Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

Book Review

The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.


Full text of review: PDF   This review is available free of charge.
Book Information:

Authors: Leonid Polterovich and Daniel Rosen
Title: Function theory on symplectic manifolds
Additional book information: CRM Monograph series, vol. 34, American Mathematical Society, Providence, Rhode Island, 2014, xii+203 pp., ISBN 978-1-4704-1693-5

References [Enhancements On Off] (What's this?)

  • [1] Johan F. Aarnes, Quasi-states and quasi-measures, Adv. Math. 86 (1991), no. 1, 41-67. MR 1097027, https://doi.org/10.1016/0001-8708(91)90035-6
  • [2] Vladimir Arnold, Sur une propriété topologique des applications globalement canoniques de la mécanique classique, C. R. Acad. Sci. Paris 261 (1965), 3719-3722 (French). MR 0193645
  • [3] Augustin Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv. 53 (1978), no. 2, 174-227 (French). MR 490874, https://doi.org/10.1007/BF02566074
  • [4] Lev Buhovsky, The $ 2/3$-convergence rate for the Poisson bracket, Geom. Funct. Anal. 19 (2010), no. 6, 1620-1649. MR 2594616, https://doi.org/10.1007/s00039-010-0045-z
  • [5] Lev Buhovsky and Yaron Ostrover, On the uniqueness of Hofer's geometry, Geom. Funct. Anal. 21 (2011), no. 6, 1296-1330. MR 2860189, https://doi.org/10.1007/s00039-011-0143-6
  • [6] Lev Buhovsky, Michael Entov, and Leonid Polterovich, Poisson brackets and symplectic invariants, Selecta Math. (N.S.) 18 (2012), no. 1, 89-157. MR 2891862, https://doi.org/10.1007/s00029-011-0068-9
  • [7] Eugenio Calabi, On the group of automorphisms of a symplectic manifold, Problems in analysis (Lectures at the Sympos. in honor of Salomon Bochner, Princeton Univ., Princeton, N.J., 1969) Princeton Univ. Press, Princeton, N.J., 1970, pp. 1-26. MR 0350776
  • [8] Franco Cardin and Claude Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations, Duke Math. J. 144 (2008), no. 2, 235-284. MR 2437680, https://doi.org/10.1215/00127094-2008-036
  • [9] C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol'd, Invent. Math. 73 (1983), no. 1, 33-49. MR 707347, https://doi.org/10.1007/BF01393824
  • [10] Y. Eliashberg, Rigidity of symplectic structures, preprint, 1981.
  • [11] Ya. M. Eliashberg, A theorem on the structure of wave fronts and its application in symplectic topology, Funktsional. Anal. i Prilozhen. 21 (1987), no. 3, 65-72, 96 (Russian). MR 911776
  • [12] Michael Entov and Leonid Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 30 (2003), 1635-1676. MR 1979584, https://doi.org/10.1155/S1073792803210011
  • [13] Michael Entov and Leonid Polterovich, $ C^0$-rigidity of Poisson brackets, Symplectic topology and measure preserving dynamical systems, Contemp. Math., vol. 512, Amer. Math. Soc., Providence, RI, 2010, pp. 25-32. MR 2605312, https://doi.org/10.1090/conm/512/10058
  • [14] Michael Entov and Leonid Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv. 81 (2006), no. 1, 75-99. MR 2208798, https://doi.org/10.4171/CMH/43
  • [15] Andrew M. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885-893. MR 0096113
  • [16] M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347.
  • [17] H. Hofer, On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), no. 1-2, 25-38. MR 1059642, https://doi.org/10.1017/S0308210500024549
  • [18] Leonid Polterovich, Quantum unsharpness and symplectic rigidity, Lett. Math. Phys. 102 (2012), no. 3, 245-264. MR 2989482, https://doi.org/10.1007/s11005-012-0564-7
  • [19] Leonid Polterovich, Symplectic geometry of quantum noise, Comm. Math. Phys. 327 (2014), no. 2, 481-519. MR 3183407, https://doi.org/10.1007/s00220-014-1937-9
  • [20] Frol Zapolsky, Quasi-states and the Poisson bracket on surfaces, J. Mod. Dyn. 1 (2007), no. 3, 465-475. MR 2318499, https://doi.org/10.3934/jmd.2007.1.465

Review Information:

Reviewer: Yakov Eliashberg
Affiliation: Department of Mathematics, Stanford University
Email: eliash@stanford.edu
Journal: Bull. Amer. Math. Soc. 54 (2017), 135-140
MSC (2010): Primary 53Dxx; Secondary 57R17, 81S10, 81P15, 22F65, 20F99, 28A10
DOI: https://doi.org/10.1090/bull/1547
Published electronically: August 17, 2016
Review copyright: © Copyright 2016 American Mathematical Society
American Mathematical Society