Robert Ghrist’s research while affiliated with William Penn University and other places

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Publications (157)


Categorical Diffusion of Weighted Lattices
  • Preprint
  • File available

January 2025

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9 Reads

Robert Ghrist

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Paige Randall North

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We introduce a categorical formalization of diffusion, motivated by data science and information dynamics. Central to our construction is the Lawvere Laplacian, an endofunctor on a product category indexed by a graph and enriched in a quantale. This framework permits the systematic study of diffusion processes on network sheaves taking values in a class of enriched categories.

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Figure 2. An origami serial chain. Spatial velocities, angular velocities, and rigid body operators are pictured (left) . The dual graph (right) is directed along the recursive equation (6). This graph is acyclic, indicating a open-chain system.
Figure 3. The degrees of freedom and constraints of the hinge model (left) and truss model (right) are displayed. The rotational velocities sum to zero around each vertex (left) and the length of the edge changes by zero (right). The data assignments of Figure 1 are divided into degrees of freedom and constraints.
Figure 7. A sketch of a cosheaf over adjacent cells. The stalks F c are vector spaces and the extension maps F d▷c are linear. There are stalks over other cells in the triangle not explicitly labeled.
Unified Origami Kinematics via Cosheaf Homology

January 2025

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10 Reads

We establish a novel local-global framework for analyzing rigid origami mechanics through cosheaf homology, proving the equivalence of truss and hinge constraint systems via an induced linear isomorphism. This approach applies to origami surfaces of various topologies, including sheets, spheres, and tori. By leveraging connecting homomorphisms from homological algebra, we link angular and spatial velocities in a novel way. Unlike traditional methods that simplify complex closed-chain systems to re-constrained tree topologies, our homological techniques enable simultaneous analysis of the entire system. This unified framework opens new avenues for homological algorithms and optimization strategies in robotic origami and beyond.


Tracking the topology of neural manifolds across populations

November 2024

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19 Reads

Proceedings of the National Academy of Sciences

Iris H. R. Yoon

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Chad Giusti

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 introduce a framework, called the method of analogous cycles, for matching topological features of neural manifolds using only observed dissimilarity matrices within and between neural populations. We demonstrate via analysis of simulations and in vivo experimental data that this method can be used to correctly identify multiple shared circular coordinate systems across both stimuli and inferred neural manifolds. Conversely, the method rejects matching features that are not intrinsic to one of the systems. Further, as this method is deterministic and does not rely on dimensionality reduction or optimization methods, it is amenable to direct mathematical investigation and interpretation in terms of the underlying neural activity. We thus propose the method of analogous cycles as a suitable foundation for a theory of cross-population analysis via neural manifolds.


Lattice-Valued Bottleneck Duality

September 2024

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13 Reads

This note reformulates certain classical combinatorial duality theorems in the context of order lattices. For source-target networks, we generalize bottleneck path-cut and flow-cut duality results to edges with capacities in a distributive lattice. For posets, we generalize a bottleneck version of Dilworth's theorem, again weighted in a distributive lattice. These results are applicable to a wide array of non-numerical network flow problems, as shown. All results, proofs, and applications were created in collaboration with AI language models. An appendix documents their role and impact.



Figure 1. The force cosheaf ℱ over the graph (left) has cosheaf boundary matrix of size 10 by 8 with nullity 1. The homology of ℱ characterizes the self-stress over edges and three degrees of freedom (two translational and one rotational of the entire structure). A local view of ℱ is highlighted (right), with the cosheaf matrices mapping edge stalks to vertex stalks "above" the graph.
Figure 2. Boundary conditions in the form of loading and reaction forces are modeled as edges connecting the truss to an external loop í µí±Œí µí±Œ.
Figure 4. The topological dual to the form diagram of Figure 1 is pictured (left), with its geometry realized (right). A local view of the position cosheaf í µí²¢í µí²¢ is highlighted (center), with cosheaf matrices mapping face stalks to edge stalks. A generator of í µí°»í µí°» 2 í µí²¢í µí²¢ provides the geometric coordinates to dual vertices on the right, characterizing a force diagram. This parallel dual realization may be derived from the self-stress of the form diagram via the graphic statics of Theorem 1.
Towards Homological Methods in Graphic Statics

July 2023

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149 Reads

Recent developments in applied algebraic topology can simplify and extend results in graphic statics - the analysis of equilibrium forces, dual diagrams, and more. The techniques introduced here are inspired by recent developments in cellular cosheaves and their homology. While the general theory has a few technical prerequisites (including homology and exact sequences), an elementary introduction based on little more than linear algebra is possible. A few classical results, such as Maxwell`s Rule and 2D graphic statics duality, are quickly derived from core ideas in algebraic topology. Contributions include: (1) a reformulation of statics and planar graphic statics in terms of cosheaves and their homology; (2) a new proof of Maxwell`s Rule in arbitrary dimensions using Euler characteristic; and (3) derivation of a novel relationship between mechanisms of the form diagram and obstructions to the generation of force diagrams. This last contribution presages deeper results beyond planar graphic statics.



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

April 2023

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88 Reads

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8 Citations

Journal of Applied and Computational Topology

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 algebra approaches for relating persistent homology classes and barcodes: persistent extension, which enumerates potential relations between homology classes from two complexes built on the same vertex set, and the method of analogous bars, which utilizes persistent extension and the witness complex built from a cross-dissimilarity measure to provide relations across systems. We provide an implementation of these methods and demonstrate their use in comparing homology classes between two samples from the same metric space and determining whether topology is maintained or destroyed under clustering and dimensionality reduction.




Citations (74)


... However, despite the success of these strategies for the resting-state connectome characterization, other high-order interaction mechanisms are still poorly studied (Battiston et al., 2020). More recently, alternative methods for exploring these interactions in the fMRI brain functional connectome have emerged, such as topological data analysis (TDA; Cassidy, Bowman, Rae, & Solo, 2018;Giusti, Ghrist, & Bassett, 2016;Lord et al., 2016;Preti, Bolton, & Ville, 2017). TDA encompasses methods aimed to characterize datasets using techniques from topology. ...

Reference:

𝓗1 persistent features of the resting-state connectome in healthy subjects
Two's company, three (or more) is a simplex: Algebraic-topological tools for understanding higher-order structure in neural data
  • Citing Preprint
  • January 2016

... Simply because certain data are better analyzed within a hyperbolic space (e.g., Google maps [50]) and often, there is no rationale to convert this representation into Euclidian space. For instance and without loss of generality, many recent applications in computer networking are represented in hyperbolic space, including : ...

Navigating the Negative Curvature of Google Maps

The Mathematical Intelligencer

... Dowker PH (Chowdhury and Mémoli 2018) is based on a Dowker complex (Dowker 1952), which is a simplicial complex that represents relations between two point clouds. Dowker complexes have been used to capture relations in molecular biology (Liu et al 2022), networks (Chowdhury and Mémoli 2018), PDF parsers (Ewing and Robinson 2021), and persistence diagrams (Yoon et al 2023). We propose using Dowker PH (Chowdhury and Mémoli 2018), a natural extension of Dowker complexes, for multispecies data. ...

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

Journal of Applied and Computational Topology

... The present work also connects to a recent model of multidimensional opinions dynamics [29], more broadly construed as "social information dynamics" [30], [31] drawing on the recently introduced theory of sheaf Laplacians [32]. Our decentralized method for arriving at (approximate) equilibrium is equivalent to max consensus [33] in the onedimensional case (d = 1). ...

Network SheafModels for Social Information Dynamics
  • Citing Conference Paper
  • December 2022

... To answer this question, the concept of sheaves provides a novel lens through which the interactions between clients in a decentralized FMTL setting can be modeled. The mathematical notion of a sheaf initially invented and developed in algebraic topology, is a framework that systematically organizes local observations in a way that allows one to make conclusions about the global consistency of such observations (Robinson, 2014;2013;Riess & Ghrist, 2022). As such requirements are a central part of FL problems, it is natural to ask how one could utilize a sheaf-based framework to find an effective solution to the above question. ...

Diffusion of Information on Networked Lattices by Gossip
  • Citing Conference Paper
  • December 2022

... Extending from vector-valued data to more general data categories poses significant challenges, especially when the data category is not abelian [21]. The introduction of the Tarski Laplacian in 2022 presented an operator on cellular sheaf cochains that take values in the category of complete order lattices with Galois connections as morphisms [19,32]. This Tarski Laplacian was shown to enact discrete-time diffusion: cochains converge to harmonic cochains over time, thanks to the Tarski Fixed Point Theorem [42]. ...

Cellular sheaves of lattices and the Tarski Laplacian
  • Citing Article
  • January 2022

Tbilisi Mathematical Journal

... In network science, sheaves provide enhanced descriptions of network structures, capturing the nature of relationships between nodes. Within this framework, opinion dynamics uses discourse sheaves to model how opinions evolve and interact within social networks [28]. Additionally, Bodnar proposed neural sheaf diffusion to learn sheaf laplacians from lower-order data, providing a novel topological perspective on heterophily and oversmoothing in GNNs [29]- [31]. ...

Opinion Dynamics on Discourse Sheaves
  • Citing Article
  • September 2021

SIAM Journal on Applied Mathematics

... Gray-scale images could be utilized for creating binary ones. The most straight forward approaches of Thresholding are replacing each one of the image pixels with black pixels in the case when the intensity of image (f) is not more than a certain fixed constant threshold (T), or white pixel in the case when the intensity of the image is more than that threshold [27]. ...

An optimal property of the hyperplane system in a finite cubing

Autonomous Robots

... Additionally, sheaves are a mathematical framework that allows one to represent not only the connections between nodes (as in the networks above), but also the specific relationship, such as linear maps, between node activities or properties that these connections represent. Thus far sheaves have been used for network coding problems (Ghrist & Hiraoka, 2011), signal processing (Robinson, 2013), and other applications (Phan-Luong, 2008), but they have great potential for use in biological research Hansen & Ghrist, 2020). Though certainly not complete, we offer the above examples to show the seemingly limitless possibilities of mathematical system representations (Battiston et al., 2020;Torres et al., 2020). ...

Opinion Dynamics on Discourse Sheaves
  • Citing Preprint
  • May 2020