Function theory on symplectic manifolds / Leonid Polterovich, Daniel Rosen

The subject of this book is the study of function theory on symplectic manifolds. Symplectic geometry arose naturally in the context of classical machanics, namely the cotangent bundle of a manifold M is a phase space of a machanical system with the configuration space M. Starting from 1980 Symplectic topology developed exponentially with the introduction of powerful new methods, such as: Gromow’s theory of pseudo-holomorphic curves, Floer homology, Hofer’s metric on the group of Hamiltonian diffeomorphisms, Gromow’s-Witten invariants, symplectic field theory and the link to Mirror Symmetry. Rigidity phenomena has ben investigated by 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 an introduction to symplectic topology. The first chapter introduces Eliashberg-Gromow C0 -rigidity theorem, Arnold’s conjecture on symplectic fixed points, Hofer’s geometry and J-holomorphic curves. Several facets of C0 -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. (Zentralblatt)

Bibliogr. p. 185-191. Index