This is an outline of what I hope to cover in the course of this semester; most of the references may be found on reserve in the library.

- Symplectic Manifolds.
^{1,2}- Basics.
- Linear algebra and symplectic forms.
- Darboux theorem and Moser stability theorem.
- Compatible almost complex structures.
- Symplectic and Lagrangian submanifolds.

- Examples.
- Cotangent bundle, Kaehler manifolds.
- Kodaira-Thurston examples.
- Gluing constructions.

- J-holomorphic curves.

- Basics.
- Contact manifolds--introduction.
- Basics, especially in dimension 3.
^{3,4}- Darboux theorem and Gray stability theorem.
- Induced foliation on an embedded surface.
- Legendrian knots; Thurson-Bennequin invariant.
- Euler class of a contact structure.

- Examples.
- R
^{3}, S^{3}, unit cotangent bundle. - All 3-manifolds have contact structures.
- Constructions of 3-manifolds: surgery, open books.
^{5} - Techniques of Lutz, and Thurston-Winkelnkemper.
^{ 6 }

- R
- Relation to symplectic manifolds and Stein manifolds.
^{ 7, 8, 9,10}- J-convexity, holomorphic and symplectic fillability.
- Morse theory of Stein manifolds--outline.

- The basic dichotomy in dimension 3: Tight vs. overtwisted contact structures.

- Basics, especially in dimension 3.
- Overtwisted structures.
- Lutz twist.
- Eliashberg's flexibility theorem: homotopic overtwisted structures are isotopic.
- Sketch of proof for B.
^{ 11, 12}

- Tight structures.
- The standard structure on S
^{3}is tight, which follows from: - Holomorphically fillable structures are tight, which follows from:
- Symplectically fillable structures are tight.
- Outline of B.
^{13}- Bishop's theorem on existence of holomorphic disks.
- Filling by holomorphic disks.
- Pseudoholomorphic disks and part C.

- Eliashberg rigidity theorem: Uniqueness for tight structures on S
^{3}.^{14}

- The standard structure on S
- Bennequin's inequality for tight contact manifolds.
^{15,16}- Gromov/Eliashberg's proof.
- Applications to knot theory.
- Comparison with taut foliations and work of Gabai/Thurston.
^{17,18}

- Sketch of relations to Seiberg-Witten theory.

^{2} B. Aebischer et al*. Symplectic Geometry*, Birkhauser
(1994), Chapters 1,2.

^{3} Aebischer et al., Chapter 8.

^{4} Mcduff-Salamon, Chapter 3.4.

^{5} D. Rolfsen, *Knots and Links*, Publish or Perish (1976).

^{6} W. Thurston and H. Winkelnkemper, On the existence of contact
forms, Proc. AMS **52 **(1975), 345-347.

^{7} Y. Eliashberg, Topological characterization of Stein manifolds
of dimension > 2, Int. J. Math. **1** (1990), 29-46.

^{8} R. Gompf, Handlebody constructions of Stein surfaces.
(Preprint, 1997).

^{9} A. Weinstein, Contact surgery and symplectic handlebodies.
Hokkaido Math. J. **20** (1991), 241-251.

^{10} E. Giroux, Topologie de contact en dimension 3, Seminaire Bourbaki, in Asterisque **216** (1993) 7-33.

^{11} Y. Eliashberg, Contact 3-manifolds twenty years since J.
Martinet's work. Ann. Inst Fourier **42** (1995), 165-192.

^{12} Giroux.

^{13} Y. Eliashberg, Filling by holomorphic discs and its
applications, in *Geometry of Low-* *Dimensional Manifolds*, London
Math. Soc. Lect. Notes **151**, Cambridge (1990), 45-67.

^{14} Giroux.

^{15} Giroux.

^{16} Y. Eliashberg, Legendrian and transversal knots in tight
contact 3-manifolds, in *Topological Methods in Modern Mathematics*,
Publish or Perish (1993).

^{17} D. Gabai, Foliations and the topology of 3-manifolds, J. Diff.
Geom. **18** (1983), 445-503.

^{18} W. Thurston, A norm for the homology of 3-manifolds, Mem. AMS
**59** (1986), no. 339.