Poisson brackets and symplectic invariants

Lev Buhovsky; Michael Entov; Leonid Polterovich

doi:10.1007/s00029-011-0068-9

Poisson brackets and symplectic invariants

Lev Buhovsky, Michael Entov, Leonid Polterovich

Tel Aviv University, Technion - Israel Institute of Technology

Research output: Contribution to journal › Article › peer-review

Abstract

We introduce new invariants associated with collections of compact subsets of a symplectic manifold. They are defined through an elementary-looking variational problem involving Poisson brackets. The proof of the non-triviality of these invariants involves various flavors of Floer theory, including the μ ³-operation in Donaldson-Fukaya category. We present applications to approximation theory on symplectic manifolds and to Hamiltonian dynamics.

Original language	English
Pages (from-to)	89-157
Number of pages	69
Journal	Selecta Mathematica, New Series
Volume	18
Issue number	1
DOIs	https://doi.org/10.1007/s00029-011-0068-9
State	Published - Mar 2012

Keywords

Donaldson-Fukaya category
Hamiltonian chord
Poisson brackets
Quasi-state
symplectic manifold

All Science Journal Classification (ASJC) codes

General Mathematics
General Physics and Astronomy

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10.1007/s00029-011-0068-9

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title = "Poisson brackets and symplectic invariants",

abstract = "We introduce new invariants associated with collections of compact subsets of a symplectic manifold. They are defined through an elementary-looking variational problem involving Poisson brackets. The proof of the non-triviality of these invariants involves various flavors of Floer theory, including the μ 3-operation in Donaldson-Fukaya category. We present applications to approximation theory on symplectic manifolds and to Hamiltonian dynamics.",

keywords = "Donaldson-Fukaya category, Hamiltonian chord, Poisson brackets, Quasi-state, symplectic manifold",

author = "Lev Buhovsky and Michael Entov and Leonid Polterovich",

note = "Funding Information: Acknowledgments We thank Richard Hind for his help with symplectic field theory and in particular for providing us a short proof of Theorem 6.1. We are grateful to Paul Seidel and Ivan Smith for useful consultations on Donaldson–Fukaya category. In particular, Smith explained to us some interesting examples where operations µ2 and µ3 do not vanish. We thank Paul Biran for a number of stimulating discussions. We thank Strom Borman and an anonymous referee for comments and corrections—in particular, Borman suggested a considerable improvement of the numerical constants in Sect. 1.5. A part of this paper was written during our stay in MSRI, Berkeley. We thank MSRI for the hospitality. The third-named author thanks the Simons Foundation for sponsoring this stay. Michael Entov was partially supported by the Israel Science Foundation grant \# 723/10 and by the Japan Technion Society Research Fund. Leonid Polterovich was partially supported by the National Science Foundation grant DMS-1006610.",

year = "2012",

month = mar,

doi = "10.1007/s00029-011-0068-9",

language = "الإنجليزيّة",

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journal = "Selecta Mathematica, New Series",

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T1 - Poisson brackets and symplectic invariants

AU - Buhovsky, Lev

AU - Entov, Michael

AU - Polterovich, Leonid

N1 - Funding Information: Acknowledgments We thank Richard Hind for his help with symplectic field theory and in particular for providing us a short proof of Theorem 6.1. We are grateful to Paul Seidel and Ivan Smith for useful consultations on Donaldson–Fukaya category. In particular, Smith explained to us some interesting examples where operations µ2 and µ3 do not vanish. We thank Paul Biran for a number of stimulating discussions. We thank Strom Borman and an anonymous referee for comments and corrections—in particular, Borman suggested a considerable improvement of the numerical constants in Sect. 1.5. A part of this paper was written during our stay in MSRI, Berkeley. We thank MSRI for the hospitality. The third-named author thanks the Simons Foundation for sponsoring this stay. Michael Entov was partially supported by the Israel Science Foundation grant # 723/10 and by the Japan Technion Society Research Fund. Leonid Polterovich was partially supported by the National Science Foundation grant DMS-1006610.

PY - 2012/3

Y1 - 2012/3

N2 - We introduce new invariants associated with collections of compact subsets of a symplectic manifold. They are defined through an elementary-looking variational problem involving Poisson brackets. The proof of the non-triviality of these invariants involves various flavors of Floer theory, including the μ 3-operation in Donaldson-Fukaya category. We present applications to approximation theory on symplectic manifolds and to Hamiltonian dynamics.

AB - We introduce new invariants associated with collections of compact subsets of a symplectic manifold. They are defined through an elementary-looking variational problem involving Poisson brackets. The proof of the non-triviality of these invariants involves various flavors of Floer theory, including the μ 3-operation in Donaldson-Fukaya category. We present applications to approximation theory on symplectic manifolds and to Hamiltonian dynamics.

KW - Donaldson-Fukaya category

KW - Hamiltonian chord

KW - Poisson brackets

KW - Quasi-state

KW - symplectic manifold

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U2 - 10.1007/s00029-011-0068-9

DO - 10.1007/s00029-011-0068-9

M3 - مقالة

SN - 1022-1824

VL - 18

SP - 89

EP - 157

JO - Selecta Mathematica, New Series

JF - Selecta Mathematica, New Series

IS - 1

ER -

Poisson brackets and symplectic invariants

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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