TY - JOUR AB - 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. AU - Bauer, Ulrich AU - Edelsbrunner, Herbert ID - 1072 IS - 5 JF - Transactions of the American Mathematical Society TI - The Morse theory of Čech and delaunay complexes VL - 369 ER - TY - JOUR AB - 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. AU - Chatterjee, Krishnendu AU - Osang, Georg F ID - 1065 JF - Information Processing Letters SN - 00200190 TI - Pushdown reachability with constant treewidth VL - 122 ER - TY - JOUR AB - 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. AU - Pranav, Pratyush AU - Edelsbrunner, Herbert AU - Van De Weygaert, Rien AU - Vegter, Gert AU - Kerber, Michael AU - Jones, Bernard AU - Wintraecken, Mathijs ID - 1022 IS - 4 JF - Monthly Notices of the Royal Astronomical Society SN - 00358711 TI - The topology of the cosmic web in terms of persistent Betti numbers VL - 465 ER - TY - JOUR AB - 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. AU - Virk, Ziga AU - Zastrow, Andreas ID - 737 JF - Topology and its Applications SN - 01668641 TI - A new topology on the universal path space VL - 231 ER - TY - CONF AB - 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. AU - Ethier, Marc AU - Jablonski, Grzegorz AU - Mrozek, Marian ID - 836 SN - 978-331956930-7 T2 - Special Sessions in Applications of Computer Algebra TI - Finding eigenvalues of self-maps with the Kronecker canonical form VL - 198 ER -