Infinite Dimensional Morse Theory and Multiple Solution Problems

I: Infinite Dimensional Morse Theory.- 1. A Review of Algebraic Topology.- 2. A Review of the Banach-Finsler Manifold.- 3. Pseudo Gradient Vector Field and the Deformation Theorems.- 4. Critical Groups and Morse Type Numbers.- 5. Gromoll-Meyer Theory.- 6. Extensions of Morse Theory.- 6.1. Morse Theory Under General Boundary Conditions.- 6.2. Morse Theory on a Locally Convex Closed Set.- 7. Equivariant Morse Theory.- 7.1. Preliminaries.- 7.2. Equivariant Deformation.- 7.3. The Splitting Theorem and the Handle Body Theorem for Critical Manifolds.- 7.4. G-Cohomology and G-Critical Groups.- II: Critical Point Theory.- 1. Topological Link.- 2. Morse Indices of Minimax Critical Points.- 2.1. Link.- 2.2. Genus and Cogenus.- 3. Connections with Other Theories.- 3.1. Degree theory.- 3.2. Ljusternik-Schnirelman Theory.- 3.3. Relative Category.- 4. Invariant Functional.- 5. Some Abstract Critical Point Theorems.- 6. Perturbation Theory.- 6.1. Perturbation on Critical Manifolds.- 6.2. Uhlenbeck’s Perturbation Method.- III: Applications to Semilinear Elliptic Boundary Value Problems.- 1. Preliminaries.- 2. Superlinear Problems.- 3. Asymptotically Linear Problems.- 3.1. Nonresonance and Resonance with the Landesman-Lazer Condition.- 3.2. Strong Resonance.- 3.3. A Bifurcation Problem.- 3.4. Jumping Nonlinearities.- 3.5. Other Examples.- 4. Bounded Nonlinearities.- 4.1. Functional Bounded From Below.- 4.2. Oscillating Nonlinearity.- 4.3. Even Functional.- 4.4. Variational Inequalities.- IV: Multiple Periodic Solutions of Hamiltonian Systems.- 1. Asymptotically Linear Systems.- 2. Reductions and Periodic Nonlinearities.- 2.1. Saddle Point Reduction.- 2.2. A Multiple Solution Theorem.- 2.3. Periodic Nonlinearity.- 3. Singular Potentials.- 4. The Multiple Pendulum Equation.- 5. Some Results on Arnold Conjectures.- 5.1. Conjectures.- 5.2. The Fixed Point Conjecture on (T2n, ?0).- 5.3. Lagrange Intersections for (?Pn, ?Pn).- V: Applications to Harmonic Maps and Minimal Surfaces.- 1. Harmonic Maps and the Heat Flow.- 2. The Morse Inequalities.- 3. Morse Decomposition.- 4. The Existence and Multiplicity for Harmonic Maps.- 5. The Plateau Problem for Minimal Surfaces.- Appendix: Witten’s Proof of the Morse Inequalities.- 1. A Review of Hodge Theory.- 2. The Witten Complex.- 3. Weak Morse Inequalities.- 4. Morse Inequalities.- References.- Index of Notation.