Function theory on symplectic manifolds.

*(English)*Zbl 1310.53002
CRM Monograph Series 34. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-1693-5/hbk). 203 p. (2014).

The subject of this book is the study of function theory on symplectic manifolds. Symplectic geometry arose naturally in the context of classical mechanics, namely the cotangent bundle of a manifold \(M\) is a phase space of a mechanical system with the configuration space \(M\). Starting from 1980 symplectic topology developed exponentially with the introduction of powerful new methods, such as: Gromov’s theory of pseudo-holomorphic curves, Floer homology, Hofer’s metric on the group of Hamiltonian diffeomorphisms, Gromov-Witten invariants, symplectic field theory and the link to mirror symmetry. Rigidity phenomena have been investigated using these new methods. In this context, function spaces exhibit interesting properties and also provide a link to quantum mechanics.

This book is a monograph on function theory on symplectic manifolds as well as an introduction to symplectic topology. The first chapter introduces Eliashberg-Gromow \(C^0\)-rigidity theorem, Arnold’s conjecture on symplectic fixed points, Hofer’s geometry and \(J\)-holomorphic curves. Several facets of \(C^0\)-robustness of the Poisson bracket are investigated in various chapters of the book. The theory of symplectic quasi-states is described in Chapter five and the applications to symplectic intersections, Lagrangian knots and Hofer’s geometry are presented in Chapter six. The last three chapters describe an introduction to Floer homology.

This book is a monograph on function theory on symplectic manifolds as well as an introduction to symplectic topology. The first chapter introduces Eliashberg-Gromow \(C^0\)-rigidity theorem, Arnold’s conjecture on symplectic fixed points, Hofer’s geometry and \(J\)-holomorphic curves. Several facets of \(C^0\)-robustness of the Poisson bracket are investigated in various chapters of the book. The theory of symplectic quasi-states is described in Chapter five and the applications to symplectic intersections, Lagrangian knots and Hofer’s geometry are presented in Chapter six. The last three chapters describe an introduction to Floer homology.

Reviewer: Antonella Nannicini (Firenze)

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53D05 | Symplectic manifolds (general theory) |

53D17 | Poisson manifolds; Poisson groupoids and algebroids |

53D40 | Symplectic aspects of Floer homology and cohomology |

57R17 | Symplectic and contact topology in high or arbitrary dimension |

81S10 | Geometry and quantization, symplectic methods |

81P15 | Quantum measurement theory, state operations, state preparations |

53D42 | Symplectic field theory; contact homology |