Lecture summaries

1/25 The syllabus.  Symplectic manifolds.  The standard symplectic form on R^{2n}.  The symplectic form on S^2.  Proof that S^{2n} for n > 1 does not admit a symplectic form.  What is known about the existence question for symplectic forms?  Symplectomorphisms.  Reference: https://people.math.ethz.ch/~acannas/Papers/lsg.pdf

1/30 Hamiltonian flows and symplectomorphisms.  The symplectomorphism group.  Cartan's magic formula.  Autonomous versus non-autonomous flows.  Situations where all symplectomorphisms are Hamiltonian.  Contact geometry.  Examples of contact manifolds.  What is known about the existence question for contact forms?  Reference: https://arxiv.org/pdf/1404.6157.pdf, https://people.math.ethz.ch/~acannas/Papers/lsg.pdf

2/1 Conservation of energy.  Conservation of volume.  Moser's trick.  Some remarks about time-varying vector fields.  Outline of the proof of Darboux's theorem.  Reference: https://people.math.ethz.ch/~acannas/Papers/lsg.pdf  

2/6 Symplectic linear algebra.  Symplectic and Lagrangian subspaces.  Every symplectic vector space has a symplectic basis.  The proof of Darboux's Theorem.  Darboux's Theorem in the contact case.  Reference: https://people.math.ethz.ch/~acannas/Papers/lsg.pdf  

2/8  Lagrangian, isotropic and co-isotropic submanifolds.  Hypersurfaces of contact type.  The Arnold conjecture.  "Everything is Lagrangian".  Lagrangian intersection phenomena.  The Weinstein conjecture.   Symplectic,  almost complex, and Riemannian geometry.  Pseudoholomorphic curves.  Reference: see the slides  I emailed to the class, or McDuff-Salamon

2/13 Polar decomposition and compatible triples.  Pseudoholomorphic curves: the question of invariance.   Contractibility. References:  https://ocw.mit.edu/courses/18-966-geometry-of-manifolds-spring-2007/ae6565bbfacae2e53361e06acc340213_lect08.pdf, https://ocw.mit.edu/courses/18-966-geometry-of-manifolds-spring-2007/resources/lect07/

2/15 Gromov's non-squeezing theorem, part 1.  Context and statement of the theorem.  Comparison with the case of volume preserving flows.  Symplectic rigidity.  Sketch of the proof.  Some facts about minimal surfaces.  A compactification trick.  References:  https://people.math.umass.edu/~wchen/705part2.pdf, https://web.math.princeton.edu/~rcabral/pdfs/minimalsurfaces.pdf, https://terrytao.wordpress.com/2009/03/29/mikhail-gromov-wins-2009-abel-prize/

2/20 Moduli space dreams.  Moduli space nightmares.  Compactness and transversality.  Reparametrization.  Bubbling.  Crash course in homology.  Reference: https://people.math.umass.edu/~wchen/705part2.pdf,

2/22 The moduli space of curves in S^2 \times T^{2n-2}, for a split almost complex structure.  More about transversality.  Somewhere injective versus multiply covered curves.  The implicit function theorem for Banach spaces.  Reference: https://people.math.umass.edu/~wchen/705part2.pdf,