1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

xii CONTENTS

• Lecture 3. The Weinstein Conjecture in the Overtwisted Case
  
  
  • An Explicit Local Bishop Family
  
  
  • The Implicit Function Theorem near an Embedded Disk
  
  
  • A Short Summary and the Prolongation of the Bishop Family
  
  
  
  


• Lecture 4. The Weinstein Conjecture in the Tight Case
  
  
  • Sketch of the Proof
  
  
  • Global Uniqueness for Families of Pseudoholomorphic Disks
  
  
  
  


• Lecture 5. Some Outlook


• Bibliography


• the Seiberg-Witten Equations on Symplectic Manifolds Michael Hutchings and Clifford Henry Taubes, An Introduction to
  
  
  • Introduction
  
  
  
  


• Lecture 1. Background from Differential Geometry
  
  
  • Vector Bundles
  
  
  • Connections
  
  
  
  


• Lecture 2. Spin and the Seiberg-Witten Equations l Self-Dual Two-Forms lll
  
  
  • Principal Bundles and Associated Bundles l
  
  
  • Spine Structures l
  
  
  • Some Group Theory l
  
  
  • The Spinor Bundles l
  
  
  • Clifford Multiplication l
  
  
  • The Spin Connection
  
  
  • The Dirac Operator
  
  
  • The Seiberg-Witten Equations l
  
  
  
  


• Lecture 3. The Seiberg-Witten Invariants
  
  
  • Gauge Transformations
  
  
  • Basic Properties of the Moduli Space
  
  
  • Why t he Conditions on b~?
  
  
  • Outline of the Proof of Compactness
  
  
  • The Seiberg-Witten Invariant
  
  
  • Examples and Applications
  
  
  
  


• Lecture 4. The Symplectic Case, Part I
  
  
  • Statement of the Theorem
    
    
    • The Canonical Spine Structure
    
    
    • Step 1: Understanding the Dirac Equation
    
    
    
    
  
  
  • Step 2: Deforming the Curvature Equation
  
  
  • Step 3: Uniqueness of the Solution
  
  
  • Appendix: An Estimate on Beta
  
  
  
  


• Lecture 5. The Symplectic Case, Part II
  
  
  • Summary of t he Last Lecture
    
    
    • Rational Gromov-Witten Invariants
    
    
    • Rational Floer Homology
    
    
    
    
  
  
  • Bibliography
  
  
  • Alexander G ivental, A Tutorial on Quantum Cohomology
    
    
    • Introduction
    
    
    
    
  
  
  • Lecture l. Moduli Spaces of Stable Maps
    
    
    • Example: Quantum Cohomology of Complex Projective Spaces
    
    
    • Stable Maps
    
    
    
    
  
  
  • Lecture 2. Gromov-Witten Invariants
  
  
  • Lecture 3. QH*(G/B) and Quantum Toda Lattices
  
  
  • Lecture 4. Singularity Theory
  
  
  • Lecture 5. Toda Lattices and the Mirror Conjecture
  
  
  • Bibliography
  
  
  • and Lagrangian Intersections Mikhail Grinberg and Robert MacPherson, Euler Characteristics
    
    
    • Introduction
    
    
    
    
  
  
  • Lecture l.
    
    
    • The Centerpiece Theorem
    
    
    • Stratifications
      
      
      • Transversality
      
      
      • The Euler Characteristic of a Constructible Function
      
      
      • Homology n-Product of Cycles
      
      
      • The Characteristic Cycle in Dimension One
      
      
      
      
    
    
    
    
  
  
  • Lecture 2.
    
    
    • The Conormal Variety to a Stratification
    
    
    • Whitney Conditions
      
      
      • Generic Covectors
      
      
      
      
    
    
    • The Half-Link
    
    
    • The Characteristic Cycle
    
    
    • Signs
    
    
    
    
  
  
  
  


• Lecture 3.
  
  
  • Classical Morse Theory
  
  
  • Stratified Morse Theory
  
  
  • Comments
  
  
  
  


• Lecture 4.
  
  
  • Standard Pairs
  
  
  • Proof of Theorem 1.1: The General Case
  
  
  • Fary Functors
  
  
  • Lecture 5. CONTENTS xi
    
    
    • Fary Functors: Comments and Examples
    
    
    • Monodromy
    
    
    • Euler Characteristics
    
    
    • Poincare-Verdier Duality
    
    
    • Morse Local Systems
    
    
    • Perverse Sheaves
    
    
    
    
  
  
  • Bibliography
  
  
  • Reduction Lisa C. Jeffrey, Hamiltonian Group Actions and Symplectic
  
  
  • Lecture l. Introduction to Hamiltonian Group Actions
    
    
    • Some Elementary Properties of Moment Maps
    
    
    • The Symplectic Quotient
    
    
    • The Normal Form Theorem
    
    
    
    
  
  
  • Lecture 2. The Geometry of the Moment Map
    
    
    • Convexity Theorems
    
    
    • The Moment Polytope
    
    
    • The Duistermaat-Heckman Theorem, Version I
    
    
    • Operations on Moment Polytopes
    
    
    • Symplectic Cutting
    
    
    • Torie Manifolds
    
    
    
    
  
  
  
  


• Lecture 3. Equivariant Cohomology and the Cartan Model
  
  
  • The Cartan Model
  
  
  • Equivariant Characteristic Classes
  
  
  • The Abelian Localization Theorem
    
    
    • Cohomology of Symplectic Quotients Lecture 4. The Duistermaat-Heckman Theorem and Applications to the
    
    
    
    
  
  
  • Stationary Phase Approximation
  
  
  • The Natural Map r;,: H(;(M)----> H *(Mred)
  
  
  • Remarks on Quantization and Representation Theory
  
  
  • Nonabelian Localization
  
  
  
  


• Lecture 5. Moduli Spaces of Vector Bundles over Riemann Surfaces
  
  
  • Prototype: The Jacobian
  
  
  • The Jacobian As an Infinite Dimensional Symplectic Quotient
  
  
  • The Moduli Space of Flat Connections on a Riemann Surface
  
  
  • The Line Bundle over t he Moduli Space of Flat Connections
  
  
  
  


• Exercises
  
  
  • 1 and 2) Hamiltonian group actions and symplectic reduction: Exercise 1 (Lectures
  
  
  • Exercise 2 (Lectures 3-5)
  
  
  • Bibliography
  
  
  
  


• and Symmetry Jerrold E. Marsden, Park City Lectures on Mechanics, Dynamics,
  
  
  • Introduction
  
  
  • Lecture 1. Reduction for Mechanical Systems with Symmetry
  
  
  • Lecture 2. Stability, Underwater Vehicle Dynamics and Phases
  
  
  • Lecture 3. Systems with Rolling Constraints and Locomotion
  
  
  • Lecture 4. Optimal Control and Stabilization of Balance Systems
  
  
  • Lecture 5. Variational Integrators
  
  
  • Bibliography