Teaching

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 (Plurisubharmonic 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.

The principle of least action in geometry and dynamics; by K.F. Siburg.

The Maslov index for paths, by Robbin and Salamon.

Area preserving homeomorphisms of open surfaces of genus zero, by Franks.

Periodic points of Hamiltonian surface diffeomorphisms, by Franks-Handel.

J-holomorphic curves and symplectic topology, by McDuff-Salamon.

Computing persistent homology, by Zomorodian and Carlsson.

TBA

Here are the exercises: the final document will be uploaded here at the end of the semester.

Here is a more detailed plan of the course with exact citations for each section: the final document will be uploaded here at the end of the semester.