Related papers: Cellular Sheaves on Higher-Dimensional Structures

Sheaf cohomology or, more generally, higher direct images of coherent sheaves along proper morphisms are central to modern algebraic geometry. However, the computation of these objects is a non-trivial and expensive task which easily…

Higher order networks are able to characterize data as different as functional brain networks, protein interaction networks and social networks beyond the framework of pairwise interactions. Most notably higher order networks include…

In many tissues, cell type varies over single-cell length-scales, creating detailed spatial heterogeneities fundamental to physiological function. To gain understanding of this relationship between tissue function and detailed structure,…

We initiate the study of sheaves on Cech closure spaces, providing a new, unified approach to sheaf theory on many of the major classes of spaces of interest to applications: topological spaces, finite simplicial complexes (seen as $T_0$…

Recently various types of topological Laplacians have been studied from the perspective of data analysis. The spectral theory of these Laplacians has significantly extended the scope of algebraic topology and data analysis. Inspired by the…

We introduce and develop the theory of metric sheaves. A metric sheaf $\A$ is defined on a topological space $X$ such that each fiber is a metric model. We describe the construction of the generic model as the quotient space of the sheaf…

Sheaf Neural Networks (SNNs) represent a powerful generalization of Graph Neural Networks (GNNs) that significantly improve our ability to model complex relational data. While directionality has been shown to substantially boost performance…

This paper introduces cellular sheaf theory to graphical methods and reciprocal constructions in structural engineering. The elementary mechanics and statics of trusses are derived from the linear algebra of sheaves and cosheaves. Further,…

This document develops general concepts useful for extracting knowledge embedded in large graphs or datasets that have pair-wise relationships, such as cause-effect-type relations. Almost no underlying assumptions are made, other than that…

Heterogeneous graphs, whose nodes and edges can belong to different types and feature spaces, arise in many real-world domains, including biology, recommendation, social networks, and computer systems. Existing heterogeneous graph neural…

Cell complexes are topological spaces constructed from simple blocks called cells. They generalize graphs, simplicial complexes, and polyhedral complexes that form important domains for practical applications. They also provide a…

We introduce a sheaf-theoretic characterization of task solvability in general distributed computing models, unifying distinct approaches to message-passing models. We establish cellular sheaves as a natural mathematical framework for…

Equipping graph neural networks with a convolution operation defined in terms of a cellular sheaf offers advantages for learning expressive representations of heterophilic graph data. The most flexible approach to constructing the sheaf is…

Membrane particles such as proteins and lipids organize into zones that perform unique functions. Here, I introduce a topological and category-theoretic framework to represent particle and zone intra-scale interactions and inter-scale…

In this paper, we try to realize the unbounded derived category of an abelian category as the homotopy category of a Quillen model structure on the category of unbounded chain complexes. We construct such a model structure based on…

We provide generalizations of the notions of Atiyah class and Kodaira-Spencer map to the case of framed sheaves. Moreover, we construct closed two-forms on the moduli spaces of framed sheaves on surfaces. As an application, we define a…

A Sheaf Neural Network (SNN) is a type of Graph Neural Network (GNN) that operates on a sheaf, an object that equips a graph with vector spaces over its nodes and edges and linear maps between these spaces. SNNs have been shown to have…

Shape graphs are complex geometrical structures commonly found in biological and anatomical systems. A shape graph is a collection of nodes, some connected by curvilinear edges with arbitrary shapes. Their high complexity stems from the…

Advances in imaging methods such as electron microscopy, tomography and other modalities are enabling high-resolution reconstructions of cellular and organelle geometries. Such advances pave the way for using these geometries for…

This paper initiates a discrete Hodge theory for cellular sheaves taking values in a category of lattices and Galois connections. The key development is the Tarski Laplacian, an endomorphism on the cochain complex whose fixed points yield a…