@article{1938, abstract = {We numerically investigate the distribution of extrema of 'chaotic' Laplacian eigenfunctions on two-dimensional manifolds. Our contribution is two-fold: (a) we count extrema on grid graphs with a small number of randomly added edges and show the behavior to coincide with the 1957 prediction of Longuet-Higgins for the continuous case and (b) we compute the regularity of their spatial distribution using discrepancy, which is a classical measure from the theory of Monte Carlo integration. The first part suggests that grid graphs with randomly added edges should behave like two-dimensional surfaces with ergodic geodesic flow; in the second part we show that the extrema are more regularly distributed in space than the grid Z2.}, author = {Pausinger, Florian and Steinerberger, Stefan}, journal = {Physics Letters, Section A}, number = {6}, pages = {535 -- 541}, publisher = {Elsevier}, title = {{On the distribution of local extrema in quantum chaos}}, doi = {10.1016/j.physleta.2014.12.010}, volume = {379}, year = {2015}, } @article{2035, abstract = {Considering a continuous self-map and the induced endomorphism on homology, we study the eigenvalues and eigenspaces of the latter. Taking a filtration of representations, we define the persistence of the eigenspaces, effectively introducing a hierarchical organization of the map. The algorithm that computes this information for a finite sample is proved to be stable, and to give the correct answer for a sufficiently dense sample. Results computed with an implementation of the algorithm provide evidence of its practical utility. }, author = {Edelsbrunner, Herbert and Jablonski, Grzegorz and Mrozek, Marian}, journal = {Foundations of Computational Mathematics}, number = {5}, pages = {1213 -- 1244}, publisher = {Springer}, title = {{The persistent homology of a self-map}}, doi = {10.1007/s10208-014-9223-y}, volume = {15}, year = {2015}, } @misc{9737, author = {Symonova, Olga and Topp, Christopher and Edelsbrunner, Herbert}, publisher = {Public Library of Science}, title = {{Root traits computed by DynamicRoots for the maize root shown in fig 2}}, doi = {10.1371/journal.pone.0127657.s001}, year = {2015}, } @article{1793, abstract = {We present a software platform for reconstructing and analyzing the growth of a plant root system from a time-series of 3D voxelized shapes. It aligns the shapes with each other, constructs a geometric graph representation together with the function that records the time of growth, and organizes the branches into a hierarchy that reflects the order of creation. The software includes the automatic computation of structural and dynamic traits for each root in the system enabling the quantification of growth on fine-scale. These are important advances in plant phenotyping with applications to the study of genetic and environmental influences on growth.}, author = {Symonova, Olga and Topp, Christopher and Edelsbrunner, Herbert}, journal = {PLoS One}, number = {6}, publisher = {Public Library of Science}, title = {{DynamicRoots: A software platform for the reconstruction and analysis of growing plant roots}}, doi = {10.1371/journal.pone.0127657}, volume = {10}, year = {2015}, } @article{1805, abstract = {We consider the problem of deciding whether the persistent homology group of a simplicial pair (K,L) can be realized as the homology H∗(X) of some complex X with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in double-struck R3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on double-struck S3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard.}, author = {Attali, Dominique and Bauer, Ulrich and Devillers, Olivier and Glisse, Marc and Lieutier, André}, journal = {Computational Geometry: Theory and Applications}, number = {8}, pages = {606 -- 621}, publisher = {Elsevier}, title = {{Homological reconstruction and simplification in R3}}, doi = {10.1016/j.comgeo.2014.08.010}, volume = {48}, year = {2015}, } @article{1828, abstract = {We construct a non-linear Markov process connected with a biological model of a bacterial genome recombination. The description of invariant measures of this process gives us the solution of one problem in elementary probability theory.}, author = {Akopyan, Arseniy and Pirogov, Sergey and Rybko, Aleksandr}, journal = {Journal of Statistical Physics}, number = {1}, pages = {163 -- 167}, publisher = {Springer}, title = {{Invariant measures of genetic recombination process}}, doi = {10.1007/s10955-015-1238-5}, volume = {160}, year = {2015}, } @article{1555, abstract = {We show that incorporating spatial dispersal of individuals into a simple vaccination epidemic model may give rise to a model that exhibits rich dynamical behavior. Using an SIVS (susceptible-infected-vaccinated-susceptible) model as a basis, we describe the spread of an infectious disease in a population split into two regions. In each subpopulation, both forward and backward bifurcations can occur. This implies that for disconnected regions the two-patch system may admit several steady states. We consider traveling between the regions and investigate the impact of spatial dispersal of individuals on the model dynamics. We establish conditions for the existence of multiple nontrivial steady states in the system, and we study the structure of the equilibria. The mathematical analysis reveals an unusually rich dynamical behavior, not normally found in the simple epidemic models. In addition to the disease-free equilibrium, eight endemic equilibria emerge from backward transcritical and saddle-node bifurcation points, forming an interesting bifurcation diagram. Stability of steady states, their bifurcations, and the global dynamics are investigated with analytical tools, numerical simulations, and rigorous set-oriented numerical computations.}, author = {Knipl, Diána and Pilarczyk, Pawel and Röst, Gergely}, issn = {1536-0040}, journal = {SIAM Journal on Applied Dynamical Systems}, number = {2}, pages = {980 -- 1017}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Rich bifurcation structure in a two patch vaccination model}}, doi = {10.1137/140993934}, volume = {14}, year = {2015}, } @article{1792, abstract = {Motivated by recent ideas of Harman (Unif. Distrib. Theory, 2010) we develop a new concept of variation of multivariate functions on a compact Hausdorff space with respect to a collection D of subsets. We prove a general version of the Koksma-Hlawka theorem that holds for this notion of variation and discrepancy with respect to D. As special cases, we obtain Koksma-Hlawka inequalities for classical notions, such as extreme or isotropic discrepancy. For extreme discrepancy, our result coincides with the usual Koksma-Hlawka theorem. We show that the space of functions of bounded D-variation contains important discontinuous functions and is closed under natural algebraic operations. Finally, we illustrate the results on concrete integration problems from integral geometry and stereology.}, author = {Pausinger, Florian and Svane, Anne}, journal = {Journal of Complexity}, number = {6}, pages = {773 -- 797}, publisher = {Academic Press}, title = {{A Koksma-Hlawka inequality for general discrepancy systems}}, doi = {10.1016/j.jco.2015.06.002}, volume = {31}, year = {2015}, } @phdthesis{1399, abstract = {This thesis is concerned with the computation and approximation of intrinsic volumes. Given a smooth body M and a certain digital approximation of it, we develop algorithms to approximate various intrinsic volumes of M using only measurements taken from its digital approximations. The crucial idea behind our novel algorithms is to link the recent theory of persistent homology to the theory of intrinsic volumes via the Crofton formula from integral geometry and, in particular, via Euler characteristic computations. Our main contributions are a multigrid convergent digital algorithm to compute the first intrinsic volume of a solid body in R^n as well as an appropriate integration pipeline to approximate integral-geometric integrals defined over the Grassmannian manifold.}, author = {Pausinger, Florian}, issn = {2663-337X}, pages = {144}, publisher = {Institute of Science and Technology Austria}, title = {{On the approximation of intrinsic volumes}}, year = {2015}, } @inbook{10817, abstract = {The Morse-Smale complex can be either explicitly or implicitly represented. Depending on the type of representation, the simplification of the Morse-Smale complex works differently. In the explicit representation, the Morse-Smale complex is directly simplified by explicitly reconnecting the critical points during the simplification. In the implicit representation, on the other hand, the Morse-Smale complex is given by a combinatorial gradient field. In this setting, the simplification changes the combinatorial flow, which yields an indirect simplification of the Morse-Smale complex. The topological complexity of the Morse-Smale complex is reduced in both representations. However, the simplifications generally yield different results. In this chapter, we emphasize properties of the two representations that cause these differences. We also provide a complexity analysis of the two schemes with respect to running time and memory consumption.}, author = {Günther, David and Reininghaus, Jan and Seidel, Hans-Peter and Weinkauf, Tino}, booktitle = {Topological Methods in Data Analysis and Visualization III.}, editor = {Bremer, Peer-Timo and Hotz, Ingrid and Pascucci, Valerio and Peikert, Ronald}, isbn = {9783319040981}, issn = {2197-666X}, pages = {135--150}, publisher = {Springer Nature}, title = {{Notes on the simplification of the Morse-Smale complex}}, doi = {10.1007/978-3-319-04099-8_9}, year = {2014}, } @inproceedings{10886, abstract = {We propose a method for visualizing two-dimensional symmetric positive definite tensor fields using the Heat Kernel Signature (HKS). The HKS is derived from the heat kernel and was originally introduced as an isometry invariant shape signature. Each positive definite tensor field defines a Riemannian manifold by considering the tensor field as a Riemannian metric. On this Riemmanian manifold we can apply the definition of the HKS. The resulting scalar quantity is used for the visualization of tensor fields. The HKS is closely related to the Gaussian curvature of the Riemannian manifold and the time parameter of the heat kernel allows a multiscale analysis in a natural way. In this way, the HKS represents field related scale space properties, enabling a level of detail analysis of tensor fields. This makes the HKS an interesting new scalar quantity for tensor fields, which differs significantly from usual tensor invariants like the trace or the determinant. A method for visualization and a numerical realization of the HKS for tensor fields is proposed in this chapter. To validate the approach we apply it to some illustrating simple examples as isolated critical points and to a medical diffusion tensor data set.}, author = {Zobel, Valentin and Reininghaus, Jan and Hotz, Ingrid}, booktitle = {Topological Methods in Data Analysis and Visualization III }, isbn = {9783319040981}, issn = {2197-666X}, pages = {249--262}, publisher = {Springer}, title = {{Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature}}, doi = {10.1007/978-3-319-04099-8_16}, year = {2014}, } @inproceedings{10892, abstract = {In this paper, 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}, booktitle = {25th International Symposium, ISAAC 2014}, isbn = {9783319130743}, issn = {1611-3349}, location = {Jeonju, Korea}, pages = {117--127}, publisher = {Springer Nature}, title = {{Planar matchings for weighted straight skeletons}}, doi = {10.1007/978-3-319-13075-0_10}, volume = {8889}, year = {2014}, } @inbook{10893, abstract = {Saddle periodic orbits are an essential and stable part of the topological skeleton of a 3D vector field. Nevertheless, there is currently no efficient algorithm to robustly extract these features. In this chapter, we present a novel technique to extract saddle periodic orbits. Exploiting the analytic properties of such an orbit, we propose a scalar measure based on the finite-time Lyapunov exponent (FTLE) that indicates its presence. Using persistent homology, we can then extract the robust cycles of this field. These cycles thereby represent the saddle periodic orbits of the given vector field. We discuss the different existing FTLE approximation schemes regarding their applicability to this specific problem and propose an adapted version of FTLE called Normalized Velocity Separation. Finally, we evaluate our method using simple analytic vector field data.}, author = {Kasten, Jens and Reininghaus, Jan and Reich, Wieland and Scheuermann, Gerik}, booktitle = {Topological Methods in Data Analysis and Visualization III }, editor = {Bremer, Peer-Timo and Hotz, Ingrid and Pascucci, Valerio and Peikert, Ronald}, isbn = {9783319040981}, issn = {2197-666X}, pages = {55--69}, publisher = {Springer}, title = {{Toward the extraction of saddle periodic orbits}}, doi = {10.1007/978-3-319-04099-8_4}, volume = {1}, year = {2014}, } @inproceedings{10894, abstract = {PHAT is a C++ library for the computation of persistent homology by matrix reduction. We aim for a simple generic design that decouples algorithms from data structures without sacrificing efficiency or user-friendliness. This makes PHAT a versatile platform for experimenting with algorithmic ideas and comparing them to state of the art implementations.}, author = {Bauer, Ulrich and Kerber, Michael and Reininghaus, Jan and Wagner, Hubert}, booktitle = {ICMS 2014: International Congress on Mathematical Software}, isbn = {9783662441985}, issn = {1611-3349}, location = {Seoul, South Korea}, pages = {137--143}, publisher = {Springer Berlin Heidelberg}, title = {{PHAT – Persistent Homology Algorithms Toolbox}}, doi = {10.1007/978-3-662-44199-2_24}, volume = {8592}, year = {2014}, } @inproceedings{2153, abstract = {We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and coker f . As an immediate corollary, we obtain a new proof of the algebraic stability theorem for persistence barcodes [5, 9], a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules. Copyright is held by the owner/author(s).}, author = {Bauer, Ulrich and Lesnick, Michael}, booktitle = {Proceedings of the Annual Symposium on Computational Geometry}, location = {Kyoto, Japan}, pages = {355 -- 364}, publisher = {ACM}, title = {{Induced matchings of barcodes and the algebraic stability of persistence}}, doi = {10.1145/2582112.2582168}, year = {2014}, } @inproceedings{2155, abstract = {Given a finite set of points in Rn and a positive radius, we study the Čech, Delaunay-Čech, alpha, and wrap complexes as instances of a generalized discrete Morse theory. We prove that the latter three complexes are simple-homotopy equivalent. Our results have applications in topological data analysis and in the reconstruction of shapes from sampled data. Copyright is held by the owner/author(s).}, author = {Bauer, Ulrich and Edelsbrunner, Herbert}, booktitle = {Proceedings of the Annual Symposium on Computational Geometry}, location = {Kyoto, Japan}, pages = {484 -- 490}, publisher = {ACM}, title = {{The morse theory of Čech and Delaunay filtrations}}, doi = {10.1145/2582112.2582167}, year = {2014}, } @inproceedings{2156, abstract = {We propose a metric for Reeb graphs, called the functional distortion distance. Under this distance, the Reeb graph is stable against small changes of input functions. At the same time, it remains discriminative at differentiating input functions. In particular, the main result is that the functional distortion distance between two Reeb graphs is bounded from below by the bottleneck distance between both the ordinary and extended persistence diagrams for appropriate dimensions. As an application of our results, we analyze a natural simplification scheme for Reeb graphs, and show that persistent features in Reeb graph remains persistent under simplification. Understanding the stability of important features of the Reeb graph under simplification is an interesting problem on its own right, and critical to the practical usage of Reeb graphs. Copyright is held by the owner/author(s).}, author = {Bauer, Ulrich and Ge, Xiaoyin and Wang, Yusu}, booktitle = {Proceedings of the Annual Symposium on Computational Geometry}, location = {Kyoto, Japan}, pages = {464 -- 473}, publisher = {ACM}, title = {{Measuring distance between Reeb graphs}}, doi = {10.1145/2582112.2582169}, year = {2014}, } @inproceedings{2177, abstract = {We give evidence for the difficulty of computing Betti numbers of simplicial complexes over a finite field. We do this by reducing the rank computation for sparse matrices with to non-zero entries to computing Betti numbers of simplicial complexes consisting of at most a constant times to simplices. Together with the known reduction in the other direction, this implies that the two problems have the same computational complexity.}, author = {Edelsbrunner, Herbert and Parsa, Salman}, booktitle = {Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms}, location = {Portland, USA}, pages = {152 -- 160}, publisher = {SIAM}, title = {{On the computational complexity of betti numbers reductions from matrix rank}}, doi = {10.1137/1.9781611973402.11}, year = {2014}, } @article{2184, abstract = {Given topological spaces X,Y, a fundamental problem of algebraic topology is understanding the structure of all continuous maps X→ Y. We consider a computational version, where X,Y are given as finite simplicial complexes, and the goal is to compute [X,Y], that is, all homotopy classes of suchmaps.We solve this problem in the stable range, where for some d ≥ 2, we have dim X ≤ 2d-2 and Y is (d-1)-connected; in particular, Y can be the d-dimensional sphere Sd. The algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplicial sets) with algorithmic tools from effective algebraic topology (locally effective simplicial sets and objects with effective homology). In contrast, [X,Y] is known to be uncomputable for general X,Y, since for X = S1 it includes a well known undecidable problem: testing triviality of the fundamental group of Y. In follow-up papers, the algorithm is shown to run in polynomial time for d fixed, and extended to other problems, such as the extension problem, where we are given a subspace A ⊂ X and a map A→ Y and ask whether it extends to a map X → Y, or computing the Z2-index-everything in the stable range. Outside the stable range, the extension problem is undecidable.}, author = {Čadek, Martin and Krcál, Marek and Matoušek, Jiří and Sergeraert, Francis and Vokřínek, Lukáš and Wagner, Uli}, journal = {Journal of the ACM}, number = {3}, publisher = {ACM}, title = {{Computing all maps into a sphere}}, doi = {10.1145/2597629}, volume = {61}, year = {2014}, } @article{1876, abstract = {We study densities of functionals over uniformly bounded triangulations of a Delaunay set of vertices, and prove that the minimum is attained for the Delaunay triangulation if this is the case for finite sets.}, author = {Dolbilin, Nikolai and Edelsbrunner, Herbert and Glazyrin, Alexey and Musin, Oleg}, issn = {1609-3321}, journal = {Moscow Mathematical Journal}, number = {3}, pages = {491 -- 504}, publisher = {Independent University of Moscow}, title = {{Functionals on triangulations of delaunay sets}}, doi = {10.17323/1609-4514-2014-14-3-491-504}, volume = {14}, year = {2014}, }