Related papers: Chromatic Alpha Complexes
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently…
We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent…
Topological complexity is a numerical homotopy invariant that measures the instability of motion planning in a space. To study the topological complexity of non-simply connected spaces, Costa and Farber introduced a cohomology class whose…
We investigate the existence of metric spaces which, for any coloring with a fixed number of colors, contain monochromatic isomorphic copies of a fixed starting space K. In the main theorem we construct such a space of size \(2^{\aleph_0}\)…
By coloring a signed graph by signed colors, one obtains the signed chromatic polynomial of the signed graph. For each signed graph we construct graded cohomology groups whose graded Euler characteristic yields the signed chromatic…
We study several different notions of algebraicity in use in stable homotopy theory and prove implications between them. The relationships between the different meanings of algebraic are unexpectedly subtle, and we illustrate this with…
We approach string phenomenology from the perspective of computational algebraic geometry, by providing new and efficient techniques for proving stability and calculating particle spectra in heterotic compactifications. This is done in the…
We continue our investigation of spaces of long embeddings (long embeddings are high-dimensional analogues of long knots). In previous work we showed that when the dimensions are in the stable range, the rational homology groups of these…
Given a compact metric space $X$, we associate to it an inverse sequence of finite $T_0$ topological spaces. The inverse limit of this inverse sequence contains a homeomorphic copy of $X$ that is a strong deformation retract. We provide a…
We suggest a diagrammatic model of computation based on an axiom of distributivity. A diagram of a decorated coloured tangle, similar to those that appear in low dimensional topology, plays the role of a circuit diagram. Equivalent diagrams…
We derive the relationship between the persistent homology barcodes of two dual filtered CW complexes. Applied to greyscale digital images, we obtain an algorithm to convert barcodes between the two different (dual) topological models of…
Motivated by the work of Cappell, Deturck, Gluch and Miller, we extend the notion of cohomology of harmonic forms (of a compact manifold with boundary) to the abstract setting of Hilbert complexes. Then, we present some geometric…
Tumor segmentation in whole-slide images of histology slides is an important step towards computer-assisted diagnosis. In this work, we propose a tumor segmentation framework based on the novel concept of persistent homology profiles…
Crystalline symmetries give rise to topological invariants that can distinguish quantum phases of matter. Understanding these in strongly interacting systems is an ongoing research direction requiring non-perturbative methods. Recent…
The notion of a coherent space is a nonlinear version of the notion of a complex Euclidean space: The vector space axioms are dropped while the notion of inner product is kept. Coherent spaces provide a setting for the study of geometry in…
This work introduces a number of algebraic topology approaches, such as multicomponent persistent homology, multi-level persistent homology and electrostatic persistence for the representation, characterization, and description of small…
In this survey, we explore recent literature on finding the cores of higher graphs using geometric and topological means. We study graphs, hypergraphs, and simplicial complexes, all of which are models of higher graphs. We study the notion…
We examine the topological characteristic cohomology classes of complexified vector bundles. In particular, all the classes coming from the real vector bundles underlying the complexification are determined.
We obtain improved upper bounds and new lower bounds on the chromatic number as a linear function of the clique number, for the intersection graphs (and their complements) of finite families of translates and homothets of a convex body in…
We compute the Balmer spectrum of the category of perfect complexes on an algebraic stack admitting a finite locally free cover by an affine scheme and identify it with the homogeneous spectrum of the cohomology ring.