@article{336ee5e904504125a53310c4adb7b951,
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{\textendash}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{\textemdash}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 = "English",
volume = "18",
pages = "89--157",
journal = "Selecta Mathematica, New Series",
issn = "1022-1824",
number = "1",
}