A Very Brief Note about Symplectic Topology

Covering:
  1. Linear symplectic topology: non-squeezing, affine-rigidity, symplectic width/capacity, Maslov index
  2. Basics on persistence module: definitions, barcode(normal form theorem), isometry theorem,
  3. Floer homology: Hofer geometry, Conley-Zehnder index, (filtered) Hamilton-Floer homology
  4. Persistent homology: Hamiltonian persistence module, dynamical stability, symplectic persistence module
  5. Application: symplectic homology, topological stability, proof of a version of non-squeezing
  6. Basics on J-hol curves: definitions, unique continuation, Carleman similarity, critical points and simple curves
  7. Moduli space: Main theorem on moduli spaces, tools used for the theorem(elliptic regularity, transversality), compactness(removal of singularity, bubbling).
One may refer to various books on symplectic topology and J-hol curve by McDuff and arxiv 1904.04044(for persistence module part) by Zhang Jun and others.