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 - 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 - 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 - Coxeter triangulations are triangulations of Euclidean space based on a single simplex. By this we mean that given an individual simplex we can recover the entire triangulation of Euclidean space by inductively reflecting in the faces of the simplex. In this paper we establish that the quality of the simplices in all Coxeter triangulations is O(1/d−−√) of the quality of regular simplex. We further investigate the Delaunay property for these triangulations. Moreover, we consider an extension of the Delaunay property, namely protection, which is a measure of non-degeneracy of a Delaunay triangulation. In particular, one family of Coxeter triangulations achieves the protection O(1/d2). We conjecture that both bounds are optimal for triangulations in Euclidean space.
AU - Choudhary, Aruni
AU - Kachanovich, Siargey
AU - Wintraecken, Mathijs
ID - 7567
JF - Mathematics in Computer Science
SN - 1661-8270
TI - Coxeter triangulations have good quality
ER -
TY - GEN
AB - Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e.manifolds defined as the zero set of some multivariate multivalued functionf:Rd→Rd−n.A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear(PL) approximation based on a triangulationTof the ambient spaceRd. In this paper, we giveconditions under which the PL-approximation of an isomanifold is topologically equivalent to theisomanifold. The conditions can always be met by taking a sufficiently fine triangulationT.
AU - Boissonnat, Jean-Daniel
AU - Wintraecken, Mathijs
ID - 7568
T2 - EUROCG 2020
TI - The topological correctness of the PL-approximation of isomanifolds
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
ER -
TY - JOUR
AB - Extending a result of Milena Radnovic and Serge Tabachnikov, we establish conditionsfor two different non-symmetric norms to define the same billiard reflection law.
AU - Akopyan, Arseniy
AU - Karasev, Roman
ID - 7791
JF - European Journal of Mathematics
SN - 2199675X
TI - When different norms lead to same billiard trajectories?
ER -
TY - JOUR
AB - We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1355–1370, ACM, New York, 2012), and the cohomology stratification algorithm given in Nanda (Found. Comput. Math. 20(2), 195–222, 2020). Additionally, we give examples of stratifications based on the geometric techniques of Breiding et al. (Rev. Mat. Complut. 31(3), 545–593, 2018), illustrating how the sheaf-theoretic approach can be used to study stratifications from both topological and geometric perspectives. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory in generalizing existing algorithms.
AU - Brown, Adam
AU - Wang, Bei
ID - 7905
JF - Discrete & Computational Geometry
SN - 0179-5376
TI - Sheaf-theoretic stratification learning from geometric and topological perspectives
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 - 978-3-99078-005-3
TI - Reconfiguration problems
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 - 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
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 - We consider the following setting: suppose that we are given a manifold M in Rd with positive reach. Moreover assume that we have an embedded simplical complex A without boundary, whose vertex set lies on the manifold, is sufficiently dense and such that all simplices in A have sufficient quality. We prove that if, locally, interiors of the projection of the simplices onto the tangent space do not intersect, then A is a triangulation of the manifold, that is, they are homeomorphic.
AU - Boissonnat, Jean-Daniel
AU - Dyer, Ramsay
AU - Ghosh, Arijit
AU - Lieutier, Andre
AU - Wintraecken, Mathijs
ID - 8248
JF - Discrete & Computational Geometry
SN - 0179-5376
TI - Local conditions for triangulating submanifolds of Euclidean space
ER -
TY - JOUR
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 one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special “basic” holes guarantee foldability.
AU - Aichholzer, Oswin
AU - Akitaya, Hugo A.
AU - Cheung, Kenneth C.
AU - Demaine, Erik D.
AU - Demaine, Martin L.
AU - Fekete, Sándor P.
AU - Kleist, Linda
AU - Kostitsyna, Irina
AU - Löffler, Maarten
AU - Masárová, Zuzana
AU - Mundilova, Klara
AU - Schmidt, Christiane
ID - 8317
JF - Computational Geometry: Theory and Applications
SN - 09257721
TI - Folding polyominoes with holes into a cube
VL - 93
ER -
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 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 - 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 - 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 - 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 - 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 - 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 -