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,

2/27 Transversality for somewhere injective  curves.  The formula for the dimension of the moduli space.  Why  don't we expect  transversality for multiply covered curves, for generic J? Reference: https://people.math.umass.edu/~wchen/705part2.pdf,

2/29 The proof of Gromov non-squeezing (modulo various details).  All the relevant curves are somewhere injective.  Bubbling can not occur.  The energy identity.  Reference: https://people.math.umass.edu/~wchen/705part2.pdf,

3/5 Definition of the Gromov-Witten invariant (in the case of Gromov's non-squeezing theorem).  Why is this well-defined?  The computation in our case gives 1.  Reference: https://people.math.umass.edu/~wchen/705part2.pdf,

3/7 Completion of the proof of Gromov non-squeezing (modulo various details).  More details, 1: Gromov compactness.  Reference: https://arxiv.org/pdf/1011.1690.pdf

3/12 More details about bubbling.  Banach spaces and Banach manifolds.  Reference: https://arxiv.org/pdf/1011.1690.pdf

3/14 More details about transversality.  The implicit function theorem for Banach spaces and the Sard-Smale theorem.  Reference: The appendix of McDuff-Salamon.  https://arxiv.org/pdf/1011.1690.pdf

Spring break!

3/26 Statement of the Weinstein conjecture.  Survey of some known results.    Floer theoretic approaches.  First attempt at a contact homology chain complex.  Reference: https://cpb-us-e1.wpmucdn.com/sites.ucsc.edu/dist/0/521/files/2017/11/contact_homology_notes-16nnaor.pdf

3/28 The expected dimension of the moduli space of cylinders.  The Conley-Zehnder index.  Reference: Reference: https://cpb-us-e1.wpmucdn.com/sites.ucsc.edu/dist/0/521/files/2017/11/contact_homology_notes-16nnaor.pdf

4/2 The first Chern class of a complex line bundle and its significance.  More about the Conley-Zehnder index.  Grading the chain complex by the Conley-Zehnder index.  Reference: https://cpb-us-e1.wpmucdn.com/sites.ucsc.edu/dist/0/521/files/2017/11/contact_homology_notes-16nnaor.pdf

4/4 SFT compactness.  Possible breakings of index 1 cylinders.  Convex and dynamically convex contact forms.  Reference: https://cpb-us-e1.wpmucdn.com/sites.ucsc.edu/dist/0/521/files/2017/11/contact_homology_notes-16nnaor.pdf

4/9 Why is the differential well-defined?  Reference: https://cpb-us-e1.wpmucdn.com/sites.ucsc.edu/dist/0/521/files/2017/11/contact_homology_notes-16nnaor.pdf

4/11 Gluing.  Reference: https://cpb-us-e1.wpmucdn.com/sites.ucsc.edu/dist/0/521/files/2017/11/contact_homology_notes-16nnaor.pdf  

4/16 Introduction to embedded contact homology (ECH).  Key properties.  The ECH index.  Reference: https://arxiv.org/pdf/1303.5789.pdf

4/18 More about the ECH index.  The index inequality.  The existence of two Reeb orbits (statement).  ECH spectral invariants.   Reference: https://arxiv.org/pdf/1303.5789.pdf

4/23 The U-map.  The ECH of S^3.  Proof of the existence of two Reeb orbits.  References: https://arxiv.org/abs/1202.4839, https://arxiv.org/pdf/1303.5789.pdf

4/25 More details about the proof of the existence of two Reeb orbits.  ECH spectral invariants.  More details about the proof of the ECH Weyl law.   References: https://arxiv.org/abs/1202.4839, https://arxiv.org/abs/1005.2260

4/30 More about the Weyl law.  Monotocity and spectral invariants.  Reference: https://arxiv.org/abs/1005.2260

5/2 Guest lecture on mirror symmetry (Thanks, Dan Pomerleano!)

5/7 Applications of the Weyl law, 1: Closing lemmas.  Reference: https://arxiv.org/abs/1508.07542, https://arxiv.org/abs/2110.02925, https://arxiv.org/abs/2110.02463

5/9 Applications of the Weyl law, 2: The Simplicity conjecture.  Reference: https://arxiv.org/abs/2001.01792,