Teaching

• Spring 2026, Introduction to symplectic geometry, USTC-IGP, Hefei.

  • Linear symplectic geometry: group Sp(2n), Maslov index, affine non-squeezing, statement of Gromov's monotonicity lemma in $\C^n$, sketch of the proof of Gromov's non-squeezing

  • Symplectic manifolds: Moser's trick, Darboux theorem, neighborhoods of symplectic and Lagrangian submanifolds

  • Symplectic maps: Weinstein chart, proof of Eliashberg-Gromov theorem via affine non-squeezing, statement of Flux conjectures

  • Hamiltonian dynamics: planar billiards and monotone twist maps, sketch of the proof of Poincaré-Birkhoff -theorem, statement of Franks theorem and HZ-conjecture, statement of Franks-Handle theorem and Conley conjecture

  • Morse theory: definition of Morse homology, persistence modules, proof of the existence of barcodes, quick review of Floer theory, idea of the proof for HZ-conjecture

  • Almost complex geometry: space of tamed complex structures, symplectic vector bundles and definition of first Chern number

  • Pseudo-holomorphic curves: Pluri-subharmonic functions, proof of Gromov's monotonicity lemma in \C^n, mean-value inequality, removable singularity theorem, minimal energy theorem, bubbling phenomenon and Gromov compactness

  • Gromov-Witten invariants and quantum homology: definitions and first examples

  • Floer theory revisited: proof of Gromov-Floer compactness, statements of gluing and transversality, definition of Floer homology, PSS isomorphism and spectral invariants, overview of some applications of spectral invariants to symplectic embeddings

  • References:

Introduction to quantitative symplectic geometry; by Zhang-Zhu.

Introduction to Symplectic topology; by McDuff-Salamon.

Here is the link to exercises: Exercises

Here is a more detailed plan of the course with exact citations for each section: Detailed Plan