TITLE: How many Lagrangian submanifolds?
QUESTION [8 upvotes]: An $n$-dimensional submanifold $L$ of a symplectic manifold $(M^{2n}, \omega)$ is called Lagrangian if $\omega|_L = 0$. I want to get some feeling about how many Lagrangian submanifolds are.
For each $\alpha \in H_n(M)$, is there a Lagrangian submanifold representing $\alpha$? Maybe it's not a good idea to distinguish Lagrangians by its homology classes. Floer homology is the same if we move Lagrangians by Hamiltonian isotopy. Is there a notion of space of Lagrangian submanifolds modulo Hamiltonian isotopy?

REPLY [4 votes]: There's an idea of Hitchin's that (all Lagrangian submanifolds)/(Hamiltonian isotopy) should be, at least approximately, the same as (all "special" Lagrangian submanifolds).  "Special" is defined with respect to some background structure on the symplectic manifold, like a metric.  Special Lagrangians make a finite-dimensional space at least locally.
A good analogy is the Hodge theorem (but Hitchin's idea doesn't work as well): the space (closed differential forms)/(exact differential forms) is the same as (harmonic differential forms), again the meaning of "harmonic" depends on some background structure, and also harmonic forms make a finite-dimensional space.
The Hitchin paper is here: 
http://arxiv.org/abs/dg-ga/9711002