نظریه تابع بر روی منیفولدهای سمپکتیک

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