Our research

The applied topology group at Oregon State University employs topology to investigate a variety of complex systems. Our work includes:

A filtered simplicial complex growing from a few triangles, with a blue-shaded square marking a new face

Methods in applied topology

We develop new mathematical frameworks and computational tools for the study of complex systems through the modern language of shape. We are particularly interested in methods that will allow us to address fundamental scientific questions, and in understanding how and why topological structure arises so often in the natural world.

A simplicial complex with a blue-shaded face drawn inside a thick looping curve that suggests a neural manifold

Topological neuroscience

In collaboration with experimental neuroscientists, we study the structure and function of in vivo and simulated neural systems with the goal of developing fundamental theory about the encoding of information and performance of computations in the brain.

However, we are always interested in Other Applications across a range of disciplines.

The Applied Topology Group is grateful to the National Science Foundation, the Air Force Office of Scientific Research, and the Air Force Research Laboratory for their prior support of our work, and to Oregon State University for their ongoing support for our work.

Methods in applied topology

Modern techniques in applied topology are evolving rapidly, and there are many open questions. We are interested in how measures of topology reflect the organization of systems, and why such structures arise in nature. To study these questions, we develop tools for interpreting topological measures in situ.

Our recent work broadly fits roughly into two categories:

1. Understanding How Topology Measures Structure in Complex Systems:

Due to computational limitations and underdeveloped mathematics, early applications of persistent homology have largely focused on statistics of persistence diagrams. Interpretation in terms of the original systems from which they were computed is usually ad hoc and prone to subtle errors. In our view, these difficulties have inhibited wide-spread application of topological methods in the sciences. One of our primary aims is thus developing rigorous, formal workflows that connect quantitative topological measures and the structure of the system they represent. Our recent work includes the Open Applied Topology software package, which provides full access to the algebraic structures underlying persistence; a user's guide for optimizing persistent cycle representatives; and, the method of analogous cycles, which matches persistent homology classes across systems using observed cross-dissimilarities.

Dowker's theorem for higher-order relations
Vin de Silva, Chad Giusti, Vladimir Itskov, Michael Robinson, Radmila Sazdanovic, Nikolas Schonsheck, Melvin Vaupel, and Iris Yoon

Journal of Applied and Computational Topology 10 (2), 2026

paper arXiv

Three polyhedral complexes side by side, each a quotient of a larger complex: a wireframe with black and grey edges, a polyhedron with green-shaded square faces, and a second wireframe.

Persistent extensions and analogous bars: data-induced relations between persistence barcodes
Hee Rhang Yoon, Robert Ghrist & Chad Giusti

Journal of Applied and Computational Topology 7 (3), 2023

paper code

One persistence barcode with its bars split by color into long bars (blue), short bars (purple) and a single distinguished bar (green) at a dashed threshold line, with an inset point cloud showing two cycle representatives.

Open Applied Topology
Based on the ExHACT project by Gregory Henselman-Petrusek, Lori Ziegelmeier, Chad Giusti

primarily supported and maintained by Pacific Northwest National Laboratory

code

A spherical point cloud rendered as a wireframe with highlighted sample points, as computed by Open Applied Topology.

Minimal Cycle Representatives in Persistent Homology Using Linear Programming: An Empirical Study With User's Guide
Lu Li, Connor Thompson, Gregory Henselman-Petrusek, Chad Giusti, Lori Ziegelmeier

Frontiers in Artificial Intelligence 4, 2021

paper code

Five panels: A to D show one planar simplicial complex with a square hole and four different dashed cycle representatives of the same class; E shows a three-step filtration in which a cycle representative grows and splits.

Eulerian Magnitude Homology: Subgraph Structure and Random Graphs
Chad Giusti and Giuliamaria Menara

arXiv:2403.09248, 2024

paper

Plots of Eulerian magnitude homology ranks against graph parameters for random graphs.

2. Developing Iterated Integrals as a Feature Set for Function Spaces

Persistent homology characterizes structure in filtered spaces. However, some data are better described in terms of functions into a state space. Function spaces are a topic of classical study in algebraic topology, and Chen's iterated integral framework for cochains on mapping spaces, recently recast as the Path Signature in the study of SDEs and ML, provides a well-founded and robust framework for studying such data. We have applied this framework to the study of time-varying persistence diagrams; and, to describe the algebraic properties of the signature transform for higher-dimensional domains.

A topological approach to mapping space signatures
Chad Giusti, Darrick Lee, Vidit Nanda, Harald Oberhauser

Advances in Applied Mathematics 163, 2025

paper

Four squares with vertices drawn solid or half-filled and some edges highlighted in magenta, labelled s2 s1(00), s1(e0), s2(e0) and g1(e0), with coordinate axes 1 and 2 at the left; terms of the cubical mapping space signature.

Signatures, Lipschitz-Free Spaces, and Paths of Persistence Diagrams
Chad Giusti and Darrick Lee

SIAM Journal on Applied Algebra and Geometry 7 (4), 2023

paper code

Three snapshots of a simulated swarm at t=3, t=75 and t=175, drawn as red points with teal motion trails, going from a tangle to a ring to parallel streams.

Iterated Integrals and Population Time Series Analysis
Chad Giusti and Darrick Lee

Topological Data Analysis (Abel Symposia 15), Springer, 2020

paper arXiv

Two pairs of periodic time series, smooth sinusoids above and a sharply reparametrized version below, sent by a correlation function to a delay plot that distinguishes them and by the second-order iterated integral to a closed loop in the plane that gives the same value, 11.6, for both.

Topological neuroscience

Topology is the mathematics of "local-to-global" inference. That is, topological tools are designed to sew together local observations to describe nonlinear global phenomena. In this light, it is perhaps less surprising that they are uniquely well-suited for describing systems like the brain, in which the simple computations performed by individual neurons are synthesized into our rich internal representations of the state of the world, and their dynamics give rise to planning and execution of intricate behaviors to interact with it. The introduction of modern topological methods to neuroscience is quite recent, and there are more questions than answers. For a brief overview of some of the big ideas we recommend the following surveys, some of which are written for non-mathematicians.

Two's Company, Three's a Simplex
Chad Giusti, Robert Ghrist & Danielle S. Bassett

Journal of Computational Neuroscience 41 (1), 2016

paper

Eight small panels showing how a brain network is built and analysed: brain regions and measures of connectivity, activity traces, an edge-only triangle beside a filled triangle, a similarity matrix binarised at a sequence of thresholds, and a plot of component count against threshold.

Topological Neuroscience: Linking Circuits to Function
Carina Curto and Nicole Sanderson

paper

Grid cell firing fields of two modules with their hexagonal tilings, each tiling identified first with a cylinder and then with a torus, and a trajectory drawn in blue across the fields, the cylinder and the torus.

What can topology tell us about the neural code?
Carina Curto

paper

Five overlapping bell-shaped tuning curves over an orientation axis with the curve centered at 45 degrees drawn bold, two oriented bars, and a neuron whose spike train is dense for one orientation and sparse for the other.

Importance of the whole: Topological data analysis for the network neuroscientist
Ann E. Sizemore, Jennifer E. Phillips-Cremins, Robert Ghrist, Danielle S. Bassett

paper

Two pale brain outlines: the left with a single curved arrow across it, the right with an orange and a grey arrow curving in from opposite directions.

Topological Adventures in Neuroscience
Kathryn Hess

paper

A hands-on tutorial on network and topological neuroscience
Eduarda Gervini Zampieri Centeno, Giulia Moreni, Chris Vriend, Linda Douw & Fernando Antônio Nóbrega Santos

paper

A translucent side view of a brain containing a network of nodes and edges, the nodes sized and colored from purple (low) to yellow (high) on a vertical color bar.

Our work

Our recent work has been focused on implications for neural computation of the classical nerve theorem of algebraic topology. In context, it effectively tells us that we can use observations of neural activity to reconstruct the topology (and some geometry) of the information encoded by a neural population. This purely mathematical fact is the reason for the appearance of so-called neural manifolds, low-dimensional subspaces of the space of population neuron activation profiles, along which activity is constrained.

In his postdoctoral work, the PI demonstrated that topological measurements of neural population activity alone were sufficient to detect or falsify the presence of a neural manifold, without reference to external stimulus or behavior. In the interim, we have been interested in understanding how these objects arise as a result of neural activity, and developing a mathematical framework for understanding the resulting computational model at the level of neural manifolds, rather than individual or collective neuronal activity. In recent work, we have constructed a method for comparing topological features across neural manifolds, or between neural manifolds and observed stimuli and behaviors; and described combinatorial foundations of convex coding, which underlies such receptive field models of neural activity.

Tracking the topology of neural manifolds across populations
Iris H.R. Yoon, Gregory Henselman-Petrusek, Yiyi Yu, Robert Ghrist, Spencer LaVere Smith, and Chad Giusti

PNAS 121 (46), 2024

paper code

Schematic of the method of analogous cycles matching a circular feature across two neural manifolds.

A No-Go Theorem for One-Layer Feedforward Networks
Chad Giusti and Vladimir Itskov

Neural Computation 26 (11), 2014

paper arXiv

A one-layer feedforward network: three input circles in an upper row, labelled y_a, connected by arrows with weights U_ia to four output circles in a lower row, labelled x_i.

On Open and Closed Convex Codes
Joshua Cruz, Chad Giusti, Vladimir Itskov, and Bill Kronholm

Discrete & Computational Geometry 61 (2), 2019

paper arXiv

Six colored regions labelled U1 to U6 arranged around a square: on the left each is an open half-disc with a dashed boundary, on the right each is a closed arc, two realizations of the same code.

Other applications

The tools we develop are not specific to neuroscience. Working with collaborators in physics, cognitive science, geography, and cell biology, we have used topological and geometric measures to study force chains in granular materials under compression, the growth of semantic networks as children learn words, maps and networks for disaster preparation and response, single-cell proteomic profiles of differentiating neurons, and the classification of networks through the lens of homology.

Leveraging Topological Data Analysis for disaster preparation and response, a cartographic perspective
Connor Progin, Carolyn Fish, Chad Giusti, David Retchless, Courtney Thatcher, and Jim Thatcher

Abstracts of the International Cartographic Association 10, 2025

paper

A road network drawn in grey on a black background, with one route highlighted as a thick teal path and a cluster of green nodes marking points along the roads to the right.

Single-Cell Proteomics Reveals Proteome Remodeling and Cellular Heterogeneity During NGF-Induced PC12 Neuronal Differentiation
Arpa Ebrahimi, Shuxin Chi, Jun Yang, Phoebe Y. Lee, Prongbaramee K. Colling, Hailey Mendenhall, Chad Giusti, David A. Hendrix, Leonard J. Foster, and Claudia S. Maier

ACS Chemical Neuroscience 17 (15), 2026

paper

A horizontal strip through a network of pathway terms drawn as colored circles joined by purple edges: brown, light green, salmon, blue and grey clusters linked by long edges, a dense dark blue cluster at the right edge, and red nodes entering from below.