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Sheaves, Cosheaves and Applications
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 combinatorial version of
How Many Directions Determine a Shape and other Sufficiency Results for Two Topological Transforms
In this paper we consider two topological transforms based on Euler calculus: the persistent homology transform (PHT) and the Euler characteristic transform (ECT). Both of these transforms are of
The fiber of the persistence map for functions on the interval
  • J. Curry
  • Mathematics
    J. Appl. Comput. Topol.
  • 19 June 2017
TLDR
Enumeration of merge trees and chiral merge trees with the same persistence makes essential use of the Elder Rule, which is given its first detailed proof in this paper.
Euler Calculus with Applications to Signals and Sensing
This article surveys the Euler calculus - an integral calculus based on Euler characteristic - and its applications to data, sensing, networks, and imaging.
Topological data analysis and cosheaves
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 using sheaves and
The Fiber of the Persistence Map
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 graph-equivalence, a
Classification of Constructible Cosheaves
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 respect to a locally
Decorated merge trees for persistent topology
TLDR
Computational frameworks for generating, visualizing and comparing decorated merge trees derived from synthetic and real data, and a novel use of Gromov-Wasserstein couplings to compute optimal merge tree alignments for a combinatorial version of the interleaving distance, are introduced.
Discrete Morse Theory for Computing Cellular Sheaf Cohomology
TLDR
An algorithm for simplifying the computation of cellular sheaf cohomology via (discrete) Morse theoretic techniques is developed, derive efficient techniques for distributed computation of (ordinary) cohmology of a cell complex.
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