Lectures on Symplectic Geometry

Symplectic Manifolds.- Symplectic Forms.- Symplectic Form on the Cotangent Bundle.- Symplectomorphisms.- Lagrangian Submanifolds.- Generating Functions.- Recurrence.- Local Forms.- Preparation for the Local Theory.- Moser Theorems.- Darboux-Moser-Weinstein Theory.- Weinstein Tubular Neighborhood Theorem.- Contact Manifolds.- Contact Forms.- Contact Dynamics.- Compatible Almost Complex Structures.- Almost Complex Structures.- Compatible Triples.- Dolbeault Theory.- Kahler Manifolds.- Complex Manifolds.- Kahler Forms.- Compact Kahler Manifolds.- Hamiltonian Mechanics.- Hamiltonian Vector Fields.- Variational Principles.- Legendre Transform.- Moment Maps.- Actions.- Hamiltonian Actions.- Symplectic Reduction.- The Marsden-Weinstein-Meyer Theorem.- Reduction.- Moment Maps Revisited.- Moment Map in Gauge Theory.- Existence and Uniqueness of Moment Maps.- Convexity.- Symplectic Toric Manifolds.- Classification of Symplectic Toric Manifolds.- Delzant Construction.- Duistermaat-Heckman Theorems.

[1]  Shlomo Sternberg,et al.  Convexity properties of the moment mapping. II , 1982 .

[2]  C. Viterbo.  A proof of Weinstein’s conjecture in ℝ 2n , 1987 .

[3]  On semifree symplectic circle actions with isolated fixed points , 1998, math/9812015.

[4]  Shiing-Shen Chern.  Complex Manifolds without Potential Theory: with an appendix on the geometry of characteristic classes , 1995 .

[5]  Two-dimensional gauge theories revisited , 1992, hep-th/9204083.

[6]  M. J. Gotay.  On coisotropic imbeddings of presymplectic manifolds , 1982 .

[7]  V. Guillemin,et al.  On the Unfolding of Folded Symplectic Structures , 2000 .

[8]  F. Pirani.  MATHEMATICAL METHODS OF CLASSICAL MECHANICS (Graduate Texts in Mathematics, 60) , 1982 .

[9]  Louis Nirenberg,et al.  Complex Analytic Coordinates in Almost Complex Manifolds , 1957 .

[10]  C. Thomas.  CONFOLIATIONS (University Lecture Series 13) , 1999 .

[11]  P. Seidel.  Lagrangian two-spheres can be symplectically knotted , 1998, math/9803083.

[12]  S. Donaldson.  Geometry of four-manifolds , 1990 .

[13]  S. Sternberg,et al.  Convexity properties of the moment mapping , 1982 .

[14]  Y. Eliashberg.  Classification of overtwisted contact structures on 3-manifolds , 1989 .

[15]  M. Hutchings.  Taubes’s proof of the Weinstein conjecture in dimension three , 2009, 0906.2444.

[16]  S. Donaldson.  Symplectic submanifolds and almost-complex geometry , 1996 .

[17]  C. Taubes.  More constraints on Symplectic forms from Seiberg-Witten invariants , 1995 .

[18]  H. Weyl.  The Classical Groups , 1939 .

[19]  Antonio Ambrosetti,et al.  Nonlinear Analysis and Semilinear Elliptic Problems , 2007 .

[20]  P. Scott,et al.  The geometries of 3-manifolds , 1983 .

[21]  C. Villani.  Topics in Optimal Transportation , 2003 .

[22]  Michèle Audin,et al.  Spinning Tops: A Course on Integrable Systems , 1996 .

[23]  A. Weinstein.  Symplectic manifolds and their lagrangian submanifolds , 1971 .

[24]  L. C. Jeffrey,et al.  Localization for nonabelian group actions , 1993 .

[25]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[26]  Michael Atiyah,et al.  Convexity and Commuting Hamiltonians , 1982 .

[27]  J. Duistermaat.  The heat kernel Lefschetz fixed point formula for the spin-c dirac operator , 1996 .

[28]  S. Donaldson.  Lefschetz Fibrations in Symplectic Geometry , 1998 .

[29]  C. Taubes.  The Seiberg-Witten and Gromov invariants , 1995 .

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

[31]  E. Lerman,et al.  Stratified symplectic spaces and reduction , 1991 .

[32]  Victor Guillemin,et al.  Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces , 1994 .

[33]  T. Delzant,et al.  Hamiltoniens périodiques et images convexes de l'application moment , 1988 .

[34]  P. Newstead.  Moduli Spaces and Vector Bundles: Geometric Invariant Theory , 2009 .

[35]  Claude Viterbo,et al.  An introduction to symplectic topology , 1991 .

[36]  Christophe Golé,et al.  SYMPLECTIC TWIST MAPS , 2001 .

[37]  S. Sternberg,et al.  Symplectic Techniques in Physics , 1984 .

[38]  Michael Atiyah,et al.  The moment map and equivariant cohomology , 1984 .

[39]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

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

[41]  S. Sternberg,et al.  Supersymmetry and equivariant de Rham theory , 1999 .

[42]  M. Chaperon.  On generating families , 1995 .

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

[44]  Non-abelian convexity by symplectic cuts , 1996, dg-ga/9603015.

[45]  C. Woodward,et al.  Spherical Varieties and Existence of Invariant K Ahler Structure , 2022 .

[46]  V. Guillemin,et al.  The relation between compact and non-compact equivariant cobordisms , 1998, math/9810101.

[47]  M. Vergne,et al.  Zeros d’un champ de vecteurs et classes caracteristiques equivariantes , 1983 .

[48]  Shaun Martin.  Symplectic quotients by a nonabelian group and by its maximal torus , 2000, math/0001002.

[49]  J. Duistermaat.  Equivariant cohomology and stationary phase , 1993 .

[50]  Transversality theory, cobordisms, and invariants of symplectic quotients , 2000, math/0001001.

[51]  H. Geiges.  Applications of contact surgery , 1997 .

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

[53]  Sergei Petrovich Novikov,et al.  Dynamical Systems IV: Symplectic Geometry and its Applications , 2010 .

[54]  S. Donaldson.  Lefschetz pencils on symplectic manifolds , 1999 .

[55]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[56]  Alberto Abbondandolo.  Morse Theory for Hamiltonian Systems , 2001 .

[57]  Kenneth R. Meyer,et al.  Symmetries and Integrals in Mechanics , 1973 .

[58]  H. Geiges,et al.  Contact geometry and complex surfaces , 1995 .

[59]  L. Hörmander,et al.  An introduction to complex analysis in several variables , 1973 .

[60]  I. Ekeland.  Convexity Methods In Hamiltonian Mechanics , 1990 .

[61]  J. Duistermaat,et al.  On the variation in the cohomology of the symplectic form of the reduced phase space , 1982 .

[62]  Raoul Bott,et al.  The Yang-Mills equations over Riemann surfaces , 1983, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[63]  H. Hofer,et al.  The weinstein conjecture in the presence of holomorphic spheres , 1992 .

[64]  Robert E. Gompf.  A new construction of symplectic manifolds , 1995 .

[65]  K. Kodaira.  ON THE STRUCTURE OF COMPACT COMPLEX ANALYTIC SURFACES. , 1964, Proceedings of the National Academy of Sciences of the United States of America.

[66]  C. Woodward.  Multiplicity-free Hamiltonian actions need not be Kähler , 1995, dg-ga/9506009.

[67]  Alan Weinstein,et al.  Geometric Models for Noncommutative Algebras , 1999 .

[68]  G. E. Bredon.  Introduction to compact transformation groups , 1972 .

[69]  C. Woodward,et al.  Hamiltonian loop group actions and Verlinde factorization , 1998 .

[70]  Allen Knutson,et al.  The cohomology ring of polygon spaces , 1997, dg-ga/9706003.

[71]  M. Gromov.  Soft and hard symplectic geometry , 1986 .

[72]  L. Polterovich,et al.  Symplectic diffeomorphisms as isometries of Hofer's norm , 1997 .

[73]  Emmanuel Giroux.  Topologie de contact en dimension 3 , 1993 .

[74]  F. Schlenk.  Embedding Problems in Symplectic Geometry , 2005 .

[75]  J. Marsden,et al.  Reduction of symplectic manifolds with symmetry , 1974 .

[76]  Kai Cieliebak,et al.  Symplectic Geometry , 1992, New Spaces in Physics.

[77]  Emmanuel Giroux.  Une structure de contact, même tendue, est plus ou moins tordue , 1994 .

[78]  I. Sata.  ON A GENERALIZATION OF THE NOTION OF MANIFOLD BY , 2004 .

[79]  P. Biran.  A stability property of symplectic packing , 1999 .

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

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

[82]  J. Brasselet.  Introduction to toric varieties , 2004 .

[83]  K. Kodaira.  ON THE STRUCTURE OF COMPACT COMPLEX ANALYTIC SURFACES, II. , 1963 .

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

[85]  J. Moser.  On the volume elements on a manifold , 1965 .

[86]  A. Gray,et al.  Compact parallelizable four-dimensional symplectic and complex manifolds , 1988 .

[87]  Johannes J. Duistermaat,et al.  On global action‐angle coordinates , 1980 .

[88]  Loring W. Tu,et al.  Differential forms in algebraic topology , 1982, Graduate texts in mathematics.

[89]  J. Latschev,et al.  Quantitative symplectic geometry , 2005 .

[90]  Ivar Ekeland,et al.  Selected new aspects of the calculus of variations in the large , 2002 .

[91]  Michèle Vergne,et al.  Heat Kernels and Dirac Operators: Grundlehren 298 , 1992 .

[92]  D. Auroux.  Asymptotically Holomorphic Families of Symplectic Submanifolds , 1997 .

[93]  C. Taubes.  THE SEIBERG-WITTEN INVARIANTS AND SYMPLECTIC FORMS , 1994 .

[94]  P. Kronheimer,et al.  Monopoles and contact structures , 1997 .

[95]  F. Kirwan.  Cohomology of Quotients in Symplectic and Algebraic Geometry. (MN-31), Volume 31 , 1984 .

[96]  D. Salamon.  Morse theory, the Conley index and Floer homology , 1990 .

[97]  A. Weinstein.  On the hypotheses of Rabinowitz' periodic orbit theorems , 1979 .

[98]  Jerrold E. Marsden,et al.  Foundations of Mechanics, Second Edition , 1987 .

[99]  C. Thomas.  INTRODUCTION TO SYMPLECTIC TOPOLOGY (Oxford Mathematical Monographs) , 1997 .

[100]  A. Weinstein.  Neighborhood classification of isotropic embeddings , 1981 .

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

[102]  H. Hofer.  Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three , 1993 .

[103]  D. Mcduff.  The local behaviour of holomorphic curves in almost complex 4-manifolds , 1991 .

[104]  D. Mcduff.  Examples of simply-connected symplectic non-Kählerian manifolds , 1984 .

[105]  W. Thurston.  Some simple examples of symplectic manifolds , 1976 .

[106]  Dusa McDuff,et al.  The geometry of symplectic energy , 1993 .

[107]  Eduard Zehnder,et al.  Symplectic Invariants and Hamiltonian Dynamics , 1994 .

[108]  Symplectic Geometry: An Introduction based on the Seminar in Bern, 1992 , 1994 .

[109]  M. Audin.  The Topology of Torus Actions on Symplectic Manifolds , 1991 .

[110]  R. S. Ward,et al.  Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces , 1999 .

[111]  Jerrold E. Marsden,et al.  Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems , 1999 .