J-holomorphic curves in symplectic geometry

Since their introduction by Gromov [4] in the mid-1980’s J-holomorphic curves have been one of the most widely used tools in symplectic geometry, leading to the formulation of various theories (Gromov-Witten invariants, quantum co-homology, various Floer homologies, symplectic field theory, Fukaya category), answers to old questions in symplectic geometry (various Arnold conjectures) and the discovery of new phenomena (e.g. non-squeezing). It was the modest aim of these lectures to explain some of the very basic underlying principles and techniques and illustrate their use in the study of Lagrange embeddings. The first lecture started with a brief introduction to symplectic geometry. I mentioned some of the typical (very general) questions, such as the Existence problem: Given an compact almost complex manifold (M,J) with a class a ∈ H2(M;R) such that a ∪ · · · ∪ a 6 = 0 ∈ H2n(M;R), does M admit a symplectic form representing this class (and which tames some almost complex structure homotopic to J)? and the Mapping problem: How special are symplectomorphisms as opposed to just volume-preserving diffeomorphisms? Can one give lower bounds on the number of fixed points of Hamiltonian diffeomorphisms in terms of the topology of M? The second half of the lecture consisted of a discussion of Lagrangian immersions and embeddings of compact manifolds into standard (R2n ∼ = Cn, ω = ∑ dxk ∧ dyk) and what one can say without the use of holomorphic curves. I stated the Gromov-Lees theorem asserting that Lagrangian immersions into Cn satisfy the h-principle, so that regular homotopy classes of Lagrange immersions are in bijective correspondence with homotopy classes of U-parallelizations of L. I also mentioned the Whitney immersions of spheres with a single double point. This was followed by a brief illustration of Givental’s construction of immersions of surfaces. I also sketched the proof of the fact that the product of a Lagrangian immersion f: V n → Cn with a Lagrangian embedding g:Wm → Cm (m ≥ 1) is homotopic in the class of Lagrangian immersions to a Lagrangian embedding of V ×W into Cn+m. I next discussed how one can remove transverse double points by Lagrange surgery. Using Whitney’s algebraic count of the number of double points of an immersion in terms of the Euler class of the normal bundle and the