TY - JOUR
AU - Pach, János
ID - 8323
JF - Discrete and Computational Geometry
SN - 01795376
TI - A farewell to Ricky Pollack
ER -
TY - JOUR
AB - Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics that are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines that are diagonally related to lines of curvature is proved theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory.
AU - Akopyan, Arseniy
AU - Bobenko, Alexander I.
AU - Schief, Wolfgang K.
AU - Techter, Jan
ID - 8338
JF - Discrete and Computational Geometry
SN - 01795376
TI - On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs
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 - 2199675X
TI - Billiards in ellipses revisited
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 - 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:1903.06981
TI - Token swapping on trees
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 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 - 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 - 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 - 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 - 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 - 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 - 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 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 - 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 - 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
TI - Multiple covers with balls
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 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 - 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 - 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 - 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 - 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 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - JOUR
AB - Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in ℝ n , we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and nonsingular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we obtain the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensions n ≤ 4.
AU - Edelsbrunner, Herbert
AU - Nikitenko, Anton
AU - Reitzner, Matthias
ID - 718
IS - 3
JF - Advances in Applied Probability
SN - 00018678
TI - Expected sizes of poisson Delaunay mosaics and their discrete Morse functions
VL - 49
ER -
TY - JOUR
AB - We generalize Brazas’ topology on the fundamental group to the whole universal path space X˜ i.e., to the set of homotopy classes of all based paths. We develop basic properties of the new notion and provide a complete comparison of the obtained topology with the established topologies, in particular with the Lasso topology and the CO topology, i.e., the topology that is induced by the compact-open topology. It turns out that the new topology is the finest topology contained in the CO topology, for which the action of the fundamental group on the universal path space is a continuous group action.
AU - Virk, Ziga
AU - Zastrow, Andreas
ID - 737
JF - Topology and its Applications
SN - 01668641
TI - A new topology on the universal path space
VL - 231
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 -
TY - JOUR
AB - Let X and Y be proper metric spaces. We show that a coarsely n-to-1 map f:X→Y induces an n-to-1 map of Higson coronas. This viewpoint turns out to be successful in showing that the classical dimension raising theorems hold in large scale; that is, if f:X→Y is a coarsely n-to-1 map between proper metric spaces X and Y then asdim(Y)≤asdim(X)+n−1. Furthermore we introduce coarsely open coarsely n-to-1 maps, which include the natural quotient maps via a finite group action, and prove that they preserve the asymptotic dimension.
AU - Austin, Kyle
AU - Virk, Ziga
ID - 521
JF - Topology and its Applications
SN - 01668641
TI - Higson compactification and dimension raising
VL - 215
ER -
TY - JOUR
AB - We study robust properties of zero sets of continuous maps f: X → ℝn. Formally, we analyze the family Z< r(f) := (g-1(0): ||g - f|| < r) of all zero sets of all continuous maps g closer to f than r in the max-norm. All of these sets are outside A := (x: |f(x)| ≥ r) and we claim that Z< r(f) is fully determined by A and an element of a certain cohomotopy group which (by a recent result) is computable whenever the dimension of X is at most 2n - 3. By considering all r > 0 simultaneously, the pointed cohomotopy groups form a persistence module-a structure leading to persistence diagrams as in the case of persistent homology or well groups. Eventually, we get a descriptor of persistent robust properties of zero sets that has better descriptive power (Theorem A) and better computability status (Theorem B) than the established well diagrams. Moreover, if we endow every point of each zero set with gradients of the perturbation, the robust description of the zero sets by elements of cohomotopy groups is in some sense the best possible (Theorem C).
AU - Franek, Peter
AU - Krcál, Marek
ID - 568
IS - 2
JF - Homology, Homotopy and Applications
SN - 15320073
TI - Persistence of zero sets
VL - 19
ER -
TY - THES
AB - The main objects considered in the present work are simplicial and CW-complexes with vertices forming a random point cloud. In particular, we consider a Poisson point process in R^n and study Delaunay and Voronoi complexes of the first and higher orders and weighted Delaunay complexes obtained as sections of Delaunay complexes, as well as the Čech complex. Further, we examine theDelaunay complex of a Poisson point process on the sphere S^n, as well as of a uniform point cloud, which is equivalent to the convex hull, providing a connection to the theory of random polytopes. Each of the complexes in question can be endowed with a radius function, which maps its cells to the radii of appropriately chosen circumspheres, called the radius of the cell. Applying and developing discrete Morse theory for these functions, joining it together with probabilistic and sometimes analytic machinery, and developing several integral geometric tools, we aim at getting the distributions of circumradii of typical cells. For all considered complexes, we are able to generalize and obtain up to constants the distribution of radii of typical intervals of all types. In low dimensions the constants can be computed explicitly, thus providing the explicit expressions for the expected numbers of cells. In particular, it allows to find the expected density of simplices of every dimension for a Poisson point process in R^4, whereas the result for R^3 was known already in 1970's.
AU - Nikitenko, Anton
ID - 6287
TI - Discrete Morse theory for random complexes
ER -
TY - CONF
AB - We show that the framework of topological data analysis can be extended from metrics to general Bregman divergences, widening the scope of possible applications. Examples are the Kullback - Leibler divergence, which is commonly used for comparing text and images, and the Itakura - Saito divergence, popular for speech and sound. In particular, we prove that appropriately generalized čech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized čech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory.
AU - Edelsbrunner, Herbert
AU - Wagner, Hubert
ID - 688
SN - 18688969
TI - Topological data analysis with Bregman divergences
VL - 77
ER -
TY - JOUR
AB - We answer a question of M. Gromov on the waist of the unit ball.
AU - Akopyan, Arseniy
AU - Karasev, Roman
ID - 707
IS - 4
JF - Bulletin of the London Mathematical Society
SN - 00246093
TI - A tight estimate for the waist of the ball
VL - 49
ER -
TY - CONF
AB - We present an efficient algorithm to compute Euler characteristic curves of gray scale images of arbitrary dimension. In various applications the Euler characteristic curve is used as a descriptor of an image. Our algorithm is the first streaming algorithm for Euler characteristic curves. The usage of streaming removes the necessity to store the entire image in RAM. Experiments show that our implementation handles terabyte scale images on commodity hardware. Due to lock-free parallelism, it scales well with the number of processor cores. Additionally, we put the concept of the Euler characteristic curve in the wider context of computational topology. In particular, we explain the connection with persistence diagrams.
AU - Heiss, Teresa
AU - Wagner, Hubert
ED - Felsberg, Michael
ED - Heyden, Anders
ED - Krüger, Norbert
ID - 833
SN - 03029743
TI - Streaming algorithm for Euler characteristic curves of multidimensional images
VL - 10424
ER -
TY - CONF
AB - Recent research has examined how to study the topological features of a continuous self-map by means of the persistence of the eigenspaces, for given eigenvalues, of the endomorphism induced in homology over a field. This raised the question of how to select dynamically significant eigenvalues. The present paper aims to answer this question, giving an algorithm that computes the persistence of eigenspaces for every eigenvalue simultaneously, also expressing said eigenspaces as direct sums of “finite” and “singular” subspaces.
AU - Ethier, Marc
AU - Jablonski, Grzegorz
AU - Mrozek, Marian
ID - 836
SN - 978-331956930-7
T2 - Special Sessions in Applications of Computer Algebra
TI - Finding eigenvalues of self-maps with the Kronecker canonical form
VL - 198
ER -
TY - CHAP
AB - The advent of high-throughput technologies and the concurrent advances in information sciences have led to a data revolution in biology. This revolution is most significant in molecular biology, with an increase in the number and scale of the “omics” projects over the last decade. Genomics projects, for example, have produced impressive advances in our knowledge of the information concealed into genomes, from the many genes that encode for the proteins that are responsible for most if not all cellular functions, to the noncoding regions that are now known to provide regulatory functions. Proteomics initiatives help to decipher the role of post-translation modifications on the protein structures and provide maps of protein-protein interactions, while functional genomics is the field that attempts to make use of the data produced by these projects to understand protein functions. The biggest challenge today is to assimilate the wealth of information provided by these initiatives into a conceptual framework that will help us decipher life. For example, the current views of the relationship between protein structure and function remain fragmented. We know of their sequences, more and more about their structures, we have information on their biological activities, but we have difficulties connecting this dotted line into an informed whole. We lack the experimental and computational tools for directly studying protein structure, function, and dynamics at the molecular and supra-molecular levels. In this chapter, we review some of the current developments in building the computational tools that are needed, focusing on the role that geometry and topology play in these efforts. One of our goals is to raise the general awareness about the importance of geometric methods in elucidating the mysterious foundations of our very existence. Another goal is the broadening of what we consider a geometric algorithm. There is plenty of valuable no-man’s-land between combinatorial and numerical algorithms, and it seems opportune to explore this land with a computational-geometric frame of mind.
AU - Edelsbrunner, Herbert
AU - Koehl, Patrice
ED - Toth, Csaba
ED - O'Rourke, Joseph
ED - Goodman, Jacob
ID - 84
T2 - Handbook of Discrete and Computational Geometry, Third Edition
TI - Computational topology for structural molecular biology
ER -
TY - JOUR
AB - We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that, for any convex shape K, there exist four points on the boundary of K such that the length of any curve passing through these points is at least half of the perimeter of K. It is also shown that the same statement does not remain valid with the additional constraint that the points are extreme points of K. Moreover, the factor ½ cannot be achieved with any fixed number of extreme points. We conclude the paper with a few other inequalities related to the perimeter of a convex shape.
AU - Akopyan, Arseniy
AU - Vysotsky, Vladislav
ID - 909
IS - 7
JF - The American Mathematical Monthly
SN - 00029890
TI - On the lengths of curves passing through boundary points of a planar convex shape
VL - 124
ER -
TY - JOUR
AB - In this article we define an algebraic vertex of a generalized polyhedron and show that the set of algebraic vertices is the smallest set of points needed to define the polyhedron. We prove that the indicator function of a generalized polytope P is a linear combination of indicator functions of simplices whose vertices are algebraic vertices of P. We also show that the indicator function of any generalized polyhedron is a linear combination, with integer coefficients, of indicator functions of cones with apices at algebraic vertices and line-cones. The concept of an algebraic vertex is closely related to the Fourier–Laplace transform. We show that a point v is an algebraic vertex of a generalized polyhedron P if and only if the tangent cone of P, at v, has non-zero Fourier–Laplace transform.
AU - Akopyan, Arseniy
AU - Bárány, Imre
AU - Robins, Sinai
ID - 1180
JF - Advances in Mathematics
SN - 00018708
TI - Algebraic vertices of non-convex polyhedra
VL - 308
ER -
TY - JOUR
AB - Phat is an open-source C. ++ library for the computation of persistent homology by matrix reduction, targeted towards developers of software for topological data analysis. We aim for a simple generic design that decouples algorithms from data structures without sacrificing efficiency or user-friendliness. We provide numerous different reduction strategies as well as data types to store and manipulate the boundary matrix. We compare the different combinations through extensive experimental evaluation and identify optimization techniques that work well in practical situations. We also compare our software with various other publicly available libraries for persistent homology.
AU - Bauer, Ulrich
AU - Kerber, Michael
AU - Reininghaus, Jan
AU - Wagner, Hubert
ID - 1433
JF - Journal of Symbolic Computation
SN - 07477171
TI - Phat - Persistent homology algorithms toolbox
VL - 78
ER -
TY - JOUR
AB - We introduce a multiscale topological description of the Megaparsec web-like cosmic matter distribution. Betti numbers and topological persistence offer a powerful means of describing the rich connectivity structure of the cosmic web and of its multiscale arrangement of matter and galaxies. Emanating from algebraic topology and Morse theory, Betti numbers and persistence diagrams represent an extension and deepening of the cosmologically familiar topological genus measure and the related geometric Minkowski functionals. In addition to a description of the mathematical background, this study presents the computational procedure for computing Betti numbers and persistence diagrams for density field filtrations. The field may be computed starting from a discrete spatial distribution of galaxies or simulation particles. The main emphasis of this study concerns an extensive and systematic exploration of the imprint of different web-like morphologies and different levels of multiscale clustering in the corresponding computed Betti numbers and persistence diagrams. To this end, we use Voronoi clustering models as templates for a rich variety of web-like configurations and the fractal-like Soneira-Peebles models exemplify a range of multiscale configurations. We have identified the clear imprint of cluster nodes, filaments, walls, and voids in persistence diagrams, along with that of the nested hierarchy of structures in multiscale point distributions. We conclude by outlining the potential of persistent topology for understanding the connectivity structure of the cosmic web, in large simulations of cosmic structure formation and in the challenging context of the observed galaxy distribution in large galaxy surveys.
AU - Pranav, Pratyush
AU - Edelsbrunner, Herbert
AU - Van De Weygaert, Rien
AU - Vegter, Gert
AU - Kerber, Michael
AU - Jones, Bernard
AU - Wintraecken, Mathijs
ID - 1022
IS - 4
JF - Monthly Notices of the Royal Astronomical Society
SN - 00358711
TI - The topology of the cosmic web in terms of persistent Betti numbers
VL - 465
ER -
TY - JOUR
AB - We consider the problem of reachability in pushdown graphs. We study the problem for pushdown graphs with constant treewidth. Even for pushdown graphs with treewidth 1, for the reachability problem we establish the following: (i) the problem is PTIME-complete, and (ii) any subcubic algorithm for the problem would contradict the k-clique conjecture and imply faster combinatorial algorithms for cliques in graphs.
AU - Chatterjee, Krishnendu
AU - Osang, Georg F
ID - 1065
JF - Information Processing Letters
SN - 00200190
TI - Pushdown reachability with constant treewidth
VL - 122
ER -
TY - JOUR
AB - Given a finite set of points in Rn and a radius parameter, we study the Čech, Delaunay–Čech, Delaunay (or alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, which are explicitly described by a single discrete gradient field.
AU - Bauer, Ulrich
AU - Edelsbrunner, Herbert
ID - 1072
IS - 5
JF - Transactions of the American Mathematical Society
TI - The Morse theory of Čech and delaunay complexes
VL - 369
ER -
TY - JOUR
AB - We introduce the Voronoi functional of a triangulation of a finite set of points in the Euclidean plane and prove that among all geometric triangulations of the point set, the Delaunay triangulation maximizes the functional. This result neither extends to topological triangulations in the plane nor to geometric triangulations in three and higher dimensions.
AU - Edelsbrunner, Herbert
AU - Glazyrin, Alexey
AU - Musin, Oleg
AU - Nikitenko, Anton
ID - 1173
IS - 5
JF - Combinatorica
SN - 02099683
TI - The Voronoi functional is maximized by the Delaunay triangulation in the plane
VL - 37
ER -