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Symplectic and Contact Geometry and Hamiltonian Dynamics

  • Mikhail B. Sevryuk
Part of the Progress in Mathematics book series (PM, volume 202)

Abstract

This is an introduction to the contributions by the lecturers at the mini-symposium on symplectic and contact geometry. We present a very general and brief account of the prehistory of the field and give references to some seminal papers and important survey works.

Keywords

Hamiltonian System Symplectic Manifold Invariant Torus Contact Manifold Contact Geometry 
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References

  1. [1]
    B. Aebischer, M. Borer, M. Kälin, Ch. Leuenberger and H. M. ReimannSymplectic GeometryProgr. Math., 124 (Birkhäuser, Basel, 1994).Google Scholar
  2. [2]
    V. I. Arnol’dSur une propri¨¦t¨¦ topologique des applications globalement canoniques de la m¨¦canique classique, C. R. Acad. Sci. Paris, 261 (1965), 3719–3722.MathSciNetzbMATHGoogle Scholar
  3. [3]
    V. I. Arnol’dThe stability problem and ergodic properties of classical dynamical systemsin: Proceedings of the Intern. Congress of Mathematicians (Moscow, 1966) (Mir, Moscow, 1968), 387–392 (in Russian).Google Scholar
  4. [4]
    V. I. Arnol’d, Acomment to H. Poincar¨¦’s paper “Sur un th¨¦or¨¨me de g¨¦om¨¦trie”in: H. Poincar¨¦, Selected Works in Three Volumes, Vol. II (Nauka, Moscow, 1972), 987–989 (in Russian).Google Scholar
  5. [5]
    V. I. Arnol’dFixed points of symplectic diffeomorphismsin: F. E. Browder, Ed., Mathematical Developments Arising from Hilbert Problems, Proc. Symp. Pure Math., 28 (A.M.S., Providence, RI, 1976), 66.Google Scholar
  6. [6]
    V. I. Arnol’dSome problems in the theory of differential equationsin: Unsolved Problems in Mechanics and Applied Mathematics (Moscow State Univ. Press, Moscow, 1977), 3–9 (in Russian).Google Scholar
  7. [7]
    V. I. Arnol’dMathematical Methods of Classical MechanicsGraduate Texts in Math., 60 (Springer-Verlag, New York, 1978) [the Russian original is of 1974, the 3rd Russian edition is of 1989].Google Scholar
  8. [8]
    V. I. Arnol’dThe first steps of symplectic topologyRussian Math. Surveys, 41 (1986), no. 6, 1–21.zbMATHCrossRefGoogle Scholar
  9. [9]
    V. I. Arnol’dSingularities of Caustics and Wave FrontsMath. Appl. (Soviet Ser.), 62 (Kluwer, Dordrecht, 1990).Google Scholar
  10. [10]
    M. Audin and J. Lafontaine, Eds.Holomorphic Curves in Symplectic GeometryProgr. Math., 117 (Birkhäuser, Basel, 1994).Google Scholar
  11. [11]
    D. AurouxAsymptotically holomorphic families of symplectic submanifoldsGeom. Funct. Anal., 7 (1997), 971–995.MathSciNetzbMATHGoogle Scholar
  12. [12]
    R. BerndtEinführung in die Symplektische Geometrie(Friedr. Vieweg & Sohn, Braunschweig, 1998). 570 M. B. SevryukGoogle Scholar
  13. [13]
    E. Bierstone, B. A. Khesin, A. G. Khovanskii and J. E. Marsden, Eds.The Arnol’dfestProceedings of a Conference in Honour of V. I. Arnol’d for his Sixtieth Birthday, Fields Inst. Comm.24 (A.M.S., Providence, RI, 2000).Google Scholar
  14. [14]
    P. BiranSymplectic packing in dimension4, Geom. Funct. Anal.7(1997), 420–437.MathSciNetzbMATHGoogle Scholar
  15. [15]
    P. Biran, Astability property of symplectic packingInv. Math.,136 (1999), 123–155 [see also Featured Review 2000b:57039 by M. Schwarz of this paper in Math. Reviews].MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    G. D. BirkhoffProof of Poincar¨¦’s geometric theorem Trans. Amer. Math. Soc.,14 (1913), 14–22.MathSciNetzbMATHGoogle Scholar
  17. [17]
    H. W. Broer, G. B. Huitema and M. B. SevryukQuasi-Periodic Motions in Families of Dynamical Systems: Order amidst ChaosLecture Notes in Math.1645 (Springer-Verlag, Berlin, 1996).Google Scholar
  18. [18]
    R. Budzyñski, S. Janeczko, W. Kondracki and A. F. Künzle, Eds.Symplectic Singularities and Geometry of Gauge FieldsBanach Center Publ.39 (Polish Acad. Sci., Inst. Math., Warsaw, 1997).Google Scholar
  19. [19]
    C. C. Conley and E. ZehnderThe Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol’dInv. Math.73 (1983), 33–49.Google Scholar
  20. [20]
    S. K. DonaldsonThe Seiberg-Witten equations and 4-manifold topologyBull. Amer. Math. Soc. (N.S.)33 (1996), 45–70 [see also Featured Review 96k:57033 by D. S. Freed of this paper in Math. Reviews].Google Scholar
  21. [21]
    S. K. DonaldsonSymplectic submanifolds and almost-complex geometryJ. Differential Geom.44 (1996), 666–705 [see also Featured Review 98h:53045 by D. Pollack of this paper in Math. Reviews].Google Scholar
  22. [22]
    S. K. DonaldsonLefschetz fibrations in symplectic geometryin: G. Fischer and U. Rehmann, Eds., Proceedings of the Intern. Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math.1998 Extra Vol. II, 309–314 (electronic).Google Scholar
  23. [23]
    Ya. M. Èliashberg, Anestimate of the number of fixed points of area-preserving transformationsPreprint (Syktyvkar, 1978) (in Russian).Google Scholar
  24. [24]
    Ya. M. ÈliashbergRigidity of symplectic and contact structuresPreprint (1981) (in Russian); see also in: Abstracts of reports to the 7th Intern. Topology Conference in Leningrad (1982).Google Scholar
  25. [25]
    Ya. M. Èliashberg and L. V. PolterovichThe problem of Lagrangian knots in four-manifoldsin [51], 313–327.Google Scholar
  26. [26]
    Ya. M. Èliashberg and W. P. ThurstonConfoliationsUniv. Lecture Series13 (A.M.S., Providence, RI, 1998).Google Scholar
  27. [27]
    Ya. M. Èliashberg and M. FraserClassification of topologically trivial Legendrian knotsin [55], 17–51.Google Scholar
  28. [28]
    Ya. M. Èliashberg and L. Traynor, Eds.Symplectic Geometry and TopologyIAS/Park City Math. Series7 (A.M.S., Providence, RI, 1999).Google Scholar
  29. [29]
    Ya. M. Èliashberg, D. B. Fuchs, T. Ratiu and A. Weinstein, Eds.Northern California Symplectic Geometry SeminarAmer. Math. Soc. Transl. Ser. 2, 196 (A.M.S., Providence, RI, 1999).Google Scholar
  30. [30]
    A. FloerAn instanton-invariant for 3-manifoldsComm. Math. Phys.118 (1988), 215–240.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    A. FloerMorse theory for Lagrangian intersectionsJ. Differential Geom.28 (1988), 513–547.MathSciNetzbMATHGoogle Scholar
  32. [32]
    A. FloerSymplectic fixed points and holomorphic spheresComm. Math. Phys.120 (1989), 575–611.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    B. FortuneA symplectic fixed point theorem forCPn, Inv. Math.81(1985), 29–46.MathSciNetzbMATHGoogle Scholar
  34. [34]
    K. Fukaya and K. OnoArnol’d conjecture and Gromov-Witten invariantTopology38 (1999), 933–1048 [see also Featured Review 2000j: 53116 by D. E. Hurtubise of this paper in Math. Reviews].MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35] K. Fukaya and K. OnoArnol’d conjecture and Gromov-Witten invariant for general symplectic manifoldsin [13], 173–190.Google Scholar
  36. [36]
    H. GeigesConstructions of contact manifoldsMath. Proc. Cambridge Philos. Soc.121 (1997), 455–464.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    V. L. Ginzburg, Anembedding S2“-I- ->R2’, 2n-1 > 7,whose Hamiltonian flow has no periodic trajectories, Internat. Math. Res. Notices,1995no. 2, 83–97 (electronic).Google Scholar
  38. [38]
    V. L. Ginzburg, Asmooth counterexample to the Hamiltonian Seifert conjecture inR6, Internat. Math. Res. Notices,1997no. 13, 641–650.MathSciNetCrossRefGoogle Scholar
  39. [39]
    V. L. GinzburgHamiltonian dynamical systems without periodic orbitsin [29], 35–48.Google Scholar
  40. [40]
    R. E. Gompf, Anew construction of symplectic manifoldsAnn. of Math. (2),142 (1995), 527–595 [see also Featured Review 96j:57025 by M. Ue of this paper in Math. Reviews].Google Scholar
  41. [41]
    M. L. GromovPseudo holomorphic curves in symplectic manifoldsInv. Math.82 (1985), 307–347.MathSciNetzbMATHGoogle Scholar
  42. [42]
    M.R.Herman,In¨¦galit¨¦s “a priori” pour des tores lagrangiens invariants par des diff¨¦omorphismes symplectiques, Inst. Hautes Etudes Sci. Publ. Math.,70(1989), 47–101.MathSciNetzbMATHCrossRefGoogle Scholar
  43. [43]
    H. Hofer and E. ZehnderSymplectic Invariants and Hamiltonian Dynamics(Birkhäuser, Basel, 1994) [see also Featured Review 96g:58001 by D. M. Burns, Jr. of this book in Math. Reviews].zbMATHCrossRefGoogle Scholar
  44. [44]
    H. Hofer, C. H. Taubes, A. Weinstein and E. Zehnder, Eds.The Floer Memorial VolumeProgr. Math.133 (Birkhäuser, Basel, 1995).Google Scholar
  45. [45]
    H. Hofer and D. A. SalamonFloer homology and Novikov ringsin [44], 483–524.Google Scholar
  46. [46]
    H. Hofer, K. Wysocki and E. ZehnderThe dynamics on three-dimensional strictly convex energy surfacesAnn. of Math. (2)148 (1998), 197–289 [see also Featured Review 99m:58089 by M. Schwarz of this paper in Math. Reviews].MathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    H. HoferDynamics topology and holomorphic curves in: G. Fischer and U. Rehmann, Eds., Proceedings of the Intern. Congress of Mathematicians, Vol. I (Berlin, 1998), Doc. Math.1998 Extra Vol. I, 255–280 (electronic).Google Scholar
  48. [48]
    J. Hurtubise, F. Lalonde and G. Sabidussi, Eds.Gauge Theory and Symplectic GeometryNATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.488 (Kluwer, Dordrecht, 1997).Google Scholar
  49. [49]
    L. A. Ibort and C. Mart¨ªnez OntalbaArnol’d’s conjecture and symplectic reductionJ. Geom. Phys.18 (1996), 25–37.MathSciNetzbMATHCrossRefGoogle Scholar
  50. [50]
    À. Jorba and J. VillanuevaOn the normal behaviour of partially elliptic lower-dimensional tori of Hamiltonian systemsNonlinearity, 10 (1997), 783–822.MathSciNetzbMATHCrossRefGoogle Scholar
  51. [51]
    W. H. Kazez, Ed.Geometric TopologyAMS/IP Stud. Adv. Math.2.1 (A.M.S., Providence, RI; Intern. Press, Cambridge, MA, 1997).Google Scholar
  52. [52]
    F. LalondeEnergy and capacities in symplectic topologyin [51], 328–374.Google Scholar
  53. [53]
    F. LalondeJ-holomorphic curves and symplectic invariantsin [48], 147–174.Google Scholar
  54. [54]
    F. LalondeNew trends in symplectic geometryC. R. Math. Rep. Acad. Sci. Canada19 (1997), 33–50.MathSciNetzbMATHGoogle Scholar
  55. [55]
    F. Lalonde, Ed.Geometry Topology and DynamicsCRM Proceedings & Lecture Notes15 (A.M.S., Providence, RI, 1998).Google Scholar
  56. [56]
    J. Li and G. TianVirtual moduli cycles and Gromov-Witten invariants of general symplectic manifoldsin [72], 47–83.Google Scholar
  57. [57]
    G. Liu and G. TianFloer homology and Arnol’d conjectureJ. Differential Geom.49 (1998), 1–74 [see also Featured Review 99m:58047 by J.-C. Sikorav of this paper in Math. Reviews].MathSciNetzbMATHGoogle Scholar
  58. [58]
    G. C. LuThe Arnol’d conjecture for a product of weakly monotone manifoldsChinese J. Math.24 (1996), 145–157.zbMATHGoogle Scholar
  59. [59]
    G. C. LuThe Arnol’d conjecture for a product of monotone manifolds and CalabiYao manifoldsActa Math. Sinica (N.S.)13 (1997), 381–388.zbMATHGoogle Scholar
  60. [60]
    D. McDuff and L. V. PolterovichSymplectic packings and algebraic geometryInv. Math.115 (1994), 405–434.MathSciNetzbMATHGoogle Scholar
  61. [61]
    D. McDuff and D. A. SalamonJ-Holomorphic Curves and Quantum CohomologyUniv. Lecture Series6 (A.M.S., Providence, RI, 1994) [see also Featured Review 95g:58026 by B. Hunt of this book in Math. Reviews].zbMATHGoogle Scholar
  62. [62]
    D. McDuff and D. A. SalamonIntroduction to Symplectic Topology(The Clarendon Press, Oxford Univ. Press, New York, 1995 [1st ed.], 1998 [2nd ed.]).Google Scholar
  63. [63]
    D. McDuffLectures on Gromov invariants for symplectic 4-manifoldsin [48], 175–210.Google Scholar
  64. [64]
    D. McDuffRecent developments in symplectic topologyin: A. Balog, G. O. H. Katona, A. Recski and D. Sz¨¢sz, Eds., Proceedings of the Second European Congress of Mathematics, Vol. II (Budapest, 1996), Progr. Math.169 (Birkhäuser, Basel, 1998), 28–42.CrossRefGoogle Scholar
  65. [65]
    D. McDuffFibrations in symplectic topologyin: G. Fischer and U. Rehmann, Eds., Proceedings of the Intern. Congress of Mathematicians, Vol. I (Berlin, 1998), Doc. Math.1998 Extra Vol. I, 339–357 (electronic).Google Scholar
  66. [66]
    D. McDuffSymplectic structures-a new approach to geometryNotices Amer. Math. Soc.45 (1998), 952–960.MathSciNetzbMATHGoogle Scholar
  67. [67]
    D. McDuffIntroduction to symplectic topologyin [28], 5–33.Google Scholar
  68. [68]
    D. McDuff, Aglimpse into symplectic geometryin: V. I. Arnol’d, M. Atiyah, P. Lax and B. Mazur, Eds., Mathematics: Frontiers and Perspectives (A.M.S., Providence, RI, 2000), 175–187.Google Scholar
  69. [69]
    K. OnoOn the Arnol’d conjecture for weakly monotone symplectic manifoldsInv. Math.119 (1995), 519–537.Google Scholar
  70. [70]
    H. PoincaréSur un th¨¦or¨¨me de om¨¦trieRend. Circ. Mat. Palermo, 33 (1912), 375–407.CrossRefGoogle Scholar
  71. [71]
    D. A. Salamon, Ed.Symplectic GeometryLondon Math. Soc. Lecture Note Series, 192 (Cambridge Univ. Press, Cambridge, 1993).Google Scholar
  72. [72]
    R. J. Stern, Ed.Topics in Symplectic 4-ManifoldsFirst Intern. Press Lecture Series, I (Intern. Press, Cambridge, MA, 1998).Google Scholar
  73. [73]
    C. H. TaubesThe Seiberg-Witten invariants and symplectic formsMath. Res. Lett., 1 (1994), 809–822.MathSciNetzbMATHGoogle Scholar
  74. [74]
    C. H. TaubesThe Seiberg-Witten and Gromov invariantsMath. Res. Lett., 2 (1995), 221–238.MathSciNetzbMATHGoogle Scholar
  75. [75]
    C. H. Taubes, SW = Gr:from the Seiberg-Witten equations to pseudo-holomorphic curvesJ. Amer. Math. Soc., 9 (1996), 845–918 [see also Featured Review 97a:57033 by D. A. Salamon of this paper in Math. Reviews].Google Scholar
  76. [76]
    C. H. TaubesCounting pseudo-holomorphic submanifolds in dimension4, J. Differential Geom., 44 (1996), 818–893 [see also Featured Review 97k:58029 by T. H. Parker of this paper in Math. Reviews].MathSciNetzbMATHGoogle Scholar
  77. [77]
    C. H. TaubesThe structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure and symplectic form on S I- xB3, Geom. Topol., 2 (1998), 221–332 (electronic) [see also Featured Review 99m:57029 by F. Lalonde of this paper in Math. Reviews].Google Scholar
  78. [78]
    C. H. TaubesThe geometry of the Seiberg-Witten invariantsin: G. Fischer and U. Rehmann, Eds., Proceedings of the Intern. Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math., 1998, Extra Vol. II, 493–504 (electronic).Google Scholar
  79. [79]
    C. B. Thomas, Ed.Contact and Symplectic GeometryPubl. Newton Inst., 8 (Cambridge Univ. Press, Cambridge, 1996).Google Scholar
  80. [80]
    A. WeinsteinLectures on Symplectic ManifoldsCBMS Regional Conference Series in Math., 29 (A.M.S., Providence, RI, 1977).Google Scholar
  81. [81]
    V. M. Zakalyukin and O. M. MyasnichenkoLagrangian singularities in symplectic reductionFunctional Anal. Appl., 32 (1998), 1–9.MathSciNetzbMATHGoogle Scholar

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© Springer Basel AG 2001

Authors and Affiliations

  • Mikhail B. Sevryuk
    • 1
  1. 1.Institute of Energy Problems of Chemical PhysicsThe Russia Academy of SciencesMoscowRussia

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