Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology

Preface.Contributors. Lectures on the Morse Complex for Infinite-Dimensional Manifolds.-1. A few facts from hyperbolic dynamics.-1.1 Adapted norms .-1.2 Linear stable and unstable spaces of an asymptotically hyperbolic path.-1.3 Morse vector fields.- 1.4 Local dynamics near a hyperbolic rest point ; 1.5 Local stable and unstable manifolds.- 1.6 The Grobman - Hartman linearization theorem.-1.7 Global stable and unstable manifolds.- 2 The Morse complex in the case of finite Morse indices.- 2.1 The Palais - Smale condition.-2.2 The Morse - Smale condition .-2.3 The assumptions .- 2.4 Forward compactness.- 2.5 Consequences of compactness and transversality.- 2.6 Cellular filtrations.- 2.7 The Morse complex.- 2.8 Representation of $\delta$ _* in terms of intersection numbers.- 2.9 How to remove the assumption (A8).- 2.10 Morse functions on Hilbert manifolds.-2.11 Basic results in transversality theory .- 2.12 Genericity of the Morse - Smale condition.-2.13 Invariance of the Morse complex.- 3 The Morse complex in the case of infinite Morse indices.- 3.1 The program.-3.2 Fredholm pairs and compact perturbations of linear subspaces.- 3.3 Finite-dimensional intersections.-3.4 Essential subbundles.- 3.5 Orientations.- 3.6 Compactness .- 3.7 Two-dimensional intersections .-3.8 The Morse complex.- Bibliographical note.- Notes on Floer Homology and Loop Space Homology.- 1 Introduction.- 2 Main result.-2.1 Loop space homology.-2.2 Floer homology for the cotangent bundle.- 3 Ring structures and ring-homomorphisms.-3.1 The pair-of-pants product.- 3.2 The ring homomorphisms between free loop space Floer homology and based loop space Floer homology and classical homology.-4 Morse-homology on the loop spaces $\Lambda$Q and $\Omega$Q, and the isomorphism.-5 Products in Morse-homology .-5.1 Ring isomorphism between Morse homology and Floer homology.- Homotopical Dynamics in Symplectic Topology.- 1 Introduction .-2 Elements of Morse theory .-2.1 Connectingmanifolds.-2.2 Operations.-3 Applications to symplectic topology.- 3.1 Bounded orbits .-3.2 Detection of pseudoholomorphic strips and Hofer's norm.- Morse Theory, Graphs, and String Topology.-1 Graphs, Morse theory, and cohomology operations.-2 String topology .-3 A Morse theoretic view of string topology.- 4 Cylindrical holomorphic curves in the cotangent bundle.- Topology of Robot Motion Planning.-1.Introduction .-2 First examples of configuration spaces .-3 Varieties of polygonal linkages.-3.1 Short and long subsets .-3.2 Poincaré polynomial of M(a) .-4 Universality theorems for configuration spaces .-5 A remark about configuration spaces in robotics .-6 The motion planning problem.-7 Tame motion planning algorithms.-8 The Schwarz genus.- 9 The second notion of topological complexity.-10 Homotopy invariance.- 11 Order of instability of a motion planning algorithm.-12 Random motion planning algorithms.- 13 Equality theorem.-14 An upper bound for TC(X).-15 A cohomological lower bound for TC(X) .-16 Examples .-17 Simultaneous control of many systems.-18 Another inequality relating TC(X) to the usual category .-19 Topological complexity of bouquets.-20 A general recipe to construct a motion planning algorithm.-21 How difficult is to avoid collisions in $\mathbb{R}$ ^m ? .-22 The case m = 2.- 23 TC(F($\mathbb{R}$ ^m ; n) in the case m $\geq$ 3 odd .- 24 Shade.-25 Illuminating the complement of the braid arrangement .-26 A quadratic motion planning algorithm in F($\mathbb{R}$ ^m ; n).-27 Configuration spaces of graphs.-28 Motion planning in projective spaces .-29 Nonsingular maps.- 30 TC(($\mathbb{R}$P ⁿ ) and the immersion problem.-31 Some open problems.- Application of Floer Homology of Langrangian Submanifolds to Symplectic Topology.- 1 Introduction.- 2 Lagrangian submanifold of $\mathbb{C}$ ⁿ .-3 Perturbing Cauchy - Riemann equation.- 4 Maslov index of Lagrangian submanifold with vanishing second Betti number.-5Floer homology and a spectral sequence .-6 Homology of loop space and Chas - Sullivan bracket .-7 Iterated integral and Gerstenhaber bracket.- 8 A$_\infty$ deformation of de Rham complex.- 9 S ¹ equivariant homology of loop space and cyclic A1 algebra .-10 L$_\infty$ structure on H(S ¹ $\times$ S ⁿ ; $\mathbb{Q}$).-11 Lagrangian submanifolds of $\mathbb{C}$ ³ .-12 Aspherical Lagrangian submanifolds .-13 Lagrangian submanifolds homotopy equivalent to S ¹ $\times$ S ^2m .-14 Lagrangian submanifolds of $\mathbb{C}$P ⁿ .- The $\mathcal{LS}$-Index: A Survey.- 1 Introduction .-2 The $\mathcal{LS}$-index.-2.1 Basic definitions and facts.-2.2 Spectra .-2.3 The $\mathcal{LS}$-index .- 3 Cohomology of spectra .-4 Attractors, repellers and Morse decompositions .- 5 Equivariant $\mathcal{LS}$-flows and the G-$\mathcal{LS}$-index.-5.1 Symmetries.-5.2 Isolating neighbourhoods and the equivariant $\mathcal{LS}$-index .-6 Applications.-6.1 A general setting .-6.2 Applications of the $\mathcal{LS}$-index .-6.3 Applications of the cohomological $\mathcal{LS}$-index .-6.4 Applications of the equivariant LS-index.- Lectures on Floer Theory and Spectral Invariants of Hamiltonian Flows.- 1 Introduction .-2 The free loop space and the action functional.-2.1 The free loop space and the S ¹ -action in general.-2.2 The free loop space of symplectic manifolds.-2.3 The Novikov covering.-2.4 Perturbed action functionals and their action spectra.-2.5 The L2-gradient flow and perturbed Cauchy - Riemann equations.-2.6 Comparison of two Cauchy - Riemann equations.-3 Floer complex and the Novikov ring.-3.1 Novikov - Floer chains and the Novikov ring.-3.2 Definition of the Floer boundary map.-3.3 Definition of the Floer chain map.-3.4 Semi-positivity and transversality.-3.5 Composition law of Floer's chain maps.-4 Energy estimates and Hofer's geometry.- 4.1 Energy estimates and the action level changes.-4.2Energy estimates and Hofer's norm.-4.3 Level changes of Floer chains under the homotopy .-4.4 The $\epsilon$-regularity type invariants .-5 Definition of spectral invariants and their axioms.-5.1 Floer complex of a small Morse function.-5.2 Definition of spectral invariants.-5.3 Axioms of spectral invariants.-6 The spectrality axiom.-6.1 A consequence of the nondegenerate spectrality axiom.-6.2 Spectrality axiom for the rational case.-6.3 Spectrality for the irrational case.-7 Pants product and the triangle inequality.-7.1 Quantum cohomology in the chain level.-7.2 Grading convention.-7.3 Hamiltonian fibrations and the pants product .- 7.4 Proof of the triangle inequality.-8 Spectral norm of Hamiltonian diffeomorphisms.-8.1 Construction of the spectral norm.-8.2 The $\epsilon$-regularity theorem and its consequences.-8.3 Proof of nondegeneracy.-9 Applications to Hofer geometry of Ham(M;$\omega$).-9.1 Quasi-autonomous Hamiltonians and the minimality conjecture.- 9.2 Length minimizing criterion via $\rho$(H; 1).-9.3 Canonical fundamental Floer cycles.-9.4 The case of autonomous Hamiltonians.-10 Remarks on the transversality for general (M;$omega$).- A Proof of the index formula.- Floer Homology, Dynamics and Groups.-1 Hamiltonian actions of finitely generated groups.-1.1 The group of Hamiltonian diffeomorphisms.-1.2 The no-torsion theorem.-1.3 Distortion in normed groups .-1.4 The No-Distortion Theorem.-1.5 The Zimmer program.- 2 Floer theory in action.-2.1 A brief sketch of Floer theory .-2.2 Width and torsion.-2.3 A geometry on Ham(M;$\omega$).-2.4 Width and distortion.-2.5 More remarks on the Zimmer program.-3 The Calabi quasi-morphism and related topics.-3.1 Extending the Calabi homomorphism.-3.2 Introducing quasi-morphisms.-3.3 Quasi-morphisms on Ham(M;$\omega$).-3.4 Distortion in Hofer's norm on Ham(M;$\omega$).- 3.5 Existence and uniqueness of Calabi quasi-morphisms.-3.6 "Hyperbolic" features of Ham(M;$\omega$)? .-3.7 From $\pi$ ₁ (M) toDiff ₀ (M;$\Omega$).- Symplectic topology and Hamilton - Jacobi equations.- 1 Introduction to symplectic geometry and generating functions.-1.1 Uniqueness and first symplectic invariants.-2 The calculus of critical level sets.-2.1 The case of GFQI.-2.2 Applications.-3 Hamilton - Jacobi equations and generating functions.-4 Coupled Hamilton - Jacobi equations.-Index.