Moduli spaces of morse functions for persistence

Catanzaro, Michael J.; Curry, Justin; Fasy, Brittany Terese; Lazovskis, Jānis; Malen, Greg; Riess, Hans; Wang, Bei; Zabka, Matthew

doi:10.1007/s41468-020-00055-x

Cited by 19 publications

(9 citation statements)

References 45 publications

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“…Finally, in §4. 4, we attempt to build a Hodge Laplacian and cohomology theory directly from a (pseudo) cochain complex, comparing and contrasting with the Tarski theory.…”

Section: Contributionsmentioning

confidence: 99%

Cellular sheaves of lattices and the Tarski Laplacian

Riess

2022

Homology, Homotopy and Applications

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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.

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“…Finally, in §4. 4, we attempt to build a Hodge Laplacian and cohomology theory directly from a (pseudo) cochain complex, comparing and contrasting with the Tarski theory.…”

Section: Contributionsmentioning

confidence: 99%

Cellular sheaves of lattices and the Tarski Laplacian

Riess

2022

Homology, Homotopy and Applications

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“…4(b–c), where a merge tree decomposes into a barcode following a branch decomposition process; and bars in a barcode can be used to assemble a (non‐unique) merge tree following a gluing process. See [CCF∗20, Cur18, KGH20] for references for the relation between a merge tree and a barcode. Note that the notions of join and split trees [CSA03] are the two forms of merge trees; a join tree is the merge tree of f and a split tree is the merge tree of – f .…”

Section: Technical Foundations On Scalar Field Topologymentioning

confidence: 99%

Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization

Yan

Masood

Sridharamurthy

et al. 2021

Computer Graphics Forum

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In topological data analysis and visualization, topological descriptors such as persistence diagrams, merge trees, contour trees, Reeb graphs, and Morse–Smale complexes play an essential role in capturing the shape of scalar field data. We present a state‐of‐the‐art report on scalar field comparison using topological descriptors. We provide a taxonomy of existing approaches based on visualization tasks associated with three categories of data: single fields, time‐varying fields, and ensembles. These tasks include symmetry detection, periodicity detection, key event/feature detection, feature tracking, clustering, and structure statistics. Our main contributions include the formulation of a set of desirable mathematical and computational properties of comparative measures, and the classification of visualization tasks and applications that are enabled by these measures.

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“…For the unit interval it is possible to count the number of path connected components in PH −1 (D) by means of the combinatorics of the barcode (Curry 2018). For higher dimensional X analyzing the fiber is a challenging problem: already for Morse functions on the 2-sphere X = S 2 new tools have been designed to describe the fiber PH −1 (D), and allowed for conjectures on the number of path connected components (Catanzaro et al 2020). However the higher dimensional homotopy groups of PH −1 (D) remain unknown, from which stems the motivation of this work.…”

Section: Introductionmentioning

confidence: 99%

Fiber of persistent homology on morse functions

Beers

Leygonie

2022

J Appl. and Comput. Topology

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Let f be a Morse function on a smooth compact manifold $$M$$ M with boundary. The path component $$\mathrm {PH}^{-1}_f(D)$$ PH f - 1 ( D ) containing f of the space of Morse functions giving rise to the same Persistent Homology $$D=\mathrm {PH}(f)$$ D = PH ( f ) is shown to be the same as the orbit of f under pre-composition $$\phi \mapsto f\circ \phi $$ ϕ ↦ f ∘ ϕ by diffeomorphisms of $$M$$ M which are isotopic to the identity. Consequently we derive topological properties of the fiber $$\mathrm {PH}^{-1}_f(D)$$ PH f - 1 ( D ) : In particular we compute its homotopy type for many compact surfaces $$M$$ M . In the 1-dimensional settings where $$M$$ M is the unit interval or the circle we extend the analysis to continuous functions and show that the fibers are made of contractible and circular components respectively.

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Moduli spaces of morse functions for persistence

Cited by 19 publications

References 45 publications

Cellular sheaves of lattices and the Tarski Laplacian

Cellular sheaves of lattices and the Tarski Laplacian

Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization

Fiber of persistent homology on morse functions

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