| Date | Topic | Main References | Further reading |
|---|---|---|---|
| September | Overview, Morse theory, Hamiltonian mechanics, phase space, symplectic manifolds and submanifolds, symplectic connections, neighborhood theorems. |
[McDuff-Salamon] Chapter 1-3, [Arnold] Chapter 8-9. | |
| Oct 3 | Action-angle coordinates, invariant tori; Integrable systems. |
[Arnold] Chapter 10. | Sardanashvily, Integrable Hamiltonian Systems |
| Oct 5 | Fubini-Study form, and Lagrangians in CP^n. | [McDuff-Salamon] 3rd version pp 166-170. | Chiang, Nonstandard Lagrangian in CP^n. |
| Oct 10, | Billards, Poincare-Birkhoff Theorem, Arnold's Conjecture. Periodic orbits as closed geodesics of Riemannian or Finsler metrics. |
And de Sliva's lecture pp 26-35. [McDuff-Salamon] Chapter 8 and 11 |
A symplectic proof of Franks theorem Cristofaro-Gardiner-Hutchings-Pomerleano, 2 or infinity Reeb orbits. |
| Oct 15 | Fall Break | Fall Break | Fall Break |
| Oct 17 | Lie group action/toric manifolds; Atiyah-Guillemin-Sternberg convexity theorem |
[McDuff-Salamon] Chapter 5 | Lerman, Symplectic cuts Non-abelian convexity by symplectic cuts |
| Oct 22 | Symplectic reduction (finite dimensional case), co-adjoint representation |
[McDuff-Salamon] Chapter 5 [Kirillov] Chapter 1 and 3 |
Woodward, Moment maps and GIT |
| Oct 24 | Donaldson's infinite-dimensional moment maps,
space of Kahler metrics, extremal Kahler metrics as critical points of the Calabi functional. |
Donaldson, Moment maps and diffeomrophism. | Xiuxiong Chen, Song Sun, Calabi flow, uniqueness of cscK metrics Compatible complex structures and Ham(X,\omega). |
| Oct 29 | Intro to symplectic embeddings and capaticites | [McDuff-Salamon] Chapter 12 | Quantitative symplectic geometry |
| Oct 31 |
Gromov compactness theorem of pseudo-holomorphic curves, a very short introduction to Gromov-Witten invariants, a sketch proof of Gromov's non-queezing Theorem |
[Audin-Lafontaine] | Gromov's 1985 paper (note: a bit sketchy) Intro to Kontsevich's formula on CP^2. Kyler Seigel's talk at Geometry seminar Nov 1st. |
| Nov 5 | Symplectic vesus complex blowup/down, symplectic cut; Kodaira-Thurston's fibration constructions; non-Kahler symplectic manifolds |
[McDuff-Salamon] Chapter 6 | Hodge theory and cohomology on non-Kahler symplectic manifolds. |
| Nov 7 | Rational elliptic surfaces, Gompf's fiber sum, K3 and elliptic fibration;
any finite presented group can be $\pi_1$ of symplectic manifolds. |
Gompf,A New Construction of Symplectic Manifolds | Quantization on E(1) and K3's. |
| Nov 12 | Stein/Weinstein domain, comparison bw complex/sympelctic convexity. |
Eliashberg, Recent advances in symplectic flexibility. Section 4-5 | From Stein to Weinstein and Back |
| Nov 14 | Contact 3-manifolds and Legendrian knots; front projection and Lagrangian projection; contact surgery Thurston-Bennequin & rotation number detect unknot in tight M^3 |
Etnyre, Legendrian and transverse knots | Thurston's work on Foliations Colin-Honda, Foliations, contact structuresin 3-d Khovanov homology is an unknot-detector |
| Nov 19 | Quantum mechanics, Von Neumann's
states and quasi-states, Gleason's theorem: quasi-states are linear. Anti Gleason phenomana: symplectic non-linear quasi-states, and its quantum information interpertation. |
Polterovich, Quantum footprint of symplectic rigidity. |
Polterovich, symplectic rigidity and quantum mechanics. |
| Nov 21 | (Pre)-quantization, real and complex polarizations. | Lerman, Geometric Quantization-A Crash Course. | Berezin-Toeplitz Quantization, a review of results. |
| Nov25-30 | Thanksgiving | Thanksgiving | Happy Holiday! |
| Dec 3 | Lagrangian foliation, almost toric manifolds, and quantization. | Geometric quantization on almost toric manifolds. | Zung, Symplectic Topology of Integrable Hamiltonian Systems Leung-Symington, Almost toric M^4. Gross, Examples of Special Lagrangian Fibrations. |
| Dec 5 | Bohr-Summerfield quantization orbits upto canonical transform i.e. Lagrangian tori up to Hamiltonian isotopy. |
Seidel's lecture, Lagrangian tori and mirror symmetry. | Vianna, Lagrangian tori and mirror symmetry. Castronovo, Exotic Lagrangian tori in Grassmannians. |
| Dec 10 | Periodic orbit revisited.
Floer theory and symplectic isotopy (of embeddings) |
Floer-Hofer-Wysocki, Applications of Symplectic Cohomology. | Hutchings, Floer homology of families I.
Khovanov-Seidel, Quivers, Floer cohomology, and braid group actions |