Symplectic geometry and topology

This is a review of the ideas underlying the application of symplectic geometry to Hamiltonian systems. The paper begins with symplectic manifolds and their Lagrangian submanifolds, covers contact manifolds and their Legendrian submanifolds, and indicates the first steps of symplectic and contact topology.

[1]  Y. Eliashberg.  Classification of contact structures on ℝ 3 , 1993 .

[2]  V. A. Vasilʹev.  Lagrange and Legendre characteristic classes , 1988 .

[3]  I. Ekeland,et al.  Symplectic topology and Hamiltonian dynamics , 1989 .

[4]  Claude Viterbo,et al.  Symplectic topology as the geometry of generating functions , 1992 .

[5]  Alan Weinstein,et al.  Lectures on Symplectic Manifolds , 1977 .

[6]  Olli Lehto.  Proceedings of the International Congress of Mathematicians : Helsinki, 1978 , 1980 .

[7]  A. Weinstein,et al.  A symplectic fixed point theorem for complex projective spaces , 1985 .

[8]  V. Arnold.  Critical Points of Smooth Functions , 1975 .

[9]  A. Givental.  Nonlinear generalization of the Maslov index , 1990 .

[10]  M. Gromov.  Pseudo holomorphic curves in symplectic manifolds , 1985 .

[11]  H. Weyl.  The Classical Groups , 1940 .

[12]  F. Laudenbach,et al.  Persistance d'intersection avec la section nulle au cours d'une isotopie hamiltonienne dans un fibré cotangent , 1985 .

[13]  R. Wells.  The Mathematical Heritage of Hermann Weyl , 1988 .

[14]  L. Polterovich,et al.  Unknottedness of Lagrangian surfaces in symplectic 4-manifolds , 1993 .

[15]  Yakov Eliashberg,et al.  Contact 3-manifolds twenty years since J. Martinet's work , 1992 .

[16]  J. Martinet.  Formes de Contact sur les Variétés de Dimension 3 , 1971 .

[17]  I. Ekeland,et al.  Syplectic topology and Hamiltonian dynamics II , 1990 .

[18]  A. Floer,et al.  An instanton-invariant for 3-manifolds , 1988 .

[19]  V. Ginzburg.  Calculation of contact and symplectic cobordism groups , 1992 .

[20]  V. I. Arnol’d.  First steps in symplectic topology , 1986 .

[21]  C. Conley,et al.  The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold , 1983 .

[22]  A. Floer.  Proof of the Arnold conjecture for surfaces and generalizations to certain Kähler manifolds , 1986 .

[23]  J. Gray.  SOME GLOBAL PROPERTIES OF CONTACT STRUCTURES , 1959 .

[24]  A. Givental.  A symplectic fixed point theorem for toric manifolds , 1995 .

[25]  W. Neumann,et al.  Casson invariant of links of singularities , 1990 .

[26]  Focal set of a surface with boundary, and caustics of groups generated by reflections Bk, Ck, and F4 , 1984 .

[27]  Victor V. Batyrev,et al.  Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties , 1993, alg-geom/9310003.

[28]  S. D. Chatterji.  Proceedings of the International Congress of Mathematicians , 1995 .

[29]  A. Givental.  Singular Lagrangian manifolds and their Lagrangian maps , 1990 .

[30]  V. I. Arnol'd,et al.  Characteristic class entering in quantization conditions , 1967 .

[31]  W. A. Arvola.  The fundamental group of the complement of an arrangement of complex hyperplanes , 1992 .

[32]  O. Shcherbak.  Wavefronts and reflection groups , 1988 .

[33]  V. I. Arnol'd,et al.  Normal forms for functions near degenerate critical points, the Weyl groups of Ak, Dk, Ek and Lagrangian singularities , 1972 .

[34]  Serge Tabachnikov.  Around four vertices , 1990 .