@article{12086,
abstract = {We present a simple algorithm for computing higher-order Delaunay mosaics that works in Euclidean spaces of any finite dimensions. The algorithm selects the vertices of the order-k mosaic from incrementally constructed lower-order mosaics and uses an algorithm for weighted first-order Delaunay mosaics as a black-box to construct the order-k mosaic from its vertices. Beyond this black-box, the algorithm uses only combinatorial operations, thus facilitating easy implementation. We extend this algorithm to compute higher-order α-shapes and provide open-source implementations. We present experimental results for properties of higher-order Delaunay mosaics of random point sets.},
author = {Edelsbrunner, Herbert and Osang, Georg F},
issn = {1432-0541},
journal = {Algorithmica},
pages = {277--295},
publisher = {Springer Nature},
title = {{A simple algorithm for higher-order Delaunay mosaics and alpha shapes}},
doi = {10.1007/s00453-022-01027-6},
volume = {85},
year = {2023},
}
@article{12287,
abstract = {We present criteria for establishing a triangulation of a manifold. Given a manifold M, a simplicial complex A, and a map H from the underlying space of A to M, our criteria are presented in local coordinate charts for M, and ensure that H is a homeomorphism. These criteria do not require a differentiable structure, or even an explicit metric on M. No Delaunay property of A is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. Because the criteria are easily verified in such a setting, they are expected to be of general use.},
author = {Boissonnat, Jean-Daniel and Dyer, Ramsay and Ghosh, Arijit and Wintraecken, Mathijs},
issn = {1432-0444},
journal = {Discrete & Computational Geometry},
keywords = {Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology, Theoretical Computer Science},
pages = {156--191},
publisher = {Springer Nature},
title = {{Local criteria for triangulating general manifolds}},
doi = {10.1007/s00454-022-00431-7},
volume = {69},
year = {2023},
}
@article{12544,
abstract = {Geometry is crucial in our efforts to comprehend the structures and dynamics of biomolecules. For example, volume, surface area, and integrated mean and Gaussian curvature of the union of balls representing a molecule are used to quantify its interactions with the water surrounding it in the morphometric implicit solvent models. The Alpha Shape theory provides an accurate and reliable method for computing these geometric measures. In this paper, we derive homogeneous formulas for the expressions of these measures and their derivatives with respect to the atomic coordinates, and we provide algorithms that implement them into a new software package, AlphaMol. The only variables in these formulas are the interatomic distances, making them insensitive to translations and rotations. AlphaMol includes a sequential algorithm and a parallel algorithm. In the parallel version, we partition the atoms of the molecule of interest into 3D rectangular blocks, using a kd-tree algorithm. We then apply the sequential algorithm of AlphaMol to each block, augmented by a buffer zone to account for atoms whose ball representations may partially cover the block. The current parallel version of AlphaMol leads to a 20-fold speed-up compared to an independent serial implementation when using 32 processors. For instance, it takes 31 s to compute the geometric measures and derivatives of each atom in a viral capsid with more than 26 million atoms on 32 Intel processors running at 2.7 GHz. The presence of the buffer zones, however, leads to redundant computations, which ultimately limit the impact of using multiple processors. AlphaMol is available as an OpenSource software.},
author = {Koehl, Patrice and Akopyan, Arseniy and Edelsbrunner, Herbert},
issn = {1549-960X},
journal = {Journal of Chemical Information and Modeling},
publisher = {American Chemical Society},
title = {{Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives}},
doi = {10.1021/acs.jcim.2c01346},
year = {2023},
}
@inproceedings{12548,
abstract = {The limited exchange between human communities is a key factor in preventing the spread of COVID-19. This paper introduces a digital framework that combines an integration of real mobility data at the country scale with a series of modeling techniques and visual capabilities that highlight mobility patterns before and during the pandemic. The findings not only significantly exhibit mobility trends and different degrees of similarities at regional and local levels but also provide potential insight into the emergence of a pandemic on human behavior patterns and their likely socio-economic impacts.},
author = {Forghani, Mohammad and Claramunt, Christophe and Karimipour, Farid and Heiler, Georg},
booktitle = {2022 IEEE International Conference on Data Mining Workshops},
issn = {2375-9259},
location = {Orlando, FL, United States},
publisher = {Institute of Electrical and Electronics Engineers},
title = {{Visual analytics of mobility network changes observed using mobile phone data during COVID-19 pandemic}},
doi = {10.1109/icdmw58026.2022.00093},
year = {2023},
}
@article{12709,
abstract = {Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness.},
author = {Corbet, René and Kerber, Michael and Lesnick, Michael and Osang, Georg F},
issn = {1432-0444},
journal = {Discrete and Computational Geometry},
publisher = {Springer Nature},
title = {{Computing the multicover bifiltration}},
doi = {10.1007/s00454-022-00476-8},
year = {2023},
}
@article{12763,
abstract = {Kleinjohann (Archiv der Mathematik 35(1):574–582, 1980; Mathematische Zeitschrift 176(3), 327–344, 1981) and Bangert (Archiv der Mathematik 38(1):54–57, 1982) extended the reach rch(S) from subsets S of Euclidean space to the reach rchM(S) of subsets S of Riemannian manifolds M, where M is smooth (we’ll assume at least C3). Bangert showed that sets of positive reach in Euclidean space and Riemannian manifolds are very similar. In this paper we introduce a slight variant of Kleinjohann’s and Bangert’s extension and quantify the similarity between sets of positive reach in Euclidean space and Riemannian manifolds in a new way: Given p∈M and q∈S, we bound the local feature size (a local version of the reach) of its lifting to the tangent space via the inverse exponential map (exp−1p(S)) at q, assuming that rchM(S) and the geodesic distance dM(p,q) are bounded. These bounds are motivated by the importance of the reach and local feature size to manifold learning, topological inference, and triangulating manifolds and the fact that intrinsic approaches circumvent the curse of dimensionality.},
author = {Boissonnat, Jean Daniel and Wintraecken, Mathijs},
issn = {2367-1734},
journal = {Journal of Applied and Computational Topology},
publisher = {Springer Nature},
title = {{The reach of subsets of manifolds}},
doi = {10.1007/s41468-023-00116-x},
year = {2023},
}
@article{12764,
abstract = {We study a new discretization of the Gaussian curvature for polyhedral surfaces. This discrete Gaussian curvature is defined on each conical singularity of a polyhedral surface as the quotient of the angle defect and the area of the Voronoi cell corresponding to the singularity. We divide polyhedral surfaces into discrete conformal classes using a generalization of discrete conformal equivalence pioneered by Feng Luo. We subsequently show that, in every discrete conformal class, there exists a polyhedral surface with constant discrete Gaussian curvature. We also provide explicit examples to demonstrate that this surface is in general not unique.},
author = {Kourimska, Hana},
issn = {1432-0444},
journal = {Discrete and Computational Geometry},
publisher = {Springer Nature},
title = {{Discrete yamabe problem for polyhedral surfaces}},
doi = {10.1007/s00454-023-00484-2},
year = {2023},
}
@article{12960,
abstract = {Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e., submanifolds of Rd defined as the zero set of some multivariate multivalued smooth function f:Rd→Rd−n, where n is the intrinsic dimension of the manifold. A natural way to approximate a smooth isomanifold M=f−1(0) is to consider its piecewise linear (PL) approximation M^
based on a triangulation T of the ambient space Rd. In this paper, we describe a simple algorithm to trace isomanifolds from a given starting point. The algorithm works for arbitrary dimensions n and d, and any precision D. Our main result is that, when f (or M) has bounded complexity, the complexity of the algorithm is polynomial in d and δ=1/D (and unavoidably exponential in n). Since it is known that for δ=Ω(d2.5), M^ is O(D2)-close and isotopic to M
, our algorithm produces a faithful PL-approximation of isomanifolds of bounded complexity in time polynomial in d. Combining this algorithm with dimensionality reduction techniques, the dependency on d in the size of M^ can be completely removed with high probability. We also show that the algorithm can handle isomanifolds with boundary and, more generally, isostratifolds. The algorithm for isomanifolds with boundary has been implemented and experimental results are reported, showing that it is practical and can handle cases that are far ahead of the state-of-the-art. },
author = {Boissonnat, Jean Daniel and Kachanovich, Siargey and Wintraecken, Mathijs},
issn = {1095-7111},
journal = {SIAM Journal on Computing},
number = {2},
pages = {452--486},
publisher = {Society for Industrial and Applied Mathematics},
title = {{Tracing isomanifolds in Rd in time polynomial in d using Coxeter–Freudenthal–Kuhn triangulations}},
doi = {10.1137/21M1412918},
volume = {52},
year = {2023},
}
@inproceedings{13048,
abstract = {In this paper we introduce a pruning of the medial axis called the (λ,α)-medial axis (axλα). We prove that the (λ,α)-medial axis of a set K is stable in a Gromov-Hausdorff sense under weak assumptions. More formally we prove that if K and K′ are close in the Hausdorff (dH) sense then the (λ,α)-medial axes of K and K′ are close as metric spaces, that is the Gromov-Hausdorff distance (dGH) between the two is 1/4-Hölder in the sense that dGH (axλα(K),axλα(K′)) ≲ dH(K,K′)1/4. The Hausdorff distance between the two medial axes is also bounded, by dH (axλα(K),λα(K′)) ≲ dH(K,K′)1/2. These quantified stability results provide guarantees for practical computations of medial axes from approximations. Moreover, they provide key ingredients for studying the computability of the medial axis in the context of computable analysis.},
author = {Lieutier, André and Wintraecken, Mathijs},
booktitle = {Proceedings of the 55th Annual ACM Symposium on Theory of Computing},
isbn = {9781450399135},
location = {Orlando, FL, United States},
pages = {1768--1776},
publisher = {Association for Computing Machinery},
title = {{Hausdorff and Gromov-Hausdorff stable subsets of the medial axis}},
doi = {10.1145/3564246.3585113},
year = {2023},
}
@article{10413,
abstract = {Motivated by the recent introduction of the intrinsic semilattice entropy, we study generalized quasi-metric semilattices and their categories. We investigate the relationship between these objects and generalized semivaluations, extending Nakamura and Schellekens' approach. Finally, we use this correspondence to compare the intrinsic semilattice entropy and the semigroup entropy induced in particular situations, like sets, torsion abelian groups and vector spaces.},
author = {Dikranjan, Dikran and Giordano Bruno, Anna and Künzi, Hans Peter and Zava, Nicolò and Toller, Daniele},
issn = {0166-8641},
journal = {Topology and its Applications},
publisher = {Elsevier},
title = {{Generalized quasi-metric semilattices}},
doi = {10.1016/j.topol.2021.107916},
volume = {309},
year = {2022},
}
@article{10754,
abstract = {Targeting dysregulated Ca2+ signaling in cancer cells is an emerging chemotherapy approach. We previously reported that store-operated Ca2+ entry (SOCE) blockers, such as RP4010, are promising antitumor drugs for esophageal cancer. As a tyrosine kinase inhibitor (TKI), afatinib received FDA approval to be used in targeted therapy for patients with EGFR mutation-positive cancers. While preclinical studies and clinical trials have shown that afatinib has benefits for esophageal cancer patients, it is not known whether a combination of afatinib and RP4010 could achieve better anticancer effects. Since TKI can alter intracellular Ca2+ dynamics through EGFR/phospholipase C-γ pathway, in this study, we evaluated the inhibitory effect of afatinib and RP4010 on intracellular Ca2+ oscillations in KYSE-150, a human esophageal squamous cell carcinoma cell line, using both experimental and mathematical simulations. Our mathematical simulation of Ca2+ oscillations could fit well with experimental data responding to afatinib or RP4010, both separately or in combination. Guided by simulation, we were able to identify a proper ratio of afatinib and RP4010 for combined treatment, and such a combination presented synergistic anticancer-effect evidence by experimental measurement of intracellular Ca2+ and cell proliferation. This intracellular Ca2+ dynamic-based mathematical simulation approach could be useful for a rapid and cost-effective evaluation of combined targeting therapy drugs.},
author = {Chang, Yan and Funk, Marah and Roy, Souvik and Stephenson, Elizabeth R and Choi, Sangyong and Kojouharov, Hristo V. and Chen, Benito and Pan, Zui},
issn = {14220067},
journal = {International Journal of Molecular Sciences},
number = {3},
publisher = {MDPI},
title = {{Developing a mathematical model of intracellular Calcium dynamics for evaluating combined anticancer effects of afatinib and RP4010 in esophageal cancer}},
doi = {10.3390/ijms23031763},
volume = {23},
year = {2022},
}
@article{10773,
abstract = {The Voronoi tessellation in Rd is defined by locally minimizing the power distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined by locally maximizing the negative power distance to other such points. We prove that the average of the two piecewise quadratic functions is piecewise linear, and that all three functions have the same critical points and values. Discretizing the two piecewise quadratic functions, we get the alpha shapes as sublevel sets of the discrete function on the Delaunay mosaic, and analogous shapes as superlevel sets of the discrete function on the Voronoi tessellation. For the same non-critical value, the corresponding shapes are disjoint, separated by a narrow channel that contains no critical points but the entire level set of the piecewise linear function.},
author = {Biswas, Ranita and Cultrera Di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
issn = {1432-0444},
journal = {Discrete and Computational Geometry},
pages = {811--842},
publisher = {Springer Nature},
title = {{Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics}},
doi = {10.1007/s00454-022-00371-2},
volume = {67},
year = {2022},
}
@inproceedings{10828,
abstract = {Digital images enable quantitative analysis of material properties at micro and macro length scales, but choosing an appropriate resolution when acquiring the image is challenging. A high resolution means longer image acquisition and larger data requirements for a given sample, but if the resolution is too low, significant information may be lost. This paper studies the impact of changes in resolution on persistent homology, a tool from topological data analysis that provides a signature of structure in an image across all length scales. Given prior information about a function, the geometry of an object, or its density distribution at a given resolution, we provide methods to select the coarsest resolution yielding results within an acceptable tolerance. We present numerical case studies for an illustrative synthetic example and samples from porous materials where the theoretical bounds are unknown.},
author = {Heiss, Teresa and Tymochko, Sarah and Story, Brittany and Garin, Adélie and Bui, Hoa and Bleile, Bea and Robins, Vanessa},
booktitle = {2021 IEEE International Conference on Big Data},
isbn = {9781665439022},
location = {Orlando, FL, United States; Virtuell},
pages = {3824--3834},
publisher = {IEEE},
title = {{The impact of changes in resolution on the persistent homology of images}},
doi = {10.1109/BigData52589.2021.9671483},
year = {2022},
}
@inproceedings{11428,
abstract = {The medial axis of a set consists of the points in the ambient space without a unique closest point on the original set. Since its introduction, the medial axis has been used extensively in many applications as a method of computing a topologically equivalent skeleton. Unfortunately, one limiting factor in the use of the medial axis of a smooth manifold is that it is not necessarily topologically stable under small perturbations of the manifold. To counter these instabilities various prunings of the medial axis have been proposed. Here, we examine one type of pruning, called burning. Because of the good experimental results, it was hoped that the burning method of simplifying the medial axis would be stable. In this work we show a simple example that dashes such hopes based on Bing’s house with two rooms, demonstrating an isotopy of a shape where the medial axis goes from collapsible to non-collapsible.},
author = {Chambers, Erin and Fillmore, Christopher D and Stephenson, Elizabeth R and Wintraecken, Mathijs},
booktitle = {38th International Symposium on Computational Geometry},
editor = {Goaoc, Xavier and Kerber, Michael},
isbn = {978-3-95977-227-3},
issn = {1868-8969},
location = {Berlin, Germany},
pages = {66:1--66:9},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{A cautionary tale: Burning the medial axis is unstable}},
doi = {10.4230/LIPIcs.SoCG.2022.66},
volume = {224},
year = {2022},
}
@book{11429,
abstract = {This book constitutes the refereed proceedings of the 18th International Symposium on Web and Wireless Geographical Information Systems, W2GIS 2022, held in Konstanz, Germany, in April 2022.
The 7 full papers presented together with 6 short papers in the volume were carefully reviewed and selected from 16 submissions. The papers cover topics that range from mobile GIS and Location-Based Services to Spatial Information Retrieval and Wireless Sensor Networks.},
editor = {Karimipour, Farid and Storandt, Sabine},
isbn = {9783031062445},
issn = {1611-3349},
pages = {153},
publisher = {Springer Nature},
title = {{Web and Wireless Geographical Information Systems}},
doi = {10.1007/978-3-031-06245-2},
volume = {13238},
year = {2022},
}
@inbook{11440,
abstract = {To compute the persistent homology of a grayscale digital image one needs to build a simplicial or cubical complex from it. For cubical complexes, the two commonly used constructions (corresponding to direct and indirect digital adjacencies) can give different results for the same image. The two constructions are almost dual to each other, and we use this relationship to extend and modify the cubical complexes to become dual filtered cell complexes. We derive a general relationship between the persistent homology of two dual filtered cell complexes, and also establish how various modifications to a filtered complex change the persistence diagram. Applying these results to images, we derive a method to transform the persistence diagram computed using one type of cubical complex into a persistence diagram for the other construction. This means software for computing persistent homology from images can now be easily adapted to produce results for either of the two cubical complex constructions without additional low-level code implementation.},
author = {Bleile, Bea and Garin, Adélie and Heiss, Teresa and Maggs, Kelly and Robins, Vanessa},
booktitle = {Research in Computational Topology 2},
editor = {Gasparovic, Ellen and Robins, Vanessa and Turner, Katharine},
isbn = {9783030955182},
pages = {1--26},
publisher = {Springer Nature},
title = {{The persistent homology of dual digital image constructions}},
doi = {10.1007/978-3-030-95519-9_1},
volume = {30},
year = {2022},
}
@article{11545,
abstract = {We classify contravariant pairings between standard Whittaker modules and Verma modules over a complex semisimple Lie algebra. These contravariant pairings are useful in extending several classical techniques for category O to the Miličić–Soergel category N . We introduce a class of costandard modules which generalize dual Verma modules, and describe canonical maps from standard to costandard modules in terms of contravariant pairings.
We show that costandard modules have unique irreducible submodules and share the same composition factors as the corresponding standard Whittaker modules. We show that costandard modules give an algebraic characterization of the global sections of costandard twisted Harish-Chandra sheaves on the associated flag variety, which are defined using holonomic duality of D-modules. We prove that with these costandard modules, blocks of category
N have the structure of highest weight categories and we establish a BGG reciprocity theorem for N .},
author = {Brown, Adam and Romanov, Anna},
issn = {0021-8693},
journal = {Journal of Algebra},
keywords = {Algebra and Number Theory},
number = {11},
pages = {145--179},
publisher = {Elsevier},
title = {{Contravariant pairings between standard Whittaker modules and Verma modules}},
doi = {10.1016/j.jalgebra.2022.06.017},
volume = {609},
year = {2022},
}
@article{11658,
abstract = {The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements.},
author = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
journal = {Leibniz International Proceedings on Mathematics},
publisher = {Schloss Dagstuhl - Leibniz Zentrum für Informatik},
title = {{Depth in arrangements: Dehn–Sommerville–Euler relations with applications}},
year = {2022},
}
@article{11660,
abstract = {We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs of theorems about the symmetry of persistence diagrams and the variation of such maps. In particular, we identify branching points and endpoints of networks as the sole source of asymmetry and relate the cycle basis in persistent homology with a version of the stable marriage problem. Our analysis provides the foundations of fast algorithms for maintaining collections of interrelated sorted lists together with their persistence diagrams. },
author = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
journal = {LIPIcs},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs}},
year = {2022},
}
@article{12307,
abstract = {Point-set topology is among the most abstract branches of mathematics in that it lacks tangible notions of distance, length, magnitude, order, and size. There is no shape, no geometry, no algebra, and no direction. Everything we are used to visualizing is gone. In the teaching and learning of mathematics, this can present a conundrum. Yet, this very property makes point set topology perfect for teaching and learning abstract mathematical concepts. It clears our minds of preconceived intuitions and expectations and forces us to think in new and creative ways. In this paper, we present guided investigations into topology through questions and thinking strategies that open up fascinating problems. They are intended for faculty who already teach or are thinking about teaching a class in topology or abstract mathematical reasoning for undergraduates. They can be used to build simple to challenging projects in topology, proofs, honors programs, and research experiences.},
author = {Shipman, Barbara A. and Stephenson, Elizabeth R},
issn = {1935-4053},
journal = {PRIMUS},
keywords = {Education, General Mathematics},
number = {5},
pages = {593--609},
publisher = {Taylor & Francis},
title = {{Tangible topology through the lens of limits}},
doi = {10.1080/10511970.2021.1872750},
volume = {32},
year = {2022},
}
@article{11938,
abstract = {A matching is compatible to two or more labeled point sets of size n with labels {1, . . . , n} if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled sets of n points in convex position there exists a compatible matching with ⌊√2n + 1 − 1⌋ edges. More generally, for any ℓ labeled point sets we construct compatible matchings of size Ω(n1/ℓ). As a corresponding upper bound, we use probabilistic arguments to show that for any ℓ given sets of n points there exists a labeling of each set such that the largest compatible matching has O(n2/(ℓ+1)) edges. Finally, we show that Θ(log n) copies of any set of n points are necessary and sufficient for the existence of labelings of these point sets such that any compatible matching consists only of a single edge.},
author = {Aichholzer, Oswin and Arroyo Guevara, Alan M and Masárová, Zuzana and Parada, Irene and Perz, Daniel and Pilz, Alexander and Tkadlec, Josef and Vogtenhuber, Birgit},
issn = {1526-1719},
journal = {Journal of Graph Algorithms and Applications},
number = {2},
pages = {225--240},
publisher = {Brown University},
title = {{On compatible matchings}},
doi = {10.7155/jgaa.00591},
volume = {26},
year = {2022},
}
@article{9649,
abstract = {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 : Rd → Rd−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 Rd. 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.},
author = {Boissonnat, Jean-Daniel and Wintraecken, Mathijs},
issn = {1615-3383},
journal = {Foundations of Computational Mathematics },
pages = {967--1012},
publisher = {Springer Nature},
title = {{The topological correctness of PL approximations of isomanifolds}},
doi = {10.1007/s10208-021-09520-0},
volume = {22},
year = {2022},
}
@article{12833,
abstract = {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.},
author = {Biniaz, Ahmad and Jain, Kshitij and Lubiw, Anna and Masárová, Zuzana and Miltzow, Tillmann and Mondal, Debajyoti and Naredla, Anurag Murty and Tkadlec, Josef and Turcotte, Alexi},
issn = {1365-8050},
journal = {Discrete Mathematics and Theoretical Computer Science},
number = {2},
publisher = {EPI Sciences},
title = {{Token swapping on trees}},
doi = {10.46298/DMTCS.8383},
volume = {24},
year = {2022},
}
@article{10208,
abstract = {It is practical to collect a huge amount of movement data and environmental context information along with the health signals of individuals because there is the emergence of new generations of positioning and tracking technologies and rapid advancements of health sensors. The study of the relations between these datasets and their sequence similarity analysis is of interest to many applications such as health monitoring and recommender systems. However, entering all movement parameters and health signals can lead to the complexity of the problem and an increase in its computational load. In this situation, dimension reduction techniques can be used to avoid consideration of simultaneous dependent parameters in the process of similarity measurement of the trajectories. The present study provides a framework, named CaDRAW, to use spatial–temporal data and movement parameters along with independent context information in the process of measuring the similarity of trajectories. In this regard, the omission of dependent movement characteristic signals is conducted by using an unsupervised feature selection dimension reduction technique. To evaluate the effectiveness of the proposed framework, it was applied to a real contextualized movement and related health signal datasets of individuals. The results indicated the capability of the proposed framework in measuring the similarity and in decreasing the characteristic signals in such a way that the similarity results -before and after reduction of dependent characteristic signals- have small differences. The mean differences between the obtained results before and after reducing the dimension were 0.029 and 0.023 for the round path, respectively.},
author = {Goudarzi, Samira and Sharif, Mohammad and Karimipour, Farid},
issn = {1868-5145},
journal = {Journal of Ambient Intelligence and Humanized Computing},
keywords = {general computer science},
pages = {2621–2635},
publisher = {Springer Nature},
title = {{A context-aware dimension reduction framework for trajectory and health signal analyses}},
doi = {10.1007/s12652-021-03569-z},
volume = {13},
year = {2022},
}
@article{10071,
author = {Adams, Henry and Kourimska, Hana and Heiss, Teresa and Percival, Sarah and Ziegelmeier, Lori},
issn = {1088-9477},
journal = {Notices of the American Mathematical Society},
number = {9},
pages = {1511--1514},
publisher = {American Mathematical Society},
title = {{How to tutorial-a-thon}},
doi = {10.1090/noti2349},
volume = {68},
year = {2021},
}
@article{10204,
abstract = {Two common representations of close packings of identical spheres consisting of hexagonal layers, called Barlow stackings, appear abundantly in minerals and metals. These motifs, however, occupy an identical portion of space and bear identical first-order topological signatures as measured by persistent homology. Here we present a novel method based on k-fold covers that unambiguously distinguishes between these patterns. Moreover, our approach provides topological evidence that the FCC motif is the more stable of the two in the context of evolving experimental sphere packings during the transition from disordered to an ordered state. We conclude that our approach can be generalised to distinguish between various Barlow stackings manifested in minerals and metals.},
author = {Osang, Georg F and Edelsbrunner, Herbert and Saadatfar, Mohammad},
issn = {1744-6848},
journal = {Soft Matter},
number = {40},
pages = {9107--9115},
publisher = {Royal Society of Chemistry },
title = {{Topological signatures and stability of hexagonal close packing and Barlow stackings}},
doi = {10.1039/d1sm00774b},
volume = {17},
year = {2021},
}
@article{10222,
abstract = {Consider a random set of points on the unit sphere in ℝd, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case d = 3, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density.},
author = {Akopyan, Arseniy and Edelsbrunner, Herbert and Nikitenko, Anton},
issn = {1944-950X},
journal = {Experimental Mathematics},
pages = {1--15},
publisher = {Taylor and Francis},
title = {{The beauty of random polytopes inscribed in the 2-sphere}},
doi = {10.1080/10586458.2021.1980459},
year = {2021},
}
@inproceedings{10367,
abstract = {How information is created, shared and consumed has changed rapidly in recent decades, in part thanks to new social platforms and technologies on the web. With ever-larger amounts of unstructured and limited labels, organizing and reconciling information from different sources and modalities is a central challenge in machine learning. This cutting-edge tutorial aims to introduce the multimodal entailment task, which can be useful for detecting semantic alignments when a single modality alone does not suffice for a whole content understanding. Starting with a brief overview of natural language processing, computer vision, structured data and neural graph learning, we lay the foundations for the multimodal sections to follow. We then discuss recent multimodal learning literature covering visual, audio and language streams, and explore case studies focusing on tasks which require fine-grained understanding of visual and linguistic semantics question answering, veracity and hatred classification. Finally, we introduce a new dataset for recognizing multimodal entailment, exploring it in a hands-on collaborative section. Overall, this tutorial gives an overview of multimodal learning, introduces a multimodal entailment dataset, and encourages future research in the topic.},
author = {Ilharco, Cesar and Shirazi, Afsaneh and Gopalan, Arjun and Nagrani, Arsha and Bratanič, Blaž and Bregler, Chris and Liu, Christina and Ferreira, Felipe and Barcik, Gabriek and Ilharco, Gabriel and Osang, Georg F and Bulian, Jannis and Frank, Jared and Smaira, Lucas and Cao, Qin and Marino, Ricardo and Patel, Roma and Leung, Thomas and Imbrasaite, Vaiva},
booktitle = {59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing, Tutorial Abstracts},
isbn = {9-781-9540-8557-2},
location = {Bangkok, Thailand},
pages = {29--30},
publisher = {Association for Computational Linguistics},
title = {{Recognizing multimodal entailment}},
doi = {10.18653/v1/2021.acl-tutorials.6},
year = {2021},
}
@article{10608,
abstract = {We consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.},
author = {Weighill, Thomas and Yamauchi, Takamitsu and Zava, Nicolò},
issn = {2199-6768},
journal = {European Journal of Mathematics},
publisher = {Springer Nature},
title = {{Coarse infinite-dimensionality of hyperspaces of finite subsets}},
doi = {10.1007/s40879-021-00515-3},
year = {2021},
}
@article{8317,
abstract = {When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special “basic” holes guarantee foldability.},
author = {Aichholzer, Oswin and Akitaya, Hugo A. and Cheung, Kenneth C. and Demaine, Erik D. and Demaine, Martin L. and Fekete, Sándor P. and Kleist, Linda and Kostitsyna, Irina and Löffler, Maarten and Masárová, Zuzana and Mundilova, Klara and Schmidt, Christiane},
issn = {09257721},
journal = {Computational Geometry: Theory and Applications},
publisher = {Elsevier},
title = {{Folding polyominoes with holes into a cube}},
doi = {10.1016/j.comgeo.2020.101700},
volume = {93},
year = {2021},
}
@phdthesis{9056,
abstract = {In this thesis we study persistence of multi-covers of Euclidean balls and the geometric structures underlying their computation, in particular Delaunay mosaics and Voronoi tessellations. The k-fold cover for some discrete input point set consists of the space where at least k balls of radius r around the input points overlap. Persistence is a notion that captures, in some sense, the topology of the shape underlying the input. While persistence is usually computed for the union of balls, the k-fold cover is of interest as it captures local density,
and thus might approximate the shape of the input better if the input data is noisy. To compute persistence of these k-fold covers, we need a discretization that is provided by higher-order Delaunay mosaics. We present and implement a simple and efficient algorithm for the computation of higher-order Delaunay mosaics, and use it to give experimental results for their combinatorial properties. The algorithm makes use of a new geometric structure, the rhomboid tiling. It contains the higher-order Delaunay mosaics as slices, and by introducing a filtration
function on the tiling, we also obtain higher-order α-shapes as slices. These allow us to compute persistence of the multi-covers for varying radius r; the computation for varying k is less straight-foward and involves the rhomboid tiling directly. We apply our algorithms to experimental sphere packings to shed light on their structural properties. Finally, inspired by periodic structures in packings and materials, we propose and implement an algorithm for periodic Delaunay triangulations to be integrated into the Computational Geometry Algorithms Library (CGAL), and discuss the implications on persistence for periodic data sets.},
author = {Osang, Georg F},
issn = {2663-337X},
pages = {134},
publisher = {IST Austria},
title = {{Multi-cover persistence and Delaunay mosaics}},
doi = {10.15479/AT:ISTA:9056},
year = {2021},
}
@inproceedings{9253,
abstract = {In March 2020, the Austrian government introduced a widespread lock-down in response to the COVID-19 pandemic. Based on subjective impressions and anecdotal evidence, Austrian public and private life came to a sudden halt. Here we assess the effect of the lock-down quantitatively for all regions in Austria and present an analysis of daily changes of human mobility throughout Austria using near-real-time anonymized mobile phone data. We describe an efficient data aggregation pipeline and analyze the mobility by quantifying mobile-phone traffic at specific point of interests (POIs), analyzing individual trajectories and investigating the cluster structure of the origin-destination graph. We found a reduction of commuters at Viennese metro stations of over 80% and the number of devices with a radius of gyration of less than 500 m almost doubled. The results of studying crowd-movement behavior highlight considerable changes in the structure of mobility networks, revealed by a higher modularity and an increase from 12 to 20 detected communities. We demonstrate the relevance of mobility data for epidemiological studies by showing a significant correlation of the outflow from the town of Ischgl (an early COVID-19 hotspot) and the reported COVID-19 cases with an 8-day time lag. This research indicates that mobile phone usage data permits the moment-by-moment quantification of mobility behavior for a whole country. We emphasize the need to improve the availability of such data in anonymized form to empower rapid response to combat COVID-19 and future pandemics.},
author = {Heiler, Georg and Reisch, Tobias and Hurt, Jan and Forghani, Mohammad and Omani, Aida and Hanbury, Allan and Karimipour, Farid},
booktitle = {2020 IEEE International Conference on Big Data},
isbn = {9781728162515},
location = {Atlanta, GA, United States},
pages = {3123--3132},
publisher = {IEEE},
title = {{Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic}},
doi = {10.1109/bigdata50022.2020.9378374},
year = {2021},
}
@inproceedings{9296,
abstract = { matching is compatible to two or more labeled point sets of size n with labels {1,…,n} if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of n points there exists a compatible matching with ⌊2n−−√⌋ edges. More generally, for any ℓ labeled point sets we construct compatible matchings of size Ω(n1/ℓ) . As a corresponding upper bound, we use probabilistic arguments to show that for any ℓ given sets of n points there exists a labeling of each set such that the largest compatible matching has O(n2/(ℓ+1)) edges. Finally, we show that Θ(logn) copies of any set of n points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge.},
author = {Aichholzer, Oswin and Arroyo Guevara, Alan M and Masárová, Zuzana and Parada, Irene and Perz, Daniel and Pilz, Alexander and Tkadlec, Josef and Vogtenhuber, Birgit},
booktitle = {15th International Conference on Algorithms and Computation},
isbn = {9783030682101},
issn = {16113349},
location = {Yangon, Myanmar},
pages = {221--233},
publisher = {Springer Nature},
title = {{On compatible matchings}},
doi = {10.1007/978-3-030-68211-8_18},
volume = {12635},
year = {2021},
}
@article{9465,
abstract = {Given a locally finite set 𝑋⊆ℝ𝑑 and an integer 𝑘≥0, we consider the function 𝐰𝑘:Del𝑘(𝑋)→ℝ on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton and Osang, Georg F},
issn = {14208997},
journal = {Journal of Geometry},
number = {1},
publisher = {Springer Nature},
title = {{A step in the Delaunay mosaic of order k}},
doi = {10.1007/s00022-021-00577-4},
volume = {112},
year = {2021},
}
@article{8773,
abstract = {Let g be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g-modules Y(χ,η) introduced by Kostant. We prove that the set of all contravariant forms on Y(χ,η) forms a vector space whose dimension is given by the cardinality of the Weyl group of g. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M(χ,η) introduced by McDowell.},
author = {Brown, Adam and Romanov, Anna},
issn = {1088-6826},
journal = {Proceedings of the American Mathematical Society},
keywords = {Applied Mathematics, General Mathematics},
number = {1},
pages = {37--52},
publisher = {American Mathematical Society},
title = {{Contravariant forms on Whittaker modules}},
doi = {10.1090/proc/15205},
volume = {149},
year = {2021},
}
@article{8940,
abstract = {We quantise Whitney’s construction to prove the existence of a triangulation for any C^2 manifold, so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is completely geometric.},
author = {Boissonnat, Jean-Daniel and Kachanovich, Siargey and Wintraecken, Mathijs},
issn = {0179-5376},
journal = {Discrete & Computational Geometry},
keywords = {Theoretical Computer Science, Computational Theory and Mathematics, Geometry and Topology, Discrete Mathematics and Combinatorics},
number = {1},
pages = {386--434},
publisher = {Springer Nature},
title = {{Triangulating submanifolds: An elementary and quantified version of Whitney’s method}},
doi = {10.1007/s00454-020-00250-8},
volume = {66},
year = {2021},
}
@article{9111,
abstract = {We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space X equipped with a continuous function f:X→R. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line R. We then introduce a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium on point-based graphics, 2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of (X,f) when it is applied to points randomly sampled from a probability density function concentrated on (X,f). Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (In: 32nd international symposium on computational geometry, volume 51 of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany, pp 53:1–53:16, 2016), we first show that the mapper graph of (X,f), a constructible R-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of (X,f) to the mapper of a super-level set of a probability density function concentrated on (X,f). Finally, building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data.},
author = {Brown, Adam and Bobrowski, Omer and Munch, Elizabeth and Wang, Bei},
issn = {2367-1726},
journal = {Journal of Applied and Computational Topology},
number = {1},
pages = {99--140},
publisher = {Springer Nature},
title = {{Probabilistic convergence and stability of random mapper graphs}},
doi = {10.1007/s41468-020-00063-x},
volume = {5},
year = {2021},
}
@article{9317,
abstract = {Given a locally finite X⊆Rd and a radius r≥0, the k-fold cover of X and r consists of all points in Rd that have k or more points of X within distance r. We consider two filtrations—one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k—and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in Rd+1 whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module of Delaunay mosaics that is isomorphic to the persistence module of the multi-covers.},
author = {Edelsbrunner, Herbert and Osang, Georg F},
issn = {1432-0444},
journal = {Discrete and Computational Geometry},
pages = {1296–1313},
publisher = {Springer Nature},
title = {{The multi-cover persistence of Euclidean balls}},
doi = {10.1007/s00454-021-00281-9},
volume = {65},
year = {2021},
}
@inproceedings{9345,
abstract = {Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functionsthat facilitates the efficient search for new materials and material properties. We prove invarianceunder isometries, continuity, and completeness in the generic case, which are necessary featuresfor the reliable comparison of crystals. The proof of continuity integrates methods from discretegeometry and lattice theory, while the proof of generic completeness combines techniques fromgeometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and relatedinclusion-exclusion formulae. We have implemented the algorithm and describe its application tocrystal structure prediction.},
author = {Edelsbrunner, Herbert and Heiss, Teresa and Kurlin , Vitaliy and Smith, Philip and Wintraecken, Mathijs},
booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
issn = {1868-8969},
location = {Virtual},
pages = {32:1--32:16},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{The density fingerprint of a periodic point set}},
doi = {10.4230/LIPIcs.SoCG.2021.32},
volume = {189},
year = {2021},
}
@inproceedings{9604,
abstract = {Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in ℝ² to ℝ³, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of n ≥ 5 points in ℝ³, the number of regions in the order-k Voronoi tessellation is N_{k-1} - binom(k,2)n + n, for 1 ≤ k ≤ n-1, in which N_{k-1} is the sum of Euler characteristics of these function’s first k-1 sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-k Voronoi tessellation.},
author = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
booktitle = {Leibniz International Proceedings in Informatics},
isbn = {9783959771849},
issn = {18688969},
location = {Online},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Counting cells of order-k voronoi tessellations in ℝ^{3} with morse theory}},
doi = {10.4230/LIPIcs.SoCG.2021.16},
volume = {189},
year = {2021},
}
@article{9602,
abstract = {An ordered graph is a graph with a linear ordering on its vertex set. We prove that for every positive integer k, there exists a constant ck > 0 such that any ordered graph G on n vertices with the property that neither G nor its complement contains an induced monotone path of size k, has either a clique or an independent set of size at least n^ck . This strengthens a result of Bousquet, Lagoutte, and Thomassé, who proved the analogous result for unordered graphs.
A key idea of the above paper was to show that any unordered graph on n vertices that does not contain an induced path of size k, and whose maximum degree is at most c(k)n for some small c(k) > 0, contains two disjoint linear size subsets with no edge between them. This approach fails for ordered graphs, because the analogous statement is false for k ≥ 3, by a construction of Fox. We provide some further examples showing that this statement also fails for ordered graphs avoiding other ordered trees.},
author = {Pach, János and Tomon, István},
issn = {0095-8956},
journal = {Journal of Combinatorial Theory. Series B},
pages = {21--37},
publisher = {Elsevier},
title = {{Erdős-Hajnal-type results for monotone paths}},
doi = {10.1016/j.jctb.2021.05.004},
volume = {151},
year = {2021},
}
@article{9821,
abstract = {Heart rate variability (hrv) is a physiological phenomenon of the variation in the length of the time interval between consecutive heartbeats. In many cases it could be an indicator of the development of pathological states. The classical approach to the analysis of hrv includes time domain methods and frequency domain methods. However, attempts are still being made to define new and more effective hrv assessment tools. Persistent homology is a novel data analysis tool developed in the recent decades that is rooted at algebraic topology. The Topological Data Analysis (TDA) approach focuses on examining the shape of the data in terms of connectedness and holes, and has recently proved to be very effective in various fields of research. In this paper we propose the use of persistent homology to the hrv analysis. We recall selected topological descriptors used in the literature and we introduce some new topological descriptors that reflect the specificity of hrv, and we discuss their relation to the standard hrv measures. In particular, we show that this novel approach provides a collection of indices that might be at least as useful as the classical parameters in differentiating between series of beat-to-beat intervals (RR-intervals) in healthy subjects and patients suffering from a stroke episode.},
author = {Graff, Grzegorz and Graff, Beata and Pilarczyk, Pawel and Jablonski, Grzegorz and Gąsecki, Dariusz and Narkiewicz, Krzysztof},
issn = {19326203},
journal = {PLoS ONE},
number = {7},
publisher = {Public Library of Science},
title = {{Persistent homology as a new method of the assessment of heart rate variability}},
doi = {10.1371/journal.pone.0253851},
volume = {16},
year = {2021},
}
@inproceedings{9824,
abstract = {We define a new compact coordinate system in which each integer triplet addresses a voxel in the BCC grid, and we investigate some of its properties. We propose a characterization of 3D discrete analytical planes with their topological features (in the Cartesian and in the new coordinate system) such as the interrelation between the thickness of the plane and the separability constraint we aim to obtain.},
author = {Čomić, Lidija and Zrour, Rita and Largeteau-Skapin, Gaëlle and Biswas, Ranita and Andres, Eric},
booktitle = {Discrete Geometry and Mathematical Morphology},
isbn = {9783030766566},
issn = {16113349},
location = {Uppsala, Sweden},
pages = {152--163},
publisher = {Springer Nature},
title = {{Body centered cubic grid - coordinate system and discrete analytical plane definition}},
doi = {10.1007/978-3-030-76657-3_10},
volume = {12708},
year = {2021},
}
@inproceedings{9605,
abstract = {Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness. },
author = {Corbet, René and Kerber, Michael and Lesnick, Michael and Osang, Georg F},
booktitle = {Leibniz International Proceedings in Informatics},
isbn = {9783959771849},
issn = {18688969},
location = {Online},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Computing the multicover bifiltration}},
doi = {10.4230/LIPIcs.SoCG.2021.27},
volume = {189},
year = {2021},
}
@inproceedings{9441,
abstract = {Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. submanifolds of ℝ^d defined as the zero set of some multivariate multivalued smooth function f: ℝ^d → ℝ^{d-n}, where n is the intrinsic dimension of the manifold. A natural way to approximate a smooth isomanifold M is to consider its Piecewise-Linear (PL) approximation M̂ based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we describe a simple algorithm to trace isomanifolds from a given starting point. The algorithm works for arbitrary dimensions n and d, and any precision D. Our main result is that, when f (or M) has bounded complexity, the complexity of the algorithm is polynomial in d and δ = 1/D (and unavoidably exponential in n). Since it is known that for δ = Ω (d^{2.5}), M̂ is O(D²)-close and isotopic to M, our algorithm produces a faithful PL-approximation of isomanifolds of bounded complexity in time polynomial in d. Combining this algorithm with dimensionality reduction techniques, the dependency on d in the size of M̂ can be completely removed with high probability. We also show that the algorithm can handle isomanifolds with boundary and, more generally, isostratifolds. The algorithm for isomanifolds with boundary has been implemented and experimental results are reported, showing that it is practical and can handle cases that are far ahead of the state-of-the-art. },
author = {Boissonnat, Jean-Daniel and Kachanovich, Siargey and Wintraecken, Mathijs},
booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
isbn = {978-3-95977-184-9},
issn = {1868-8969},
location = {Virtual},
pages = {17:1--17:16},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Tracing isomanifolds in Rd in time polynomial in d using Coxeter-Freudenthal-Kuhn triangulations}},
doi = {10.4230/LIPIcs.SoCG.2021.17},
volume = {189},
year = {2021},
}
@article{10867,
abstract = {In this paper we find a tight estimate for Gromov’s waist of the balls in spaces of constant curvature, deduce the estimates for the balls in Riemannian manifolds with upper bounds on the curvature (CAT(ϰ)-spaces), and establish similar result for normed spaces.},
author = {Akopyan, Arseniy and Karasev, Roman},
issn = {1687-0247},
journal = {International Mathematics Research Notices},
keywords = {General Mathematics},
number = {3},
pages = {669--697},
publisher = {Oxford University Press},
title = {{Waist of balls in hyperbolic and spherical spaces}},
doi = {10.1093/imrn/rny037},
volume = {2020},
year = {2020},
}
@inbook{74,
abstract = {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.},
author = {Akopyan, Arseniy and Karasev, Roman},
booktitle = {Geometric Aspects of Functional Analysis},
editor = {Klartag, Bo'az and Milman, Emanuel},
isbn = {9783030360191},
issn = {16179692},
pages = {1--27},
publisher = {Springer Nature},
title = {{Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures}},
doi = {10.1007/978-3-030-36020-7_1},
volume = {2256},
year = {2020},
}
@phdthesis{7460,
abstract = {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.},
author = {Ölsböck, Katharina},
issn = {2663-337X},
keywords = {shape reconstruction, hole manipulation, ordered complexes, Alpha complex, Wrap complex, computational topology, Bregman geometry},
pages = {155},
publisher = {IST Austria},
title = {{The hole system of triangulated shapes}},
doi = {10.15479/AT:ISTA:7460},
year = {2020},
}
@article{7567,
abstract = {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.},
author = {Choudhary, Aruni and Kachanovich, Siargey and Wintraecken, Mathijs},
issn = {1661-8289},
journal = {Mathematics in Computer Science},
pages = {141--176},
publisher = {Springer Nature},
title = {{Coxeter triangulations have good quality}},
doi = {10.1007/s11786-020-00461-5},
volume = {14},
year = {2020},
}
@article{7554,
abstract = {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 ${R}^n$.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton},
issn = {10957219},
journal = {Theory of Probability and its Applications},
number = {4},
pages = {595--614},
publisher = {SIAM},
title = {{Weighted Poisson–Delaunay mosaics}},
doi = {10.1137/S0040585X97T989726},
volume = {64},
year = {2020},
}
@article{7666,
abstract = {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.},
author = {Edelsbrunner, Herbert and Ölsböck, Katharina},
issn = {14320444},
journal = {Discrete and Computational Geometry},
pages = {759--775},
publisher = {Springer Nature},
title = {{Tri-partitions and bases of an ordered complex}},
doi = {10.1007/s00454-020-00188-x},
volume = {64},
year = {2020},
}
@article{7791,
abstract = {Extending a result of Milena Radnovic and Serge Tabachnikov, we establish conditionsfor two different non-symmetric norms to define the same billiard reflection law.},
author = {Akopyan, Arseniy and Karasev, Roman},
issn = {2199-6768},
journal = {European Journal of Mathematics},
publisher = {Springer Nature},
title = {{When different norms lead to same billiard trajectories?}},
doi = {10.1007/s40879-020-00405-0},
year = {2020},
}
@article{7905,
abstract = {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.},
author = {Brown, Adam and Wang, Bei},
issn = {0179-5376},
journal = {Discrete and Computational Geometry},
pages = {1166--1198},
publisher = {Springer Nature},
title = {{Sheaf-theoretic stratification learning from geometric and topological perspectives}},
doi = {10.1007/s00454-020-00206-y},
volume = {65},
year = {2020},
}
@phdthesis{7944,
abstract = {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.},
author = {Masárová, Zuzana},
isbn = {978-3-99078-005-3},
issn = {2663-337X},
keywords = {reconfiguration, reconfiguration graph, triangulations, flip, constrained triangulations, shellability, piecewise-linear balls, token swapping, trees, coloured weighted token swapping},
pages = {160},
publisher = {IST Austria},
title = {{Reconfiguration problems}},
doi = {10.15479/AT:ISTA:7944},
year = {2020},
}
@article{7962,
abstract = {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.},
author = {Pach, János and Reed, Bruce and Yuditsky, Yelena},
issn = {14320444},
journal = {Discrete and Computational Geometry},
number = {4},
pages = {888--917},
publisher = {Springer Nature},
title = {{Almost all string graphs are intersection graphs of plane convex sets}},
doi = {10.1007/s00454-020-00213-z},
volume = {63},
year = {2020},
}
@inproceedings{8135,
abstract = {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.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton and Ölsböck, Katharina and Synak, Peter},
booktitle = {Topological Data Analysis},
isbn = {9783030434076},
issn = {21978549},
pages = {181--218},
publisher = {Springer Nature},
title = {{Radius functions on Poisson–Delaunay mosaics and related complexes experimentally}},
doi = {10.1007/978-3-030-43408-3_8},
volume = {15},
year = {2020},
}
@article{8163,
abstract = {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.},
author = {Vegter, Gert and Wintraecken, Mathijs},
issn = {1588-2896},
journal = {Studia Scientiarum Mathematicarum Hungarica},
number = {2},
pages = {193--199},
publisher = {AKJournals},
title = {{Refutation of a claim made by Fejes Tóth on the accuracy of surface meshes}},
doi = {10.1556/012.2020.57.2.1454},
volume = {57},
year = {2020},
}
@article{8248,
abstract = {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.},
author = {Boissonnat, Jean-Daniel and Dyer, Ramsay and Ghosh, Arijit and Lieutier, Andre and Wintraecken, Mathijs},
issn = {0179-5376},
journal = {Discrete and Computational Geometry},
pages = {666--686},
publisher = {Springer Nature},
title = {{Local conditions for triangulating submanifolds of Euclidean space}},
doi = {10.1007/s00454-020-00233-9},
volume = {66},
year = {2020},
}
@article{8323,
author = {Pach, János},
issn = {14320444},
journal = {Discrete and Computational Geometry},
pages = {571--574},
publisher = {Springer Nature},
title = {{A farewell to Ricky Pollack}},
doi = {10.1007/s00454-020-00237-5},
volume = {64},
year = {2020},
}
@article{8338,
abstract = {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.},
author = {Akopyan, Arseniy and Bobenko, Alexander I. and Schief, Wolfgang K. and Techter, Jan},
issn = {1432-0444},
journal = {Discrete and Computational Geometry},
pages = {938--976},
publisher = {Springer Nature},
title = {{On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs}},
doi = {10.1007/s00454-020-00240-w},
volume = {66},
year = {2020},
}
@inproceedings{8580,
abstract = {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.},
author = {Graff, Grzegorz and Graff, Beata and Jablonski, Grzegorz and Narkiewicz, Krzysztof},
booktitle = {11th Conference of the European Study Group on Cardiovascular Oscillations: Computation and Modelling in Physiology: New Challenges and Opportunities, },
isbn = {9781728157511},
location = {Pisa, Italy},
publisher = {IEEE},
title = {{The application of persistent homology in the analysis of heart rate variability}},
doi = {10.1109/ESGCO49734.2020.9158054},
year = {2020},
}
@inproceedings{8703,
abstract = {Even though Delaunay originally introduced his famous triangulations in the case of infinite point sets with translational periodicity, a software that computes such triangulations in the general case is not yet available, to the best of our knowledge. Combining and generalizing previous work, we present a practical algorithm for computing such triangulations. The algorithm has been implemented and experiments show that its performance is as good as the one of the CGAL package, which is restricted to cubic periodicity. },
author = {Osang, Georg F and Rouxel-Labbé, Mael and Teillaud, Monique},
booktitle = {28th Annual European Symposium on Algorithms},
isbn = {9783959771627},
issn = {18688969},
location = {Virtual, Online; Pisa, Italy},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Generalizing CGAL periodic Delaunay triangulations}},
doi = {10.4230/LIPIcs.ESA.2020.75},
volume = {173},
year = {2020},
}
@article{9157,
abstract = {Representing an atom by a solid sphere in 3-dimensional Euclidean space, we get the space-filling diagram of a molecule by taking the union. Molecular dynamics simulates its motion subject to bonds and other forces, including the solvation free energy. The morphometric approach [12, 17] writes the latter as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted mean curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [3], and the weighted Gaussian curvature [1], this yields the derivative of the morphometric expression of the solvation free energy.},
author = {Akopyan, Arseniy and Edelsbrunner, Herbert},
issn = {2544-7297},
journal = {Computational and Mathematical Biophysics},
number = {1},
pages = {51--67},
publisher = {Walter de Gruyter},
title = {{The weighted mean curvature derivative of a space-filling diagram}},
doi = {10.1515/cmb-2020-0100},
volume = {8},
year = {2020},
}
@article{9156,
abstract = {The morphometric approach [11, 14] writes the solvation free energy as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted Gaussian curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [4], and the weighted mean curvature in [1], this yields the derivative of the morphometric expression of solvation free energy.},
author = {Akopyan, Arseniy and Edelsbrunner, Herbert},
issn = {2544-7297},
journal = {Computational and Mathematical Biophysics},
number = {1},
pages = {74--88},
publisher = {Walter de Gruyter},
title = {{The weighted Gaussian curvature derivative of a space-filling diagram}},
doi = {10.1515/cmb-2020-0101},
volume = {8},
year = {2020},
}
@article{9249,
abstract = {Rhombic dodecahedron is a space filling polyhedron which represents the close packing of spheres in 3D space and the Voronoi structures of the face centered cubic (FCC) lattice. In this paper, we describe a new coordinate system where every 3-integer coordinates grid point corresponds to a rhombic dodecahedron centroid. In order to illustrate the interest of the new coordinate system, we propose the characterization of 3D digital plane with its topological features, such as the interrelation between the thickness of the digital plane and the separability constraint we aim to obtain. We also present the characterization of 3D digital lines and study it as the intersection of multiple digital planes. Characterization of 3D digital sphere with relevant topological features is proposed as well along with the 48-symmetry appearing in the new coordinate system.},
author = {Biswas, Ranita and Largeteau-Skapin, Gaëlle and Zrour, Rita and Andres, Eric},
issn = {2353-3390},
journal = {Mathematical Morphology - Theory and Applications},
number = {1},
pages = {143--158},
publisher = {De Gruyter},
title = {{Digital objects in rhombic dodecahedron grid}},
doi = {10.1515/mathm-2020-0106},
volume = {4},
year = {2020},
}
@inproceedings{9299,
abstract = {We call a multigraph non-homotopic if it can be drawn in the plane in such a way that no two edges connecting the same pair of vertices can be continuously transformed into each other without passing through a vertex, and no loop can be shrunk to its end-vertex in the same way. It is easy to see that a non-homotopic multigraph on n>1 vertices can have arbitrarily many edges. We prove that the number of crossings between the edges of a non-homotopic multigraph with n vertices and m>4n edges is larger than cm2n for some constant c>0 , and that this bound is tight up to a polylogarithmic factor. We also show that the lower bound is not asymptotically sharp as n is fixed and m⟶∞ .},
author = {Pach, János and Tardos, Gábor and Tóth, Géza},
booktitle = {28th International Symposium on Graph Drawing and Network Visualization},
isbn = {9783030687656},
issn = {1611-3349},
location = {Virtual, Online},
pages = {359--371},
publisher = {Springer Nature},
title = {{Crossings between non-homotopic edges}},
doi = {10.1007/978-3-030-68766-3_28},
volume = {12590},
year = {2020},
}
@article{9630,
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},
issn = {1920180X},
journal = {Journal of Computational Geometry},
number = {2},
pages = {162--182},
publisher = {Carleton University},
title = {{Topological data analysis in information space}},
doi = {10.20382/jocg.v11i2a7},
volume = {11},
year = {2020},
}
@inproceedings{7952,
abstract = {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. },
author = {Boissonnat, Jean-Daniel and Wintraecken, Mathijs},
booktitle = {36th International Symposium on Computational Geometry},
isbn = {978-3-95977-143-6},
issn = {1868-8969},
location = {Zürich, Switzerland},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{The topological correctness of PL-approximations of isomanifolds}},
doi = {10.4230/LIPIcs.SoCG.2020.20},
volume = {164},
year = {2020},
}
@article{8538,
abstract = {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.},
author = {Akopyan, Arseniy and Schwartz, Richard and Tabachnikov, Serge},
issn = {2199-6768},
journal = {European Journal of Mathematics},
publisher = {Springer Nature},
title = {{Billiards in ellipses revisited}},
doi = {10.1007/s40879-020-00426-9},
year = {2020},
}
@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{5678,
abstract = {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.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton},
issn = {14320444},
journal = {Discrete and Computational Geometry},
number = {4},
pages = {865–878},
publisher = {Springer},
title = {{Poisson–Delaunay Mosaics of Order k}},
doi = {10.1007/s00454-018-0049-2},
volume = {62},
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 = {223–256},
publisher = {Carleton University},
title = {{Simplices modelled on spaces of constant curvature}},
doi = {10.20382/jocg.v10i1a9},
volume = {10},
year = {2019},
}
@inproceedings{6628,
abstract = {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.},
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 für 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, André and Wintraecken, Mathijs},
issn = {2367-1734},
journal = {Journal of Applied and Computational Topology},
number = {1-2},
pages = {29–58},
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 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.},
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 Schläfli 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{7216,
abstract = {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.},
author = {Osang, Georg F and Cook, James and Fabrikant, Alex and Gruteser, Marco},
booktitle = {2019 IEEE Intelligent Transportation Systems Conference},
isbn = {9781538670248},
location = {Auckland, New Zealand},
publisher = {IEEE},
title = {{LiveTraVeL: Real-time matching of transit vehicle trajectories to transit routes at scale}},
doi = {10.1109/ITSC.2019.8917514},
year = {2019},
}
@article{6608,
abstract = {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.},
author = {Edelsbrunner, Herbert and Ölsböck, 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{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 Löffler, Maarten and Masárová, 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},
}
@unpublished{7950,
abstract = {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.},
author = {Biniaz, Ahmad and Jain, Kshitij and Lubiw, Anna and Masárová, Zuzana and Miltzow, Tillmann and Mondal, Debajyoti and Naredla, Anurag Murty and Tkadlec, Josef and Turcotte, Alexi},
booktitle = {arXiv},
title = {{Token swapping on trees}},
year = {2019},
}
@article{106,
abstract = {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.},
author = {Akopyan, Arseniy and Petrunin, Anton},
journal = {Mathematical Intelligencer},
number = {3},
pages = {26 -- 31},
publisher = {Springer},
title = {{Long geodesics on convex surfaces}},
doi = {10.1007/s00283-018-9795-5},
volume = {40},
year = {2018},
}
@article{1064,
abstract = {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.},
author = {Akopyan, Arseniy and Balitskiy, Alexey and Grigorev, Mikhail},
issn = {14320444},
journal = {Discrete & Computational Geometry},
number = {4},
pages = {1001--1009},
publisher = {Springer},
title = {{On the circle covering theorem by A.W. Goodman and R.E. Goodman}},
doi = {10.1007/s00454-017-9883-x},
volume = {59},
year = {2018},
}
@inproceedings{188,
abstract = {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.},
author = {Edelsbrunner, Herbert and Virk, Ziga and Wagner, Hubert},
location = {Budapest, Hungary},
pages = {35:1 -- 35:13},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Smallest enclosing spheres and Chernoff points in Bregman geometry}},
doi = {10.4230/LIPIcs.SoCG.2018.35},
volume = {99},
year = {2018},
}
@inproceedings{193,
abstract = {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.},
author = {Alwen, Joel F and Gazi, Peter and Kamath Hosdurg, Chethan and Klein, Karen and Osang, Georg F and Pietrzak, Krzysztof Z and Reyzin, Lenoid and Rolinek, Michal and Rybar, Michal},
booktitle = {Proceedings of the 2018 on Asia Conference on Computer and Communication Security},
location = {Incheon, Republic of Korea},
pages = {51 -- 65},
publisher = {ACM},
title = {{On the memory hardness of data independent password hashing functions}},
doi = {10.1145/3196494.3196534},
year = {2018},
}
@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},
}
@article{312,
abstract = {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.},
author = {Edelsbrunner, Herbert and Iglesias Ham, Mabel},
issn = {08954801},
journal = {SIAM J Discrete Math},
number = {1},
pages = {750 -- 782},
publisher = {Society for Industrial and Applied Mathematics },
title = {{On the optimality of the FCC lattice for soft sphere packing}},
doi = {10.1137/16M1097201},
volume = {32},
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 = {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.},
author = {Edelsbrunner, Herbert and Iglesias Ham, Mabel},
journal = {Computational Geometry: Theory and Applications},
pages = {119 -- 133},
publisher = {Elsevier},
title = {{Multiple covers with balls I: Inclusion–exclusion}},
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{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},
}
@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},
}
@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 m≥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{87,
abstract = {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.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton},
journal = {Annals of Applied Probability},
number = {5},
pages = {3215 -- 3238},
publisher = {Institute of Mathematical Statistics},
title = {{Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics}},
doi = {10.1214/18-AAP1389},
volume = {28},
year = {2018},
}
@inproceedings{187,
abstract = {Given a locally finite X ⊆ ℝd and a radius r ≥ 0, the k-fold cover of X and r consists of all points in ℝd that have k or more points of X within distance r. We consider two filtrations - one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k - and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in ℝd+1 whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module from Delaunay mosaics that is isomorphic to the persistence module of the multi-covers. },
author = {Edelsbrunner, Herbert and Osang, Georg F},
location = {Budapest, Hungary},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{The multi-cover persistence of Euclidean balls}},
doi = {10.4230/LIPIcs.SoCG.2018.34},
volume = {99},
year = {2018},
}
@article{1022,
abstract = {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.},
author = {Pranav, Pratyush and Edelsbrunner, Herbert and Van De Weygaert, Rien and Vegter, Gert and Kerber, Michael and Jones, Bernard and Wintraecken, Mathijs},
issn = {00358711},
journal = {Monthly Notices of the Royal Astronomical Society},
number = {4},
pages = {4281 -- 4310},
publisher = {Oxford University Press},
title = {{The topology of the cosmic web in terms of persistent Betti numbers}},
doi = {10.1093/mnras/stw2862},
volume = {465},
year = {2017},
}
@article{1065,
abstract = {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.},
author = {Chatterjee, Krishnendu and Osang, Georg F},
issn = {00200190},
journal = {Information Processing Letters},
pages = {25 -- 29},
publisher = {Elsevier},
title = {{Pushdown reachability with constant treewidth}},
doi = {10.1016/j.ipl.2017.02.003},
volume = {122},
year = {2017},
}