110 citations · 163 across the 22 of their papers we have counts for
18 papers · 1 filter
Counting Barcodes with the same Betti Curve
Henry Ashley, Håvard Bakke Bjerkevik, Justin Curry +2
This paper considers an important inverse problem in topological data analysis (TDA): How many different barcodes produce the same Betti curve? Equivalently, given a function $β\co…
Kan Approximations of the Persistent Homology Transform
Shreya Arya, Justin Curry
The persistent homology transform (PHT) of a subset is a map from the unit sphere to the space of persist…
Exploring the Stratified Space Structure of an RL Game with the Volume Growth Transform
Justin Curry, Brennan Lagasse, Ngoc B. Lam +3
In this work, we explore the structure of the embedding space of a transformer model trained for playing a particular reinforcement learning (RL) game. Specifically, we investigate…
Topologically Attributed Graphs for Shape Discrimination
Justin Curry, Washington Mio, Tom Needham +2
In this paper we introduce a novel family of attributed graphs for the purpose of shape discrimination. Our graphs typically arise from variations on the Mapper graph construction,…
Convergence of Leray Cosheaves for Decorated Mapper Graphs
Justin Curry, Washington Mio, Tom Needham +2
We introduce decorated mapper graphs as a generalization of mapper graphs capable of capturing more topological information of a data set. A decorated mapper graph can be viewed as…
A Sheaf-Theoretic Construction of Shape Space
Shreya Arya, Justin Curry, Sayan Mukherjee
We present a sheaf-theoretic construction of shape space -- the space of all shapes. We do this by describing a homotopy sheaf on the poset category of constructible sets, where ea…
A Lattice-Theoretic Perspective on the Persistence Map
Brendan Mallery, Adélie Garin, Justin Curry
We provide a naturally isomorphic description of the persistence map from merge trees to barcodes in terms of a monotone map from the partition lattice to the subset lattice. Our d…
From Trees to Barcodes and Back Again II: Combinatorial and Probabilistic Aspects of a Topological Inverse Problem
Justin Curry, Jordan DeSha, Adélie Garin +3
In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered con…
Decorated Merge Trees for Persistent Topology
Justin Curry, Haibin Hang, Washington Mio +2
This paper introduces decorated merge trees (DMTs) as a novel invariant for persistent spaces. DMTs combine both and information into a single data structure that disti…
Moduli Spaces of Morse Functions for Persistence
Michael J. Catanzaro, Justin Curry, Brittany Terese Fasy +5
We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classe…
When Left and Right Turns Inside Out: A Geometric and Categorical Introduction to an Inverse Problem in Persistence
Justin Curry
In this paper we introduce the problem of counting embedded spheres in R^3 whose projection to the z-axis yields a level set barcode of a particular type. Two embedded spheres are…
How Many Directions Determine a Shape and other Sufficiency Results for Two Topological Transforms
Justin Curry, Sayan Mukherjee, Katharine Turner
In this paper we consider two topological transforms that are popular in applied topology: the Persistent Homology Transform (PHT) and the Euler Characteristic Transform (ECT). Bot…
The Fiber of the Persistence Map for Functions on the Interval
Justin Curry
In this paper we study functions on the interval that have the same persistent homology. By introducing an equivalence relation modeled after topological conjugacy, which we call g…
Dualities between Cellular Sheaves and Cosheaves
Justin Curry
This paper affirms a conjecture of MacPherson: that the derived category of cellular sheaves is equivalent to the derived category of cellular cosheaves. We give a self-contained t…
Classification of Constructible Cosheaves
Justin Curry, Amit Patel
In this paper we prove an equivalence theorem originally observed by Robert MacPherson. On one side of the equivalence is the category of cosheaves that are constructible with resp…
Topological Data Analysis and Cosheaves
Justin Curry
This paper contains an expository account of persistent homology and its usefulness for topological data analysis. An alternative foundation for level-set persistence is presented…
Discrete Morse theory for computing cellular sheaf cohomology
Justin Curry, Robert Ghrist, Vidit Nanda
Sheaves and sheaf cohomology are powerful tools in computational topology, greatly generalizing persistent homology. We develop an algorithm for simplifying the computation of cell…
Sheaves, Cosheaves and Applications
Justin Curry
This thesis develops the theory of sheaves and cosheaves with an eye towards applications in science and engineering. To provide a theory that is computable, we focus on a combinat…