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.
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References
B. Aebischer, M. Borer, M. Kälin, Ch. Leuenberger and H. M. ReimannSymplectic GeometryProgr. Math., 124 (Birkhäuser, Basel, 1994).
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.
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).
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).
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.
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).
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].
V. I. Arnol’dThe first steps of symplectic topologyRussian Math. Surveys, 41 (1986), no. 6, 1–21.
V. I. Arnol’dSingularities of Caustics and Wave FrontsMath. Appl. (Soviet Ser.), 62 (Kluwer, Dordrecht, 1990).
M. Audin and J. Lafontaine, Eds.Holomorphic Curves in Symplectic GeometryProgr. Math., 117 (Birkhäuser, Basel, 1994).
D. AurouxAsymptotically holomorphic families of symplectic submanifoldsGeom. Funct. Anal., 7 (1997), 971–995.
R. BerndtEinführung in die Symplektische Geometrie(Friedr. Vieweg & Sohn, Braunschweig, 1998). 570 M. B. Sevryuk
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).
P. BiranSymplectic packing in dimension4, Geom. Funct. Anal.7(1997), 420–437.
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].
G. D. BirkhoffProof of Poincar¨¦’s geometric theorem Trans. Amer. Math. Soc.,14 (1913), 14–22.
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).
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).
C. C. Conley and E. ZehnderThe Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol’dInv. Math.73 (1983), 33–49.
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].
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].
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).
Ya. M. Èliashberg, Anestimate of the number of fixed points of area-preserving transformationsPreprint (Syktyvkar, 1978) (in Russian).
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).
Ya. M. Èliashberg and L. V. PolterovichThe problem of Lagrangian knots in four-manifoldsin [51], 313–327.
Ya. M. Èliashberg and W. P. ThurstonConfoliationsUniv. Lecture Series13 (A.M.S., Providence, RI, 1998).
Ya. M. Èliashberg and M. FraserClassification of topologically trivial Legendrian knotsin [55], 17–51.
Ya. M. Èliashberg and L. Traynor, Eds.Symplectic Geometry and TopologyIAS/Park City Math. Series7 (A.M.S., Providence, RI, 1999).
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).
A. FloerAn instanton-invariant for 3-manifoldsComm. Math. Phys.118 (1988), 215–240.
A. FloerMorse theory for Lagrangian intersectionsJ. Differential Geom.28 (1988), 513–547.
A. FloerSymplectic fixed points and holomorphic spheresComm. Math. Phys.120 (1989), 575–611.
B. FortuneA symplectic fixed point theorem forCPn, Inv. Math.81(1985), 29–46.
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].
[35] K. Fukaya and K. OnoArnol’d conjecture and Gromov-Witten invariant for general symplectic manifoldsin [13], 173–190.
H. GeigesConstructions of contact manifoldsMath. Proc. Cambridge Philos. Soc.121 (1997), 455–464.
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).
V. L. Ginzburg, Asmooth counterexample to the Hamiltonian Seifert conjecture inR6, Internat. Math. Res. Notices,1997no. 13, 641–650.
V. L. GinzburgHamiltonian dynamical systems without periodic orbitsin [29], 35–48.
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].
M. L. GromovPseudo holomorphic curves in symplectic manifoldsInv. Math.82 (1985), 307–347.
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.
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].
H. Hofer, C. H. Taubes, A. Weinstein and E. Zehnder, Eds.The Floer Memorial VolumeProgr. Math.133 (Birkhäuser, Basel, 1995).
H. Hofer and D. A. SalamonFloer homology and Novikov ringsin [44], 483–524.
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].
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).
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).
L. A. Ibort and C. Mart¨ªnez OntalbaArnol’d’s conjecture and symplectic reductionJ. Geom. Phys.18 (1996), 25–37.
À. Jorba and J. VillanuevaOn the normal behaviour of partially elliptic lower-dimensional tori of Hamiltonian systemsNonlinearity, 10 (1997), 783–822.
W. H. Kazez, Ed.Geometric TopologyAMS/IP Stud. Adv. Math.2.1 (A.M.S., Providence, RI; Intern. Press, Cambridge, MA, 1997).
F. LalondeEnergy and capacities in symplectic topologyin [51], 328–374.
F. LalondeJ-holomorphic curves and symplectic invariantsin [48], 147–174.
F. LalondeNew trends in symplectic geometryC. R. Math. Rep. Acad. Sci. Canada19 (1997), 33–50.
F. Lalonde, Ed.Geometry Topology and DynamicsCRM Proceedings & Lecture Notes15 (A.M.S., Providence, RI, 1998).
J. Li and G. TianVirtual moduli cycles and Gromov-Witten invariants of general symplectic manifoldsin [72], 47–83.
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].
G. C. LuThe Arnol’d conjecture for a product of weakly monotone manifoldsChinese J. Math.24 (1996), 145–157.
G. C. LuThe Arnol’d conjecture for a product of monotone manifolds and CalabiYao manifoldsActa Math. Sinica (N.S.)13 (1997), 381–388.
D. McDuff and L. V. PolterovichSymplectic packings and algebraic geometryInv. Math.115 (1994), 405–434.
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].
D. McDuff and D. A. SalamonIntroduction to Symplectic Topology(The Clarendon Press, Oxford Univ. Press, New York, 1995 [1st ed.], 1998 [2nd ed.]).
D. McDuffLectures on Gromov invariants for symplectic 4-manifoldsin [48], 175–210.
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.
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).
D. McDuffSymplectic structures-a new approach to geometryNotices Amer. Math. Soc.45 (1998), 952–960.
D. McDuffIntroduction to symplectic topologyin [28], 5–33.
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.
K. OnoOn the Arnol’d conjecture for weakly monotone symplectic manifoldsInv. Math.119 (1995), 519–537.
H. PoincaréSur un th¨¦or¨¨me de om¨¦trieRend. Circ. Mat. Palermo, 33 (1912), 375–407.
D. A. Salamon, Ed.Symplectic GeometryLondon Math. Soc. Lecture Note Series, 192 (Cambridge Univ. Press, Cambridge, 1993).
R. J. Stern, Ed.Topics in Symplectic 4-ManifoldsFirst Intern. Press Lecture Series, I (Intern. Press, Cambridge, MA, 1998).
C. H. TaubesThe Seiberg-Witten invariants and symplectic formsMath. Res. Lett., 1 (1994), 809–822.
C. H. TaubesThe Seiberg-Witten and Gromov invariantsMath. Res. Lett., 2 (1995), 221–238.
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].
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].
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].
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).
C. B. Thomas, Ed.Contact and Symplectic GeometryPubl. Newton Inst., 8 (Cambridge Univ. Press, Cambridge, 1996).
A. WeinsteinLectures on Symplectic ManifoldsCBMS Regional Conference Series in Math., 29 (A.M.S., Providence, RI, 1977).
V. M. Zakalyukin and O. M. MyasnichenkoLagrangian singularities in symplectic reductionFunctional Anal. Appl., 32 (1998), 1–9.
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Sevryuk, M.B. (2001). Symplectic and Contact Geometry and Hamiltonian Dynamics. In: Casacuberta, C., Miró-Roig, R.M., Verdera, J., Xambó-Descamps, S. (eds) European Congress of Mathematics. Progress in Mathematics, vol 202. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8266-8_49
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