目錄
Preface
Chapter 1 Symplectic Manifold
1.1 Symplectic Structure
1.2 Examples of Symplectic Manifolds
1.3 Almost Complex Structure
1.4 J-Holomorphic Curve (Part One)
1.5 Symplectomorphism
1.6 Lagrangian Submanifold
1.7 Hamiltonian Diffeomorphism
Chapter 2 Physics in Symplectic Topology
2.1 Lagrangian Mechanics
2.2 Hamiltonian Mechanics
2.3 Poisson Bracket
2.4 Group Action
2.5 Moment Map
2.6 Integrable System
2.7 Noether's Principle
2.8 Local Description
Chapter 3 Contact Geometry
3.1 Contact Structure
3.2 Contact Manifold
3.3 J-Holomorphic Curve (Part Two)
3.4 Legendrian Submanifolds
3.5 Contact Hamiltonian Dynamics
3.6 Contact Poisson Bracket
3.7 Orderability
Chapter 4 Moser's Argument
4.1 Moser's Trick
4.2 Stability Theorems
4.3 Neighborhood Theorems
Chapter 5 Floer Theory
5.1 Morse Theory
5.2 Persistence Module
5.3 Homological Rigidity
5.4 ∞-dimensional Morse Theory
5.5 Indices
Chapter 6 Quantitative Characterizations
6.1 Hofer's Geometry
6.2 Chaotic Hamiltonian Dynamics
6.3 Symplectic Homology
6.4 Symplectic Capacity
Chapter 7 Miscellaneous
7.1 Floer Continuation Map
7.2 Toric Domains
7.3 Shape Invariant
Chapter 8 Quantum Cohomology
8.1 Novikov Field (Ring)
8.2 Non-Archimedean Linear Algebra
8.3 Stable Map
8.4 Quantum Cohomology Group
8.5 Gromov-Witten Invariant
8.6 Gromov-Witten Potential
8.7 Back to Quantum Cohomology
Chapter 9 Hard Stories on Lagrangians
9.1 More Examples
9.2 A∞-algebra and Potential Function
9.3 Distance Between Lagrangians (Small Scale)
9.4 Distance Between Lagrangians (Large Scale)
9.5 Lagrangian Cobordism
9.6 Triangulated Category
9.7 Neck-Stretching
Chapter 10 A Different Language—Sheaf
10.1 Generating Function
10.2 A Crash Course on Sheaf
10.3 Sheaf Cohomology
10.4 Derived Category and Derived Functor
10.5 i-th Derived Functor And Its Computation
10.6 Singular Support (I)
10.7 Singular Support (II)
10.8 Arnold-Givental's Conjecture
Appendix Exams
I Midterm Exam (MATH6435P - USTC, Spring 2023)
II Final Exam (MATH6435P - USTC, Spring 2023)
III Midterm Exam (MATH6437P - USTC, Spring 2024)
IV Final Exam (MATH6437P - USTC, Spring 2024)
Bibliography
Index
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