@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}, } @article{909, abstract = {We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that, for any convex shape K, there exist four points on the boundary of K such that the length of any curve passing through these points is at least half of the perimeter of K. It is also shown that the same statement does not remain valid with the additional constraint that the points are extreme points of K. Moreover, the factor &#xbd; cannot be achieved with any fixed number of extreme points. We conclude the paper with a few other inequalities related to the perimeter of a convex shape.}, author = {Akopyan, Arseniy and Vysotsky, Vladislav}, issn = {0002-9890}, journal = {The American Mathematical Monthly}, number = {7}, pages = {588 -- 596}, publisher = {Mathematical Association of America}, title = {{On the lengths of curves passing through boundary points of a planar convex shape}}, doi = {10.4169/amer.math.monthly.124.7.588}, volume = {124}, 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}, } @article{521, abstract = {Let X and Y be proper metric spaces. We show that a coarsely n-to-1 map f:X→Y induces an n-to-1 map of Higson coronas. This viewpoint turns out to be successful in showing that the classical dimension raising theorems hold in large scale; that is, if f:X→Y is a coarsely n-to-1 map between proper metric spaces X and Y then asdim(Y)≤asdim(X)+n−1. Furthermore we introduce coarsely open coarsely n-to-1 maps, which include the natural quotient maps via a finite group action, and prove that they preserve the asymptotic dimension.}, author = {Austin, Kyle and Virk, Ziga}, issn = {0166-8641}, journal = {Topology and its Applications}, pages = {45 -- 57}, publisher = {Elsevier}, title = {{Higson compactification and dimension raising}}, doi = {10.1016/j.topol.2016.10.005}, volume = {215}, year = {2017}, } @article{568, abstract = {We study robust properties of zero sets of continuous maps f: X → ℝn. Formally, we analyze the family Z< r(f) := (g-1(0): ||g - f|| < r) of all zero sets of all continuous maps g closer to f than r in the max-norm. All of these sets are outside A := (x: |f(x)| ≥ r) and we claim that Z< r(f) is fully determined by A and an element of a certain cohomotopy group which (by a recent result) is computable whenever the dimension of X is at most 2n - 3. By considering all r > 0 simultaneously, the pointed cohomotopy groups form a persistence module-a structure leading to persistence diagrams as in the case of persistent homology or well groups. Eventually, we get a descriptor of persistent robust properties of zero sets that has better descriptive power (Theorem A) and better computability status (Theorem B) than the established well diagrams. Moreover, if we endow every point of each zero set with gradients of the perturbation, the robust description of the zero sets by elements of cohomotopy groups is in some sense the best possible (Theorem C).}, author = {Franek, Peter and Krcál, Marek}, issn = {1532-0073}, journal = {Homology, Homotopy and Applications}, number = {2}, pages = {313 -- 342}, publisher = {International Press}, title = {{Persistence of zero sets}}, doi = {10.4310/HHA.2017.v19.n2.a16}, volume = {19}, year = {2017}, } @inbook{5803, abstract = {Different distance metrics produce Voronoi diagrams with different properties. It is a well-known that on the (real) 2D plane or even on any 3D plane, a Voronoi diagram (VD) based on the Euclidean distance metric produces convex Voronoi regions. In this paper, we first show that this metric produces a persistent VD on the 2D digital plane, as it comprises digitally convex Voronoi regions and hence correctly approximates the corresponding VD on the 2D real plane. Next, we show that on a 3D digital plane D, the Euclidean metric spanning over its voxel set does not guarantee a digital VD which is persistent with the real-space VD. As a solution, we introduce a novel concept of functional-plane-convexity, which is ensured by the Euclidean metric spanning over the pedal set of D. Necessary proofs and some visual result have been provided to adjudge the merit and usefulness of the proposed concept.}, author = {Biswas, Ranita and Bhowmick, Partha}, booktitle = {Combinatorial image analysis}, isbn = {978-3-319-59107-0}, issn = {0302-9743}, location = {Plovdiv, Bulgaria}, pages = {93--104}, publisher = {Springer Nature}, title = {{Construction of persistent Voronoi diagram on 3D digital plane}}, doi = {10.1007/978-3-319-59108-7_8}, volume = {10256}, 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 = {1868-8969}, 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 = {0024-6093}, journal = {Bulletin of the London Mathematical Society}, number = {4}, pages = {690 -- 693}, publisher = {Wiley}, title = {{A tight estimate for the waist of the ball }}, doi = {10.1112/blms.12062}, volume = {49}, year = {2017}, } @phdthesis{6287, abstract = {The main objects considered in the present work are simplicial and CW-complexes with vertices forming a random point cloud. In particular, we consider a Poisson point process in R^n and study Delaunay and Voronoi complexes of the first and higher orders and weighted Delaunay complexes obtained as sections of Delaunay complexes, as well as the Čech complex. Further, we examine theDelaunay complex of a Poisson point process on the sphere S^n, as well as of a uniform point cloud, which is equivalent to the convex hull, providing a connection to the theory of random polytopes. Each of the complexes in question can be endowed with a radius function, which maps its cells to the radii of appropriately chosen circumspheres, called the radius of the cell. Applying and developing discrete Morse theory for these functions, joining it together with probabilistic and sometimes analytic machinery, and developing several integral geometric tools, we aim at getting the distributions of circumradii of typical cells. For all considered complexes, we are able to generalize and obtain up to constants the distribution of radii of typical intervals of all types. In low dimensions the constants can be computed explicitly, thus providing the explicit expressions for the expected numbers of cells. In particular, it allows to find the expected density of simplices of every dimension for a Poisson point process in R^4, whereas the result for R^3 was known already in 1970's.}, author = {Nikitenko, Anton}, issn = {2663-337X}, pages = {86}, publisher = {Institute of Science and Technology Austria}, title = {{Discrete Morse theory for random complexes }}, doi = {10.15479/AT:ISTA:th_873}, 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 = {0001-8678}, 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 = {0166-8641}, 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{1433, abstract = {Phat is an open-source C. ++ library for the computation of persistent homology by matrix reduction, targeted towards developers of software for topological data analysis. We aim for a simple generic design that decouples algorithms from data structures without sacrificing efficiency or user-friendliness. We provide numerous different reduction strategies as well as data types to store and manipulate the boundary matrix. We compare the different combinations through extensive experimental evaluation and identify optimization techniques that work well in practical situations. We also compare our software with various other publicly available libraries for persistent homology.}, author = {Bauer, Ulrich and Kerber, Michael and Reininghaus, Jan and Wagner, Hubert}, issn = { 0747-7171}, journal = {Journal of Symbolic Computation}, pages = {76 -- 90}, publisher = {Academic Press}, title = {{Phat - Persistent homology algorithms toolbox}}, doi = {10.1016/j.jsc.2016.03.008}, volume = {78}, year = {2017}, } @article{1149, abstract = {We study the usefulness of two most prominent publicly available rigorous ODE integrators: one provided by the CAPD group (capd.ii.uj.edu.pl), the other based on the COSY Infinity project (cosyinfinity.org). Both integrators are capable of handling entire sets of initial conditions and provide tight rigorous outer enclosures of the images under a time-T map. We conduct extensive benchmark computations using the well-known Lorenz system, and compare the computation time against the final accuracy achieved. We also discuss the effect of a few technical parameters, such as the order of the numerical integration method, the value of T, and the phase space resolution. We conclude that COSY may provide more precise results due to its ability of avoiding the variable dependency problem. However, the overall cost of computations conducted using CAPD is typically lower, especially when intervals of parameters are involved. Moreover, access to COSY is limited (registration required) and the rigorous ODE integrators are not publicly available, while CAPD is an open source free software project. Therefore, we recommend the latter integrator for this kind of computations. Nevertheless, proper choice of the various integration parameters turns out to be of even greater importance than the choice of the integrator itself. © 2016 IMACS. Published by Elsevier B.V. All rights reserved.}, author = {Miyaji, Tomoyuki and Pilarczyk, Pawel and Gameiro, Marcio and Kokubu, Hiroshi and Mischaikow, Konstantin}, journal = {Applied Numerical Mathematics}, pages = {34 -- 47}, publisher = {Elsevier}, title = {{A study of rigorous ODE integrators for multi scale set oriented computations}}, doi = {10.1016/j.apnum.2016.04.005}, volume = {107}, year = {2016}, } @article{1222, abstract = {We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason—the problem of “super resolution of images.” We have found optimal arrangements for N=6, 7 and 8 circles. Surprisingly, for the case N=7 there are three different optimal arrangements. Our proof is based on a computer enumeration of toroidal irreducible contact graphs.}, author = {Musin, Oleg and Nikitenko, Anton}, journal = {Discrete & Computational Geometry}, number = {1}, pages = {1 -- 20}, publisher = {Springer}, title = {{Optimal packings of congruent circles on a square flat torus}}, doi = {10.1007/s00454-015-9742-6}, volume = {55}, year = {2016}, } @article{1360, abstract = {We apply the technique of Károly Bezdek and Daniel Bezdek to study billiard trajectories in convex bodies, when the length is measured with a (possibly asymmetric) norm. We prove a lower bound for the length of the shortest closed billiard trajectory, related to the non-symmetric Mahler problem. With this technique we are able to give short and elementary proofs to some known results. }, author = {Akopyan, Arseniy and Balitskiy, Alexey and Karasev, Roman and Sharipova, Anastasia}, journal = {Proceedings of the American Mathematical Society}, number = {10}, pages = {4501 -- 4513}, publisher = {American Mathematical Society}, title = {{Elementary approach to closed billiard trajectories in asymmetric normed spaces}}, doi = {10.1090/proc/13062}, volume = {144}, year = {2016}, } @article{1408, abstract = {The concept of well group in a special but important case captures homological properties of the zero set of a continuous map (Formula presented.) on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within (Formula presented.) distance r from f for a given (Formula presented.). The main drawback of the approach is that the computability of well groups was shown only when (Formula presented.) or (Formula presented.). Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of (Formula presented.) by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and (Formula presented.), our approximation of the (Formula presented.)th well group is exact. For the second part, we find examples of maps (Formula presented.) with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.}, author = {Franek, Peter and Krcál, Marek}, journal = {Discrete & Computational Geometry}, number = {1}, pages = {126 -- 164}, publisher = {Springer}, title = {{On computability and triviality of well groups}}, doi = {10.1007/s00454-016-9794-2}, volume = {56}, year = {2016}, } @inproceedings{1237, abstract = {Bitmap images of arbitrary dimension may be formally perceived as unions of m-dimensional boxes aligned with respect to a rectangular grid in ℝm. Cohomology and homology groups are well known topological invariants of such sets. Cohomological operations, such as the cup product, provide higher-order algebraic topological invariants, especially important for digital images of dimension higher than 3. If such an operation is determined at the level of simplicial chains [see e.g. González-Díaz, Real, Homology, Homotopy Appl, 2003, 83-93], then it is effectively computable. However, decomposing a cubical complex into a simplicial one deleteriously affects the efficiency of such an approach. In order to avoid this overhead, a direct cubical approach was applied in [Pilarczyk, Real, Adv. Comput. Math., 2015, 253-275] for the cup product in cohomology, and implemented in the ChainCon software package [http://www.pawelpilarczyk.com/chaincon/]. We establish a formula for the Steenrod square operations [see Steenrod, Annals of Mathematics. Second Series, 1947, 290-320] directly at the level of cubical chains, and we prove the correctness of this formula. An implementation of this formula is programmed in C++ within the ChainCon software framework. We provide a few examples and discuss the effectiveness of this approach. One specific application follows from the fact that Steenrod squares yield tests for the topological extension problem: Can a given map A → Sd to a sphere Sd be extended to a given super-complex X of A? In particular, the ROB-SAT problem, which is to decide for a given function f: X → ℝm and a value r > 0 whether every g: X → ℝm with ∥g - f ∥∞ ≤ r has a root, reduces to the extension problem.}, author = {Krcál, Marek and Pilarczyk, Pawel}, location = {Marseille, France}, pages = {140 -- 151}, publisher = {Springer}, title = {{Computation of cubical Steenrod squares}}, doi = {10.1007/978-3-319-39441-1_13}, volume = {9667}, year = {2016}, } @article{1252, abstract = {We study the homomorphism induced in homology by a closed correspondence between topological spaces, using projections from the graph of the correspondence to its domain and codomain. We provide assumptions under which the homomorphism induced by an outer approximation of a continuous map coincides with the homomorphism induced in homology by the map. In contrast to more classical results we do not require that the projection to the domain have acyclic preimages. Moreover, we show that it is possible to retrieve correct homological information from a correspondence even if some data is missing or perturbed. Finally, we describe an application to combinatorial maps that are either outer approximations of continuous maps or reconstructions of such maps from a finite set of data points.}, author = {Harker, Shaun and Kokubu, Hiroshi and Mischaikow, Konstantin and Pilarczyk, Pawel}, issn = {1088-6826}, journal = {Proceedings of the American Mathematical Society}, number = {4}, pages = {1787 -- 1801}, publisher = {American Mathematical Society}, title = {{Inducing a map on homology from a correspondence}}, doi = {10.1090/proc/12812}, volume = {144}, year = {2016}, } @article{1254, abstract = {We use rigorous numerical techniques to compute a lower bound for the exponent of expansivity outside a neighborhood of the critical point for thousands of intervals of parameter values in the quadratic family. We first compute a radius of the critical neighborhood outside which the map is uniformly expanding. This radius is taken as small as possible, yet large enough for our numerical procedure to succeed in proving that the expansivity exponent outside this neighborhood is positive. Then, for each of the intervals, we compute a lower bound for this expansivity exponent, valid for all the parameters in that interval. We illustrate and study the distribution of the radii and the expansivity exponents. The results of our computations are mathematically rigorous. The source code of the software and the results of the computations are made publicly available at http://www.pawelpilarczyk.com/quadratic/.}, author = {Golmakani, Ali and Luzzatto, Stefano and Pilarczyk, Pawel}, journal = {Experimental Mathematics}, number = {2}, pages = {116 -- 124}, publisher = {Taylor and Francis}, title = {{Uniform expansivity outside a critical neighborhood in the quadratic family}}, doi = {10.1080/10586458.2015.1048011}, volume = {25}, year = {2016}, } @article{1272, abstract = {We study different means to extend offsetting based on skeletal structures beyond the well-known constant-radius and mitered offsets supported by Voronoi diagrams and straight skeletons, for which the orthogonal distance of offset elements to their respective input elements is constant and uniform over all input elements. Our main contribution is a new geometric structure, called variable-radius Voronoi diagram, which supports the computation of variable-radius offsets, i.e., offsets whose distance to the input is allowed to vary along the input. We discuss properties of this structure and sketch a prototype implementation that supports the computation of variable-radius offsets based on this new variant of Voronoi diagrams.}, author = {Held, Martin and Huber, Stefan and Palfrader, Peter}, journal = {Computer-Aided Design and Applications}, number = {5}, pages = {712 -- 721}, publisher = {Taylor and Francis}, title = {{Generalized offsetting of planar structures using skeletons}}, doi = {10.1080/16864360.2016.1150718}, volume = {13}, year = {2016}, }