1,189 research outputs found
Three-dimensional topology-based analysis segments volumetric and spatiotemporal fluorescence microscopy
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
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Notes on Factorization Algebras and TQFTs
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, -algebras, vertex algebras, and the functorial
perspective on field theories.Comment: 42 pages. Comments welcome
Torsion volume forms
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
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
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
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
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 -vectors and study their
automorphism groups.Comment: 40 pages, 22 figure
Euler Characteristic Tools For Topological Data Analysis
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
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