Persistence Modules with Operators in Morse and Floer Theory

Polterovich, Leonid; Shelukhin, Egor; Stojisavljević, Vukašin

doi:10.17323/1609-4514-2017-17-4-757-786

Cited by 31 publications

(36 citation statements)

References 54 publications

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“…This map is Lipschitz with respect to the L 1,∞ -distance on H = H M , and the bottleneck distance on the space barcodes of barcodes. This observation was used in [76], in [3,40,78,92,99,105] and more recently in [18,31,60,66,93,95] to produce various quantitative results in symplectic topology. Set barcodes ′ for the quotient space of barcodes with respect to the isometric R-action by shifts.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Viterbo conjecture for Zoll symmetric spaces

Shelukhin¹

2018

Preprint

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We prove a conjecture of Viterbo from 2007 on the existence of a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in unit cotangent disk bundles, for bases given by compact rank one symmetric spaces S n , RP n , CP n , HP n , n ≥ 1. We discuss generalizations and give applications, in particular to C 0 symplectic topology. Our key methods, which are of independent interest, consist of a reinterpretation of the spectral norm via the asymptotic behavior of a family of cones of filtered morphisms, and a quantitative deformation argument for Floer persistence modules, that allows to excise a divisor.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Floer homology as a module over quantum homology. In the absolute case, as discussed in detail in [78], an element α M ∈ QH m (M ) \ {0} gives, for H ∈ H, and r ∈ Z, a ∈ R a map…”

Section: 23mentioning

confidence: 99%

Viterbo conjecture for Zoll symmetric spaces

Shelukhin¹

2018

Preprint

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“…For instance, in terms of the objects from dynamics, the well-known Hofer distance d Hofer is defined between Hamiltonian diffeomorphisms of a symplectic manifold, which is based on the fundamental work [17,19]. This led to an influential research direction called Hofer geometry [21], as well as far-reaching applications to Hamiltonian dynamic ( [24,25,32,34]). For another instance, inspired by Ostrover and Polterovich, the recent work [15,31,33] studied quantitative comparisons between symplectic objects constructed from a geometric perspective, that is, star-shaped domains of a Liouville manifold.…”

Section: Motivationmentioning

confidence: 99%

Relative growth rate and contact Banach–Mazur distance

Rosen

Zhang

2021

Geom Dedicata

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In this paper, we define non-linear versions of Banach-Mazur distance in the contact geometry set-up, called contact Banach-Mazur distances and denoted by d CBM . Explicitly, we consider the following two set-ups, either on a contact manifold W × S 1 where W is a Liouville manifold, or a closed Liouville-fillable contact manifold M. The inputs of d CBM are different in these two cases. In the former case the inputs are (contact) star-shaped domains of W × S 1 which correspond to the homotopy classes of positive contact isotopies, and in the latter case the inputs are contact 1-forms of M inducing the same contact structure. In particular, the contact Banach-Mazur distance d CBM defined in the former case is motivated by the concept, relative growth rate, which was originally defined and studied by Eliashberg-Polterovich. The main results are the large-scale geometric properties in terms of d CBM . In addition, we propose a quantitative comparison between elements in a certain subcategory of the derived categories of sheaves of modules (over certain topological spaces). This is based on several important properties of the singular support of sheaves and Guillermou-Kashiwara-Schapira's sheaf quantization.

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“…Extending κ[R + ]-linearly defines F (X, l) ϕ on all of F n (X, l). Note that F (X, l) ϕ is a graded map due to the degree shifts in (7). In particular, the face maps on F (X, l) are given on generators by…”

Section: Remark 32mentioning

confidence: 99%

“…}, the ring κ[R + ] is not a PID. Related Künneth formulas have been proven in [7] and [1]. Theorem 1.4.…”

mentioning

confidence: 98%

Persistent homology of the sum metric

Carlsson¹,

Filippenko²

2020

Journal of Pure and Applied Algebra

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Given finite metric spaces (X, d X ) and (Y, d Y ), we investigate the persistent homology P H * (X × Y ) of the Cartesian product X × Y equipped with the sum metric d X +d Y . Interpreting persistent homology as a module over a polynomial ring, one might expect the usual Künneth short exact sequence to hold. We prove that it holds for P H 0 and P H 1 , and we illustrate with the Hamming cube {0, 1} k that it fails for P H n , n ≥ 2. For n = 2, the prediction for P H 2 (X × Y ) from the expected Künneth short exact sequence has a natural surjection onto P H 2 (X × Y ). We compute the nontrivial kernel of this surjection for the splitting of Hamming cubes {0, 1} k = {0, 1} k−1 × {0, 1}. For all n ≥ 0, the interleaving distance between the prediction for P H n (X × Y ) and the true persistent homology is bounded above by the minimum of the diameters of X and Y . As preliminary results of independent interest, we establish an algebraic Künneth formula for simplicial modules over the ring κ[R + ] of polynomials with coefficients in a field κ and exponents in R + = [0, ∞), as well as a Künneth formula for the persistent homology of R + -filtered simplicial sets -both of these Künneth formulas hold in all homological dimensions n ≥ 0.

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Persistence Modules with Operators in Morse and Floer Theory

Cited by 31 publications

References 54 publications

Viterbo conjecture for Zoll symmetric spaces

Viterbo conjecture for Zoll symmetric spaces

Relative growth rate and contact Banach–Mazur distance

Persistent homology of the sum metric

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