The following subjects will be considered:

• Almost complex structures in R²ⁿ.
• Existence of almost complex structures on symplectic manifolds.
• Complex structures on the sphere: Beltrami equation, Ahlfors-Bers-Vekua theory.
• Complex structures on a Riemannian surface.
• Integrable complex structures, the Newlander-Nirenberg theorem.
• (Pseudo-) holomorphic curves, local equation and local existence theorem.
• Inner regularity of holomorphic curves.
• Sequences of holomorphic curves.
• Local structure of a holomorphic curve: isolation of critical points and of intersections, tangent cones, Michalef-White theorem.
• Gromov-Schwarz lemma and area estimates.
• Removable singularities of holomorphic curves.
• Plurisubharmonic functions and totally real sets on an almost complex manifold.
• Boundary behavior of holomorphic curves near a totally real submanifold.
• Gromov's rigidity theorems in symplectic geometry.
• Gromov's existence theorem for discs attached to a Lagrangian submanifold of Cⁿ.
• Meromorphic hulls of symplectic spheres in a Kähler manifold.

M. Gromov. Pseudoholomorphic curves in symplectic manifolds, Invent. Math., 82, (1985), 307-347.

M. Audin, J. Lafontaine, Eds. Holomorphic curves in symplectic geometry, Birkhäuser, Basel, 1994.

D. McDuff, D. Salamon. J-holomorphic curves and quantum cohomology, AMS, Providence, RI, 1994.

The course is now scheduled for 9.30 AM Monday, Wednesday, and Friday, but it is expected that a later hour will be chosen at the first meeting of the class.

If you have questions about the course, please consult Lee Stout.