while making a good use of the available space? Fixing a particular criterion, what would be the optimal sphere configuration? The first result of this thesis are short formulas for the computation of volumes covered by at least k of the balls. The formulas exploit information contained in the order-k Voronoi diagrams and its closely related Level-k complex. The used complexes lead to a natural generalization into poset diagrams, a theoretical formalism that contains the order-k and degree-k diagrams as special cases. In parallel, we define different criteria to determine what could be considered an optimal arrangement from a geometrical point of view. Fixing a criterion, we find optimal soft packing configurations in 2D and 3D where the ball centers lie on a lattice. As a last step, we use tools from computational topology on real physical data, to show the potentials of higher-order diagrams in the description of melting crystals. The results of the experiments leaves us with an open window to apply the theories developed in this thesis in real applications. AU - Iglesias Ham, Mabel ID - 201 SN - 2663-337X TI - Multiple covers with balls ER - TY - CONF AB - 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 from Delaunay mosaics that is isomorphic to the persistence module of the multi-covers. AU - Edelsbrunner, Herbert AU - Osang, Georg F ID - 187 TI - The multi-cover persistence of Euclidean balls VL - 99 ER - TY - JOUR AB - We consider families of confocal conics and two pencils of Apollonian circles having the same foci. We will show that these families of curves generate trivial 3-webs and find the exact formulas describing them. AU - Akopyan, Arseniy ID - 692 IS - 1 JF - Geometriae Dedicata TI - 3-Webs generated by confocal conics and circles VL - 194 ER - TY - JOUR AB - Inside a two-dimensional region (``cake""), there are m nonoverlapping tiles of a certain kind (``toppings""). We want to expand the toppings while keeping them nonoverlapping, and possibly add some blank pieces of the same ``certain kind,"" such that the entire cake is covered. How many blanks must we add? We study this question in several cases: (1) The cake and toppings are general polygons. (2) The cake and toppings are convex figures. (3) The cake and toppings are axis-parallel rectangles. (4) The cake is an axis-parallel rectilinear polygon and the toppings are axis-parallel rectangles. In all four cases, we provide tight bounds on the number of blanks. AU - Akopyan, Arseniy AU - Segal Halevi, Erel ID - 58 IS - 3 JF - SIAM Journal on Discrete Mathematics TI - Counting blanks in polygonal arrangements VL - 32 ER - TY - JOUR AB - We consider congruences of straight lines in a plane with the combinatorics of the square grid, with all elementary quadrilaterals possessing an incircle. It is shown that all the vertices of such nets (we call them incircular or IC-nets) lie on confocal conics. Our main new results are on checkerboard IC-nets in the plane. These are congruences of straight lines in the plane with the combinatorics of the square grid, combinatorially colored as a checkerboard, such that all black coordinate quadrilaterals possess inscribed circles. We show how this larger class of IC-nets appears quite naturally in Laguerre geometry of oriented planes and spheres and leads to new remarkable incidence theorems. Most of our results are valid in hyperbolic and spherical geometries as well. We present also generalizations in spaces of higher dimension, called checkerboard IS-nets. The construction of these nets is based on a new 9 inspheres incidence theorem. AU - Akopyan, Arseniy AU - Bobenko, Alexander ID - 458 IS - 4 JF - Transactions of the American Mathematical Society TI - Incircular nets and confocal conics VL - 370 ER - TY - JOUR AB - The goal of this article is to introduce the reader to the theory of intrinsic geometry of convex surfaces. We illustrate the power of the tools by proving a theorem on convex surfaces containing an arbitrarily long closed simple geodesic. Let us remind ourselves that a curve in a surface is called geodesic if every sufficiently short arc of the curve is length minimizing; if, in addition, it has no self-intersections, we call it simple geodesic. A tetrahedron with equal opposite edges is called isosceles. The axiomatic method of Alexandrov geometry allows us to work with the metrics of convex surfaces directly, without approximating it first by a smooth or polyhedral metric. Such approximations destroy the closed geodesics on the surface; therefore it is difficult (if at all possible) to apply approximations in the proof of our theorem. On the other hand, a proof in the smooth or polyhedral case usually admits a translation into Alexandrov’s language; such translation makes the result more general. In fact, our proof resembles a translation of the proof given by Protasov. Note that the main theorem implies in particular that a smooth convex surface does not have arbitrarily long simple closed geodesics. However we do not know a proof of this corollary that is essentially simpler than the one presented below. AU - Akopyan, Arseniy AU - Petrunin, Anton ID - 106 IS - 3 JF - Mathematical Intelligencer TI - Long geodesics on convex surfaces VL - 40 ER - TY - JOUR AB - Inclusion–exclusion is an effective method for computing the volume of a union of measurable sets. We extend it to multiple coverings, proving short inclusion–exclusion formulas for the subset of Rn covered by at least k balls in a finite set. We implement two of the formulas in dimension n=3 and report on results obtained with our software. AU - Edelsbrunner, Herbert AU - Iglesias Ham, Mabel ID - 530 JF - Computational Geometry: Theory and Applications TI - Multiple covers with balls I: Inclusion–exclusion VL - 68 ER - TY - CONF AB - We show attacks on five data-independent memory-hard functions (iMHF) that were submitted to the password hashing competition (PHC). Informally, an MHF is a function which cannot be evaluated on dedicated hardware, like ASICs, at significantly lower hardware and/or energy cost than evaluating a single instance on a standard single-core architecture. Data-independent means the memory access pattern of the function is independent of the input; this makes iMHFs harder to construct than data-dependent ones, but the latter can be attacked by various side-channel attacks. Following [Alwen-Blocki'16], we capture the evaluation of an iMHF as a directed acyclic graph (DAG). The cumulative parallel pebbling complexity of this DAG is a measure for the hardware cost of evaluating the iMHF on an ASIC. Ideally, one would like the complexity of a DAG underlying an iMHF to be as close to quadratic in the number of nodes of the graph as possible. Instead, we show that (the DAGs underlying) the following iMHFs are far from this bound: Rig.v2, TwoCats and Gambit each having an exponent no more than 1.75. Moreover, we show that the complexity of the iMHF modes of the PHC finalists Pomelo and Lyra2 have exponents at most 1.83 and 1.67 respectively. To show this we investigate a combinatorial property of each underlying DAG (called its depth-robustness. By establishing upper bounds on this property we are then able to apply the general technique of [Alwen-Block'16] for analyzing the hardware costs of an iMHF. AU - Alwen, Joel F AU - Gazi, Peter AU - Kamath Hosdurg, Chethan AU - Klein, Karen AU - Osang, Georg F AU - Pietrzak, Krzysztof Z AU - Reyzin, Lenoid AU - Rolinek, Michal AU - Rybar, Michal ID - 193 T2 - Proceedings of the 2018 on Asia Conference on Computer and Communication Security TI - On the memory hardness of data independent password hashing functions ER - TY - JOUR AB - Motivated by biological questions, we study configurations of equal spheres that neither pack nor cover. Placing their centers on a lattice, we define the soft density of the configuration by penalizing multiple overlaps. Considering the 1-parameter family of diagonally distorted 3-dimensional integer lattices, we show that the soft density is maximized at the FCC lattice. AU - Edelsbrunner, Herbert AU - Iglesias Ham, Mabel ID - 312 IS - 1 JF - SIAM J Discrete Math SN - 08954801 TI - On the optimality of the FCC lattice for soft sphere packing VL - 32 ER - TY - JOUR AB - We give a simple proof of T. Stehling's result [4], whereby in any normal tiling of the plane with convex polygons with number of sides not less than six, all tiles except a finite number are hexagons. AU - Akopyan, Arseniy ID - 409 IS - 4 JF - Comptes Rendus Mathematique SN - 1631073X TI - On the number of non-hexagons in a planar tiling VL - 356 ER - TY - JOUR AB - Using the geodesic distance on the n-dimensional sphere, we study the expected radius function of the Delaunay mosaic of a random set of points. Specifically, we consider the partition of the mosaic into intervals of the radius function and determine the expected number of intervals whose radii are less than or equal to a given threshold. We find that the expectations are essentially the same as for the Poisson–Delaunay mosaic in n-dimensional Euclidean space. Assuming the points are not contained in a hemisphere, the Delaunay mosaic is isomorphic to the boundary complex of the convex hull in Rn+1, so we also get the expected number of faces of a random inscribed polytope. As proved in Antonelli et al. [Adv. in Appl. Probab. 9–12 (1977–1980)], an orthant section of the n-sphere is isometric to the standard n-simplex equipped with the Fisher information metric. It follows that the latter space has similar stochastic properties as the n-dimensional Euclidean space. Our results are therefore relevant in information geometry and in population genetics. AU - Edelsbrunner, Herbert AU - Nikitenko, Anton ID - 87 IS - 5 JF - Annals of Applied Probability TI - Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics VL - 28 ER - TY - JOUR AB - We prove that any cyclic quadrilateral can be inscribed in any closed convex C1-curve. The smoothness condition is not required if the quadrilateral is a rectangle. AU - Akopyan, Arseniy AU - Avvakumov, Sergey ID - 6355 JF - Forum of Mathematics, Sigma SN - 2050-5094 TI - Any cyclic quadrilateral can be inscribed in any closed convex smooth curve VL - 6 ER - TY - JOUR AB - In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erdős: Given a family of (round) disks of radii r1, … , rn in the plane, it is always possible to cover them by a disk of radius R= ∑ ri, provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially the same idea may work for different analogues and generalizations of their result. In particular, we prove the following: Given a family of positive homothetic copies of a fixed convex body K⊂ Rd with homothety coefficients τ1, … , τn> 0 , it is always possible to cover them by a translate of d+12(∑τi)K, provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets. AU - Akopyan, Arseniy AU - Balitskiy, Alexey AU - Grigorev, Mikhail ID - 1064 IS - 4 JF - Discrete & Computational Geometry SN - 01795376 TI - On the circle covering theorem by A.W. Goodman and R.E. Goodman VL - 59 ER - TY - GEN AB - We prove that any convex body in the plane can be partitioned into m convex parts of equal areas and perimeters for any integer m≥2; this result was previously known for prime powers m=pk. We also give a higher-dimensional generalization. AU - Akopyan, Arseniy AU - Avvakumov, Sergey AU - Karasev, Roman ID - 75 TI - Convex fair partitions into arbitrary number of pieces ER - TY - JOUR AB - We introduce planar matchings on directed pseudo-line arrangements, which yield a planar set of pseudo-line segments such that only matching-partners are adjacent. By translating the planar matching problem into a corresponding stable roommates problem we show that such matchings always exist. Using our new framework, we establish, for the first time, a complete, rigorous definition of weighted straight skeletons, which are based on a so-called wavefront propagation process. We present a generalized and unified approach to treat structural changes in the wavefront that focuses on the restoration of weak planarity by finding planar matchings. AU - Biedl, Therese AU - Huber, Stefan AU - Palfrader, Peter ID - 481 IS - 3-4 JF - International Journal of Computational Geometry and Applications TI - Planar matchings for weighted straight skeletons VL - 26 ER -