@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}, issn = {2663-337X}, pages = {171}, publisher = {Institute of Science and Technology Austria}, title = {{Multiple covers with balls}}, doi = {10.15479/AT:ISTA:th_1026}, 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generated by confocal conics and circles}}, doi = {10.1007/s10711-017-0265-6}, volume = {194}, year = {2018}, } @unpublished{75, abstract = {We prove that any convex body in the plane can be partitioned into m convex parts of equal areas and perimeters for any integer m≥2; this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}, publisher = {arXiv}, title = {{Convex fair partitions into arbitrary number of pieces}}, doi = {10.48550/arXiv.1804.03057}, 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{6355, abstract = {We prove that any cyclic quadrilateral can be inscribed in any closed convex C1-curve. The smoothness condition is not required if the quadrilateral is a rectangle.}, author = {Akopyan, Arseniy and Avvakumov, Sergey}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Any cyclic quadrilateral can be inscribed in any closed convex smooth curve}}, doi = {10.1017/fms.2018.7}, volume = {6}, year = {2018}, } @article{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}, } @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{1072, abstract = {Given a finite set of points in Rn and a radius parameter, we study the Čech, Delaunay–Čech, Delaunay (or alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, which are explicitly described by a single discrete gradient field.}, author = {Bauer, Ulrich and Edelsbrunner, Herbert}, journal = {Transactions of the American Mathematical Society}, number = {5}, pages = {3741 -- 3762}, publisher = {American Mathematical Society}, title = {{The Morse theory of Čech and delaunay complexes}}, doi = {10.1090/tran/6991}, volume = {369}, year = {2017}, } @article{1173, abstract = {We introduce the Voronoi functional of a triangulation of a finite set of points in the Euclidean plane and prove that among all geometric triangulations of the point set, the Delaunay triangulation maximizes the functional. This result neither extends to topological triangulations in the plane nor to geometric triangulations in three and higher dimensions.}, author = {Edelsbrunner, Herbert and Glazyrin, Alexey and Musin, Oleg and Nikitenko, Anton}, issn = {02099683}, journal = {Combinatorica}, number = {5}, pages = {887 -- 910}, publisher = {Springer}, title = {{The Voronoi functional is maximized by the Delaunay triangulation in the plane}}, doi = {10.1007/s00493-016-3308-y}, volume = {37}, year = {2017}, } @inproceedings{833, abstract = {We present an efficient algorithm to compute Euler characteristic curves of gray scale images of arbitrary dimension. In various applications the Euler characteristic curve is used as a descriptor of an image. Our algorithm is the first streaming algorithm for Euler characteristic curves. The usage of streaming removes the necessity to store the entire image in RAM. Experiments show that our implementation handles terabyte scale images on commodity hardware. Due to lock-free parallelism, it scales well with the number of processor cores. Additionally, we put the concept of the Euler characteristic curve in the wider context of computational topology. In particular, we explain the connection with persistence diagrams.}, author = {Heiss, Teresa and Wagner, Hubert}, editor = {Felsberg, Michael and Heyden, Anders and Krüger, Norbert}, issn = {03029743}, location = {Ystad, Sweden}, pages = {397 -- 409}, publisher = {Springer}, title = {{Streaming algorithm for Euler characteristic curves of multidimensional images}}, doi = {10.1007/978-3-319-64689-3_32}, volume = {10424}, year = {2017}, } @inproceedings{836, abstract = {Recent research has examined how to study the topological features of a continuous self-map by means of the persistence of the eigenspaces, for given eigenvalues, of the endomorphism induced in homology over a field. This raised the question of how to select dynamically significant eigenvalues. The present paper aims to answer this question, giving an algorithm that computes the persistence of eigenspaces for every eigenvalue simultaneously, also expressing said eigenspaces as direct sums of “finite” and “singular” subspaces.}, author = {Ethier, Marc and Jablonski, Grzegorz and Mrozek, Marian}, booktitle = {Special Sessions in Applications of Computer Algebra}, isbn = {978-331956930-7}, location = {Kalamata, Greece}, pages = {119 -- 136}, publisher = {Springer}, title = {{Finding eigenvalues of self-maps with the Kronecker canonical form}}, doi = {10.1007/978-3-319-56932-1_8}, volume = {198}, year = {2017}, } @inbook{84, abstract = {The advent of high-throughput technologies and the concurrent advances in information sciences have led to a data revolution in biology. This revolution is most significant in molecular biology, with an increase in the number and scale of the “omics” projects over the last decade. Genomics projects, for example, have produced impressive advances in our knowledge of the information concealed into genomes, from the many genes that encode for the proteins that are responsible for most if not all cellular functions, to the noncoding regions that are now known to provide regulatory functions. Proteomics initiatives help to decipher the role of post-translation modifications on the protein structures and provide maps of protein-protein interactions, while functional genomics is the field that attempts to make use of the data produced by these projects to understand protein functions. The biggest challenge today is to assimilate the wealth of information provided by these initiatives into a conceptual framework that will help us decipher life. For example, the current views of the relationship between protein structure and function remain fragmented. We know of their sequences, more and more about their structures, we have information on their biological activities, but we have difficulties connecting this dotted line into an informed whole. We lack the experimental and computational tools for directly studying protein structure, function, and dynamics at the molecular and supra-molecular levels. In this chapter, we review some of the current developments in building the computational tools that are needed, focusing on the role that geometry and topology play in these efforts. One of our goals is to raise the general awareness about the importance of geometric methods in elucidating the mysterious foundations of our very existence. Another goal is the broadening of what we consider a geometric algorithm. There is plenty of valuable no-man’s-land between combinatorial and numerical algorithms, and it seems opportune to explore this land with a computational-geometric frame of mind.}, author = {Edelsbrunner, Herbert and Koehl, Patrice}, booktitle = {Handbook of Discrete and Computational Geometry, Third Edition}, editor = {Toth, Csaba and O'Rourke, Joseph and Goodman, Jacob}, pages = {1709 -- 1735}, publisher = {Taylor & Francis}, title = {{Computational topology for structural molecular biology}}, doi = {10.1201/9781315119601}, year = {2017}, } @inproceedings{688, abstract = {We show that the framework of topological data analysis can be extended from metrics to general Bregman divergences, widening the scope of possible applications. Examples are the Kullback - Leibler divergence, which is commonly used for comparing text and images, and the Itakura - Saito divergence, popular for speech and sound. In particular, we prove that appropriately generalized čech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized čech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory. }, author = {Edelsbrunner, Herbert and Wagner, Hubert}, issn = {18688969}, location = {Brisbane, Australia}, pages = {391--3916}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Topological data analysis with Bregman divergences}}, doi = {10.4230/LIPIcs.SoCG.2017.39}, volume = {77}, year = {2017}, } @article{707, abstract = {We answer a question of M. Gromov on the waist of the unit ball.}, author = {Akopyan, Arseniy and Karasev, Roman}, issn = {00246093}, journal = {Bulletin of the London Mathematical Society}, number = {4}, pages = {690 -- 693}, publisher = {Wiley-Blackwell}, title = {{A tight estimate for the waist of the ball }}, doi = {10.1112/blms.12062}, volume = {49}, year = {2017}, } @article{718, abstract = {Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in ℝ n , we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and nonsingular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we obtain the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensions n ≤ 4.}, author = {Edelsbrunner, Herbert and Nikitenko, Anton and Reitzner, Matthias}, issn = {00018678}, journal = {Advances in Applied Probability}, number = {3}, pages = {745 -- 767}, publisher = {Cambridge University Press}, title = {{Expected sizes of poisson Delaunay mosaics and their discrete Morse functions}}, doi = {10.1017/apr.2017.20}, volume = {49}, year = {2017}, } @article{737, abstract = {We generalize Brazas’ topology on the fundamental group to the whole universal path space X˜ i.e., to the set of homotopy classes of all based paths. We develop basic properties of the new notion and provide a complete comparison of the obtained topology with the established topologies, in particular with the Lasso topology and the CO topology, i.e., the topology that is induced by the compact-open topology. It turns out that the new topology is the finest topology contained in the CO topology, for which the action of the fundamental group on the universal path space is a continuous group action.}, author = {Virk, Ziga and Zastrow, Andreas}, issn = {01668641}, journal = {Topology and its Applications}, pages = {186 -- 196}, publisher = {Elsevier}, title = {{A new topology on the universal path space}}, doi = {10.1016/j.topol.2017.09.015}, volume = {231}, year = {2017}, } @article{481, abstract = {We introduce planar matchings on directed pseudo-line arrangements, which yield a planar set of pseudo-line segments such that only matching-partners are adjacent. By translating the planar matching problem into a corresponding stable roommates problem we show that such matchings always exist. Using our new framework, we establish, for the first time, a complete, rigorous definition of weighted straight skeletons, which are based on a so-called wavefront propagation process. We present a generalized and unified approach to treat structural changes in the wavefront that focuses on the restoration of weak planarity by finding planar matchings.}, author = {Biedl, Therese and Huber, Stefan and Palfrader, Peter}, journal = {International Journal of Computational Geometry and Applications}, number = {3-4}, pages = {211 -- 229}, publisher = {World Scientific Publishing}, title = {{Planar matchings for weighted straight skeletons}}, doi = {10.1142/S0218195916600050}, volume = {26}, year = {2017}, }