Function theory on symplectic manifolds

Preface
Three wonders of symplectic geometry
First wonder: C0-rigidity
Second wonder: Arnold's conjecture
Mathematical model of classical mechanics
Fixed points of Hamiltonian diffeomorphisms
Third wonder: Hofer's metric
The group of Hamiltonian diffeomorphisms
Ham as a Lie group
Hofer's metric on Ham(M, )
A small scale in symplectic topology
The universal cover Ham"0365Ham(M,)
More examples: Kähler manifolds
J-holomorphic curves
Marsden-Weinstein reduction
C0-rigidity of the Poisson bracket
Rigidity of the Poisson bracket
The first proof (via Hofer's geodesics)
The second proof (via Hofer's displacement energy)
Rigidity of symplectomorphisms
Higher Poisson brackets
Quasi-morphisms
Homomorphisms up to a bounded error
Quasi-morphisms and irreversible dynamics
The Poincaré rotation number
The Maslov quasi-morphism
Quasi-morphisms and invariant pseudo-norms
Subadditive spectral invariants
The Calabi homomorphism
The action spectrum
Subadditive spectral invariants
Spectral width of a subset
Partial symplectic quasi-states
The Poisson bracket inequality
Two classes of subadditive spectral invariants
Calabi quasi-morphism
Symplectic quasi-states and quasi-measures
Symplectic quasi-states
Quasi-states and the quantum-classical correspondence
Topological quasi-states
Quasi-measures
Reduction of symplectic quasi-states
Lie quasi-states
Applications of partial symplectic quasi-states
Symplectic intersections
The non-displaceable fiber theorem
Superheavy subsets
Examples of Poisson commutative subspaces
Lagrangian knots
Applications to Hofer's geometry
Growth of one-parameter subgroups
Hofer's geometry of curves on surfaces
A Poisson bracket invariant of quadruples
An invariant of quadruples
Basic properties of pb4
pb4 and symplectic quasi-states
A dynamical interpretation of pb4
pb4 and deformations of the symplectic form
Topological preliminaries
A lower bound
Quadrilaterals on surfaces
An effect of stabilization
Symplectic approximation theory
The profile function
Behavior at zero
A lower bound via symplectic quasi-states
A lower bound via pb4
Geometry of covers and quantum noise
Prelude: covers vs. packings in symplectic topology
A Poisson bracket invariant of covers
The Berezin-Toeplitz quantization
Operational quantum mechanics
Preliminaries on POVMs
Naimark's dilation theorem
The noise operator
Uncertainty jump for joint measurements
Smearing of POVMs
Measuring noise
An unsharpness principle
Classical and quantum registration procedures
Geometry of overlaps and pb4
Preliminaries from Morse theory
Spectral numbers of functions
Morse homology
Canonical morphisms
Ring structure
Morse-Novikov homology
An overview of Floer theory
Spherically monotone symplectic manifolds
The least action principle
The Floer equation
The dimension of moduli spaces
Compactness breaking mechanism
The Floer complex
Ring structure
Constructing subadditive spectral invariants
Quantum homology
The Frobenius structure
Non-Archimedean geometry of `39`42`"613A``45`47`"603AQH
The PSS isomorphism
Spectral invariants in Floer theory
Subadditive spectral invariants revisited
Bibliography
Notation Index
Subject Index
Name Index