Robert Ghrist

Cellular sheaves of lattices and the Tarski Laplacian

Homology Homotopy and Applications · 2022 · 15 citations

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 cohomology that agrees with the global section functor in degree zero. This has immediate applications in consensus and distributed optimization problems over networks and broader potential applications.

Opinion Dynamics on Discourse Sheaves

arXiv (Cornell University) · 2021 · 11 citations

We introduce a novel class of Laplacians and diffusion dynamics on discourse sheaves as a model for network dynamics, with application to opinion dynamics on social networks. These sheaves are algebraic data structures tethered to a network (or more general space) that can represent various modes of communication, including selective opinion modulation and lying. After introducing the sheaf model, we develop a sheaf Laplacian in this context and show how to evolve both opinions and communication…

Persistent extensions and analogous bars: data-induced relations between persistence barcodes

Journal of Applied and Computational Topology · 2023-04-15 · 9 citations

Abstract A central challenge in topological data analysis is the interpretation of barcodes. The classical algebraic-topological approach to interpreting homology classes is to build maps to spaces whose homology carries semantics we understand and then to appeal to functoriality. However, we often lack such maps in real data; instead, we must rely on a cross-dissimilarity measure between our observations of a system and a reference. In this paper, we develop a pair of computational homological…

Diffusion of Information on Networked Lattices by Gossip

2022 IEEE 61st Conference on Decision and Control (CDC) · 2022-12-06 · 7 citations

We study time-dependent dynamics on a network of order lattices, where structure-preserving lattice maps are used to fuse lattice-valued data over vertices and edges. The principal contribution is a novel asynchronous Laplacian, generalizing the usual graph Laplacian, adapted to a network of heterogeneous lattices. This resulting gossip algorithm is shown to converge asymptotically to stable "harmonic" distributions of lattice data. This general theorem is applicable to several general problems,…

Tracking the topology of neural manifolds across populations

Proceedings of the National Academy of Sciences · 2024-11-08 · 3 citations

Neural manifolds summarize the intrinsic structure of the information encoded by a population of neurons. Advances in experimental techniques have made simultaneous recordings from multiple brain regions increasingly commonplace, raising the possibility of studying how these manifolds relate across populations. However, when the manifolds are nonlinear and possibly code for multiple unknown variables, it is challenging to extract robust and falsifiable information about their relationships. We i…