Persistent homology and Floer-Novikov theory
By Michael Usher, Jun Zhang. 2015
115 citations recorded by the source corpus.
We construct "barcodes" for the chain complexes over Novikov rings that arise in Novikov's Morse theory for closed one-forms and in Floer theory on not-necessarily-monotone symplectic manifolds. In the case of classical Morse theory these coincide with the barcodes familiar from persistent homology. Our barcodes completely characterize the filtered chain homotopy type of the chain complex; in particular they subsume in a natural way previous filtered Floer-theoretic invariants such as boundary depth and torsion exponents, and also reflect information about spectral invariants. We moreover prove a continuity result which is a natural analogue both of the classical bottleneck stability theorem in persistent homology and of standard continuity results for spectral invariants, and we use this to prove a C^0-robustness result for the fixed points of Hamiltonian diffeomorphisms. Our approach, which is rather different from the standard methods of persistent homology, is based on a non-Archimedean singular value decomposition for the boundary operator of the chain complex.
Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/9b20a1073afccd5ea920bbb6ccd6015b6f5d363c
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In Computational Theory and Mathematics
In Geometry and Topology
- E. Shelukhin
- A. Kislev
- Vukavsin Stojisavljevi'c
- L. Polterovich
- Michael Usher
- Michael Entov
- Yong-Geun Oh
- F. Zapolsky
In Mathematical Physics
- E. Shelukhin
- A. Kislev
- Vukavsin Stojisavljevi'c
- L. Polterovich
- Michael Entov
- Michael Usher
- Yong-Geun Oh
- F. Zapolsky
Citations in the corpus
- Bounds on spectral norms and barcodes (2018, background): “[116]) also applies for the case of piecewise linear radially symmetric functions.”
- Persistence modules with operators in Morse and Floer theory (2017, methodology): “This definition of chain complex filtered by function is the special case of the definition of Floer-type complex over Novikov field ΛK,Γ given in [56] in case Γ = {0} and valuation on K is trivial.”