Bounds on spectral norms and barcodes

Kislev, Asaf; Shelukhin, Egor

doi:10.2140/gt.2021.25.3257

Cited by 19 publications

(15 citation statements)

References 105 publications

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“…Polterovich, Shelukhin and Stojisavljevic introduce a new structure to persistence homology for Floer theory, coming from the quantum cup product [23]. Buhovsky, Humiliere and Seyfaddini [3] apply the barcodes technology of Kislev and Shelukhin [20] to study Hamiltonian (non-smooth) homeomorphisms. All of the above are in the symplectic setting, focusing on quantitative results for Hamiltonians.…”

Section: The Oscillation Of a Contact Hamiltonian Hmentioning

confidence: 99%

The persistence of the Chekanov–Eliashberg algebra

Rizell

Sullivan

2020

Sel. Math. New Ser.

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We apply the barcodes of persistent homology theory to the c Chekanov–Eliashberg algebra of a Legendrian submanifold to deduce displacement energy bounds for arbitrary Legendrians. We do not require the full Chekanov–Eliashberg algebra to admit an augmentation as we linearize the algebra only below a certain action level. As an application we show that it is not possible to $$C^0$$ C 0 -approximate a stabilized Legendrian by a Legendrian that admits an augmentation.

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Section: The Oscillation Of a Contact Hamiltonian Hmentioning

confidence: 99%

The persistence of the Chekanov–Eliashberg algebra

Rizell

Sullivan

2020

Sel. Math. New Ser.

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“…Note that it is enough to prove the first inequality, the second follows by using c − ( L 1 , L 2 ) = −c + ( L 2 , L 1 ). We shall use the inequality from Lemma A.1 in Appendix A (this is essentially in proposition 36 from [KS22]), and apply it to F H * ( L 1 , L 2 ). This yields…”

Section: The Inverse Reduction Inequality and Applicationsmentioning

confidence: 99%

“…The goal of this appendix is to make the paper more self-contained by giving a proof of the Kislev-Shelukhin inequality. There is nothing new here compared to [KS22] except that we reduced their proof to an abstract result on persistence modules endowed with a ring structure, but this is already implicit in their work.…”

Section: Appendix a The Kislev-shelukhin Inequalitymentioning

confidence: 99%

Inverse reduction inequalities for spectral numbers and applications

Viterbo¹

2022

Preprint

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Our main result is the proof of an inequality between the spectral numbers of a Lagrangian and the spectral numbers of its reductions, going in the opposite direction to the classical inequality (see e.g [Vit92]). This has applications to the "Geometrically bounded Lagrangians are spectrally bounded" conjecture from [Vit08] and to the structure of elements in the γ-completion of the set of exact Lagrangians (see [Vit22]). We also investigate the local path-connectedness of the set of Hamiltonian diffeomorphisms with the spectral metric. Friday 25 th March, 2022, 00:48 CONTENTS 1. Introduction 1 2. Comments and acknowledgements 3 3. Basic definitions and notations 4 4. Spectral invariants for sheaves and Lagrangians 5 5. The inverse reduction inequality and applications 8 6. A remarkable property of the spectral distance 17 7. Spectral boundedness of geometrically bounded Lagrangians 21 8. The non-compact case 25 9. The spectral distance and locally path-connectedness 26 Appendix A. The Kislev-Shelukhin inequality 29 References 32

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“…(3) According to the continuity of the spectral distance in terms of the C 0 -distance, proved in [BHS21], an element in the group of Hamiltonian homeomorphisms, that is a C 0 -limit of elements of DHam c (M , ω), belongs to DHam(M,ω). Moreover an element in L(T * N ) (or DHam(M,ω)) has a barcode, as follows from the Kislev-Shelukhin theorem (see [KS22] and Appendix in [Vit22]) and was pointed out in [BHS21].…”

Section: Z) the Action Does Is The Obvious Action On The Grading (And...mentioning

confidence: 99%

“…Notice that to a pair L 1 , L 2 in L(M,ω) we may associate a Floer homology as a filtered vector space. Indeed, by the Kislev-Shelukhin inequality (see [KS22], or [Vit22], Appendix), the bottleneck distance between the persistence module, denoted…”

Section: Proofmentioning

confidence: 99%

On the supports in the Humilière completion and $γ$-coisotropic sets (with an Appendix joint with Vincent Humilière)

Viterbo¹

2022

Preprint

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The symplectic spectral metric on the set of Lagrangian submanifolds or Hamiltonian maps can be used to define a completion of these spaces. For an element of such a completion, we define its γ-support. We also define the notion of γ-coisotropic set, and prove that a γ-support must be γ-coisotropic toghether with many properties of the γ-support and γ-coisotropic sets. We give examples of Lagrangians in the completion having large γ-support and we study those (called "regular Lagrangians") having small γ-support. We compare the notion of γ-coisotropy with other notions of isotropy. In a joint Appendix with V. Humilière, we connect the γ-support with an extension of the notion of Birkhoff attractor of a dissipative map to higher dimension.

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Bounds on spectral norms and barcodes

Cited by 19 publications

References 105 publications

The persistence of the Chekanov–Eliashberg algebra

The persistence of the Chekanov–Eliashberg algebra

Inverse reduction inequalities for spectral numbers and applications

On the supports in the Humilière completion and $γ$-coisotropic sets (with an Appendix joint with Vincent Humilière)

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