@article{5678,
abstract = {The order-k Voronoi tessellation of a locally finite set đââđ decomposes âđ 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.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton},
issn = {14320444},
journal = {Discrete and Computational Geometry},
number = {4},
pages = {865â878},
publisher = {Springer},
title = {{PoissonâDelaunay Mosaics of Order k}},
doi = {10.1007/s00454-018-0049-2},
volume = {62},
year = {2019},
}
@article{6050,
abstract = {We answer a question of David Hilbert: given two circles it is not possible in general to construct their centers using only a straightedge. On the other hand, we give infinitely many families of pairs of circles for which such construction is possible. },
author = {Akopyan, Arseniy and Fedorov, Roman},
journal = {Proceedings of the American Mathematical Society},
pages = {91--102},
publisher = {AMS},
title = {{Two circles and only a straightedge}},
doi = {10.1090/proc/14240},
volume = {147},
year = {2019},
}
@article{6515,
abstract = {We give non-degeneracy criteria for Riemannian simplices based on simplices in spaces of constant sectional curvature. It extends previous work on Riemannian simplices, where we developed Riemannian simplices with respect to Euclidean reference simplices. The criteria we give in this article are in terms of quality measures for spaces of constant curvature that we develop here. We see that simplices in spaces that have nearly constant curvature, are already non-degenerate under very weak quality demands. This is of importance because it allows for sampling of Riemannian manifolds based on anisotropy of the manifold and not (absolute) curvature.},
author = {Dyer, Ramsay and Vegter, Gert and Wintraecken, Mathijs},
issn = {1920-180X},
journal = {Journal of Computational Geometry },
number = {1},
pages = {223â256},
publisher = {Carleton University},
title = {{Simplices modelled on spaces of constant curvature}},
doi = {10.20382/jocg.v10i1a9},
volume = {10},
year = {2019},
}
@article{6608,
abstract = {We use the canonical bases produced by the tri-partition algorithm in (Edelsbrunner and ĂlsbĂ¶ck, 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.},
author = {Edelsbrunner, Herbert and ĂlsbĂ¶ck, Katharina},
journal = {Computer Aided Geometric Design},
pages = {1--15},
publisher = {Elsevier},
title = {{Holes and dependences in an ordered complex}},
doi = {10.1016/j.cagd.2019.06.003},
volume = {73},
year = {2019},
}
@inproceedings{6628,
abstract = {Fejes TĂłth [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/(dâ1), where m is the number of vertices and d is the dimension of Euclidean space. Moreover the pro-portionality constant can be expressed in terms of the Gaussian curvature, an intrinsic quantity. In this short note, we prove the extrinsic nature of this constant for manifolds of sufficiently high codimension. We do so by constructing an family of isometric embeddings of the flat torus in Euclidean space.},
author = {Vegter, Gert and Wintraecken, Mathijs},
booktitle = {The 31st Canadian Conference in Computational Geometry},
location = {Edmonton, Canada},
pages = {275--279},
title = {{The extrinsic nature of the Hausdorff distance of optimal triangulations of manifolds}},
year = {2019},
}
@article{6634,
abstract = {In this paper we prove several new results around Gromov's waist theorem. We give a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk-Ulam-Crofton technique, consider waists of real and complex projective spaces, flat tori, convex bodies in Euclidean space; and establish waist-type results in terms of the Hausdorff measure.},
author = {Akopyan, Arseniy and Hubard, Alfredo and Karasev, Roman},
journal = {Topological Methods in Nonlinear Analysis},
number = {2},
pages = {457--490},
publisher = {Akademicka Platforma Czasopism},
title = {{Lower and upper bounds for the waists of different spaces}},
doi = {10.12775/TMNA.2019.008},
volume = {53},
year = {2019},
}
@inproceedings{6648,
abstract = {Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory
needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable the usage of existing computational topology software in this context.},
author = {Edelsbrunner, Herbert and Virk, Ziga and Wagner, Hubert},
booktitle = {35th International Symposium on Computational Geometry},
isbn = {9783959771047},
location = {Portland, OR, United States},
pages = {31:1--31:14},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum fĂŒr Informatik},
title = {{Topological data analysis in information space}},
doi = {10.4230/LIPICS.SOCG.2019.31},
volume = {129},
year = {2019},
}
@article{6671,
abstract = {In this paper we discuss three results. The first two concern general sets of positive reach: we first characterize the reach of a closed set by means of a bound on the metric distortion between the distance measured in the ambient Euclidean space and the shortest path distance measured in the set. Secondly, we prove that the intersection of a ball with radius less than the reach with the set is geodesically convex, meaning that the shortest path between any two points in the intersection lies itself in the intersection. For our third result we focus on manifolds with positive reach and give a bound on the angle between tangent spaces at two different points in terms of the reach and the distance between the two points.},
author = {Boissonnat, Jean-Daniel and Lieutier, AndrĂ© and Wintraecken, Mathijs},
issn = {2367-1734},
journal = {Journal of Applied and Computational Topology},
number = {1-2},
pages = {29â58},
publisher = {Springer Nature},
title = {{The reach, metric distortion, geodesic convexity and the variation of tangent spaces}},
doi = {10.1007/s41468-019-00029-8},
volume = {3},
year = {2019},
}
@article{6756,
abstract = {We study the topology generated by the temperature fluctuations of the cosmic microwave background (CMB) radiation, as quantified by the number of components and holes, formally given by the Betti numbers, in the growing excursion sets. We compare CMB maps observed by the Planck satellite with a thousand simulated maps generated according to the ÎCDM 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 7.33Â°. The survey of the CMB over đ2 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 âmasksâ is of importance, we introduce the concept of relative homology. The parametric Ï2-test shows differences between observations and simulations, yielding p-values at percent to less than permil levels roughly between 2 and 7Â°, with the difference in the number of components and holes peaking at more than 3Ï sporadically at these scales. The highest observed deviation between the observations and simulations for b0 and b1 is approximately between 3Ï and 4Ï at scales of 3â7Â°. There are reports of mildly unusual behaviour of the Euler characteristic at 3.66Â° in the literature, computed from independent measurements of the CMB temperature fluctuations by Planckâs predecessor, the Wilkinson Microwave Anisotropy Probe (WMAP) satellite. The mildly anomalous behaviour of the Euler characteristic is phenomenologically related to the strongly anomalous behaviour of components and holes, or the zeroth and first Betti numbers, respectively. Further, since these topological descriptors show consistent anomalous behaviour over independent measurements of Planck and WMAP, instrumental and systematic errors may be an unlikely source. These are also the scales at which the observed maps exhibit low variance compared to the simulations, and approximately the range of scales at which the power spectrum exhibits a dip with respect to the theoretical model. Non-parametric tests show even stronger differences at almost all scales. Crucially, Gaussian simulations based on power-spectrum matching the characteristics of the observed dipped power spectrum are not able to resolve the anomaly. Understanding the origin of the anomalies in the CMB, whether cosmological in nature or arising due to late-time effects, is an extremely challenging task. Regardless, beyond the trivial possibility that this may still be a manifestation of an extreme Gaussian case, these observations, along with the super-horizon scales involved, may motivate the study of primordial non-Gaussianity. Alternative scenarios worth exploring may be models with non-trivial topology, including topological defect models.},
author = {Pranav, Pratyush and Adler, Robert J. and Buchert, Thomas and Edelsbrunner, Herbert and Jones, Bernard J.T. and Schwartzman, Armin and Wagner, Hubert and Van De Weygaert, Rien},
issn = {14320746},
journal = {Astronomy and Astrophysics},
publisher = {EDP Sciences},
title = {{Unexpected topology of the temperature fluctuations in the cosmic microwave background}},
doi = {10.1051/0004-6361/201834916},
volume = {627},
year = {2019},
}
@article{6793,
abstract = {The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here, we give a simple geometric proof of Regge symmetries in Euclidean, spherical, and hyperbolic geometry.},
author = {Akopyan, Arseniy and Izmestiev, Ivan},
issn = {14692120},
journal = {Bulletin of the London Mathematical Society},
number = {5},
pages = {765--775},
publisher = {London Mathematical Society},
title = {{The Regge symmetry, confocal conics, and the SchlĂ€fli formula}},
doi = {10.1112/blms.12276},
volume = {51},
year = {2019},
}
@article{6828,
abstract = {In this paper we construct a family of exact functors from the category of Whittaker modules of the simple complex Lie algebra of type to the category of finite-dimensional modules of the graded affine Hecke algebra of type . Using results of Backelin [2] and of Arakawa-Suzuki [1], we prove that these functors map standard modules to standard modules (or zero) and simple modules to simple modules (or zero). Moreover, we show that each simple module of the graded affine Hecke algebra appears as the image of a simple Whittaker module. Since the Whittaker category contains the BGG category as a full subcategory, our results generalize results of Arakawa-Suzuki [1], which in turn generalize Schur-Weyl duality between finite-dimensional representations of and representations of the symmetric group .},
author = {Brown, Adam},
issn = {0021-8693},
journal = {Journal of Algebra},
pages = {261--289},
publisher = {Elsevier},
title = {{Arakawa-Suzuki functors for Whittaker modules}},
doi = {10.1016/j.jalgebra.2019.07.027},
volume = {538},
year = {2019},
}
@inproceedings{6989,
abstract = {When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with hole(s) to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special simple holes guarantee foldability. },
author = {Aichholzer, Oswin and Akitaya, Hugo A and Cheung, Kenneth C and Demaine, Erik D and Demaine, Martin L and Fekete, Sandor P and Kleist, Linda and Kostitsyna, Irina and LĂ¶ffler, Maarten and MasĂĄrovĂĄ, Zuzana and Mundilova, Klara and Schmidt, Christiane},
booktitle = {Proceedings of the 31st Canadian Conference on Computational Geometry},
location = {Edmonton, Canada},
pages = {164--170},
publisher = {Canadian Conference on Computational Geometry},
title = {{Folding polyominoes with holes into a cube}},
year = {2019},
}
@phdthesis{201,
abstract = {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.},
author = {Iglesias Ham, Mabel},
pages = {171},
publisher = {IST Austria},
title = {{Multiple covers with balls}},
doi = {10.15479/AT:ISTA:th_1026},
year = {2018},
}
@unpublished{75,
abstract = {We prove that any convex body in the plane can be partitioned into m convex parts of equal areas and perimeters for any integer mâ„2; this result was previously known for prime powers m=pk. We also give a higher-dimensional generalization.},
author = {Akopyan, Arseniy and Avvakumov, Sergey and Karasev, Roman},
pages = {11},
publisher = {arXiv},
title = {{Convex fair partitions into arbitrary number of pieces}},
year = {2018},
}
@article{409,
abstract = {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.},
author = {Akopyan, Arseniy},
issn = {1631073X},
journal = {Comptes Rendus Mathematique},
number = {4},
pages = {412--414},
publisher = {Elsevier},
title = {{On the number of non-hexagons in a planar tiling}},
doi = {10.1016/j.crma.2018.03.005},
volume = {356},
year = {2018},
}
@article{458,
abstract = {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.},
author = {Akopyan, Arseniy and Bobenko, Alexander},
journal = {Transactions of the American Mathematical Society},
number = {4},
pages = {2825 -- 2854},
publisher = {American Mathematical Society},
title = {{Incircular nets and confocal conics}},
doi = {10.1090/tran/7292},
volume = {370},
year = {2018},
}
@article{530,
abstract = {Inclusionâexclusion is an effective method for computing the volume of a union of measurable sets. We extend it to multiple coverings, proving short inclusionâexclusion 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.},
author = {Edelsbrunner, Herbert and Iglesias Ham, Mabel},
journal = {Computational Geometry: Theory and Applications},
pages = {119 -- 133},
publisher = {Elsevier},
title = {{Multiple covers with balls I: Inclusionâexclusion}},
doi = {10.1016/j.comgeo.2017.06.014},
volume = {68},
year = {2018},
}
@article{58,
abstract = {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.},
author = {Akopyan, Arseniy and Segal Halevi, Erel},
journal = {SIAM Journal on Discrete Mathematics},
number = {3},
pages = {2242 -- 2257},
publisher = {Society for Industrial and Applied Mathematics },
title = {{Counting blanks in polygonal arrangements}},
doi = {10.1137/16M110407X},
volume = {32},
year = {2018},
}
@article{6355,
abstract = {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.},
author = {Akopyan, Arseniy and Avvakumov, Sergey},
issn = {2050-5094},
journal = {Forum of Mathematics, Sigma},
publisher = {Cambridge University Press},
title = {{Any cyclic quadrilateral can be inscribed in any closed convex smooth curve}},
doi = {10.1017/fms.2018.7},
volume = {6},
year = {2018},
}
@article{692,
abstract = {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.},
author = {Akopyan, Arseniy},
journal = {Geometriae Dedicata},
number = {1},
pages = {55 -- 64},
publisher = {Springer},
title = {{3-Webs generated by confocal conics and circles}},
doi = {10.1007/s10711-017-0265-6},
volume = {194},
year = {2018},
}