1,189 research outputs found

    Three-dimensional topology-based analysis segments volumetric and spatiotemporal fluorescence microscopy

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    Image analysis techniques provide objective and reproducible statistics for interpreting microscopy data. At higher dimensions, three-dimensional (3D) volumetric and spatiotemporal data highlight additional properties and behaviors beyond the static 2D focal plane. However, increased dimensionality carries increased complexity, and existing techniques for general segmentation of 3D data are either primitive, or highly specialized to specific biological structures. Borrowing from the principles of 2D topological data analysis (TDA), we formulate a 3D segmentation algorithm that implements persistent homology to identify variations in image intensity. From this, we derive two separate variants applicable to spatial and spatiotemporal data, respectively. We demonstrate that this analysis yields both sensitive and specific results on simulated data and can distinguish prominent biological structures in fluorescence microscopy images, regardless of their shape. Furthermore, we highlight the efficacy of temporal TDA in tracking cell lineage and the frequency of cell and organelle replication

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Notes on Factorization Algebras and TQFTs

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    These are notes from talks given at a spring school on topological quantum field theory in Nova Scotia during May of 2023. The aim is to introduce the reader to the role of factorization algebras and related concepts in field theory. In particular, we discuss the relationship between factorization algebras, En\mathbb{E}_n-algebras, vertex algebras, and the functorial perspective on field theories.Comment: 42 pages. Comments welcome

    Torsion volume forms

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    We introduce volume forms on mapping stacks in derived algebraic geometry using a parametrized version of the Reidemeister-Turaev torsion. In the case of derived loop stacks we describe this volume form in terms of the Todd class. In the case of mapping stacks from surfaces, we compare it to the symplectic volume form. As an application of these ideas, we construct canonical orientation data for cohomological DT invariants of closed oriented 3-manifolds.Comment: 58 page

    Spherically-symmetric geometries in a matter reference frame as quantum gravity condensates

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    Candidate microstates of a spherically symmetric geometry are constructed in the group field theory formalism for quantum gravity, for models including both quantum geometric and scalar matter degrees of freedom. The latter are used as a material reference frame to define the spacetime localization of the various elements of quantum geometry. By computing quantum geometric observables, we then match the quantum states with a spherically symmetric classical geometry, written in a suitable matter reference frame.Comment: 27 pages, 1 figur

    A unified framework for Simplicial Kuramoto models

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    Simplicial Kuramoto models have emerged as a diverse and intriguing class of models describing oscillators on simplices rather than nodes. In this paper, we present a unified framework to describe different variants of these models, categorized into three main groups: "simple" models, "Hodge-coupled" models, and "order-coupled" (Dirac) models. Our framework is based on topology, discrete differential geometry as well as gradient flows and frustrations, and permits a systematic analysis of their properties. We establish an equivalence between the simple simplicial Kuramoto model and the standard Kuramoto model on pairwise networks under the condition of manifoldness of the simplicial complex. Then, starting from simple models, we describe the notion of simplicial synchronization and derive bounds on the coupling strength necessary or sufficient for achieving it. For some variants, we generalize these results and provide new ones, such as the controllability of equilibrium solutions. Finally, we explore a potential application in the reconstruction of brain functional connectivity from structural connectomes and find that simple edge-based Kuramoto models perform competitively or even outperform complex extensions of node-based models.Comment: 36 pages, 11 figure

    An exterior calculus framework for polytopal methods

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    We develop in this work the first polytopal complexes of differential forms. These complexes, inspired by the Discrete De Rham and the Virtual Element approaches, are discrete versions of the de Rham complex of differential forms built on meshes made of general polytopal elements. Both constructions benefit from the high-level approach of polytopal methods, which leads, on certain meshes, to leaner constructions than the finite element method. We establish commutation properties between the interpolators and the discrete and continuous exterior derivatives, prove key polynomial consistency results for the complexes, and show that their cohomologies are isomorphic to the cohomology of the continuous de Rham complex

    Generalized Heawood Graphs and Triangulations of Tori

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    The Heawood graph is a remarkable graph that played a fundamental role in the development of the theory of graph colorings on surfaces in the 19th and 20th centuries. Based on permutahedral tilings, we introduce a generalization of the classical Heawood graph indexed by a sequence of positive integers. We show that the resulting generalized Heawood graphs are toroidal graphs, which are dual to higher dimensional triangulated tori. We also present explicit combinatorial formulas for their ff-vectors and study their automorphism groups.Comment: 40 pages, 22 figure

    Euler Characteristic Tools For Topological Data Analysis

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    In this article, we study Euler characteristic techniques in topological data analysis. Pointwise computing the Euler characteristic of a family of simplicial complexes built from data gives rise to the so-called Euler characteristic profile. We show that this simple descriptor achieve state-of-the-art performance in supervised tasks at a very low computational cost. Inspired by signal analysis, we compute hybrid transforms of Euler characteristic profiles. These integral transforms mix Euler characteristic techniques with Lebesgue integration to provide highly efficient compressors of topological signals. As a consequence, they show remarkable performances in unsupervised settings. On the qualitative side, we provide numerous heuristics on the topological and geometric information captured by Euler profiles and their hybrid transforms. Finally, we prove stability results for these descriptors as well as asymptotic guarantees in random settings.Comment: 39 page

    Applications of matter reference frames in quantum gravity

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