Here is a trivial example that I read from a survey article written by Arnold in the late 80s.
Consider $T^*S^1$, the cotangent bundle of $S^1$ which we can identify with the product $\newcommand{\bR}{\mathbb{R}}$ $S^1\times\bR$. I will denote the obvious coordinates on this cylinder by $(\theta, t)$.
Like any cotangent bundle, $T^*S^1$ carries a symplectic structure, and in this case, any curve on this symplectic manifold is a lagrangian submanifold. However, there are curves, and there are curves.
Take for example the curves $C_\tau:=\lbrace t=\tau\rbrace$, $\tau$ a nonzero constant, which are disjoint from the zero section and are deformations of the zero section via the symplectic flow
$$ (\theta,t)\mapsto \Phi_\tau(\theta,t)=(\theta,t+\tau). $$
Consider next a smooth function
$$ S^1\ni\theta\mapsto f(\theta). $$
Its differential is a section of $T^*S^1$, and its graph $\Gamma_{df}=(\theta,f'(\theta))$ intersects the zero section along the critical points of $f$.
The lagrangian $\Gamma_{df}$ is a rather special deformation of the zero section: it is a Hamiltonian deformation, the points of intersection of $\Gamma_{df}|$ correspond to the periodic orbits of the Hamiltonian deformation.
Why is this fascinating? Certain pairs of lagrangian subspaces intersect in more points than predicted by topology alone, which is in itself an indication that symplectic topology is a bit more rigid than smooth topology alone.
How does the above trivial example fit the general picture?
A lagrangian submanifold $L$ of a symplectic manifold has a tubular neighborhood symplectomorphic to $T^* L$. Thus the case of cotangent bundles can be viewed as local situations of the more general cases. of lagrangian submanifolds and their hamiltonian perturbations.
Given a Hamiltonian flow $\Phi_t$ on a symplectic manifold $X$, the graph of the time $1$-map is a lagrangian submanifold in $X\times X$. Its fixed points correspond to the intersection of the graph with the diagonal in $X\times X$, which is another lagrangian submanifold. Thus the problem of intersection of lagrangian submanifolds contains as a special case the problem of existence of periodic solutions of hamiltonian systems.
Leaving aside the mysterious rigidity of symplectic topology alluded to above, the problem of existence of periodic orbits of hamiltonian systems has fascinated many classics, such as Poincare, because of it's obvious connection to the many body problem and the philosophical question: does the history of our planetary system repeat itself?
Assume dim$(M) \geq 6$ and $M$ is simply connected.
By cancelling Morse critical points for a Morse function $f$ on $M$ one can get the number of critical points equal to the minimum number of generators needed in $C_* (M)$ to generate $H_* (M)$. I.e. the sum of the Betti numbers plus 2 for each torsion generator. This is also the number of cells in a minimal cell structure (CW-complex) for $M$.
This means that if one can define Floer homology with $\mathbb{Z}$ coefficients and the PSS isomorphism works with $\mathbb{Z}$ coefficients, then the strong Arnold conjecture is true.
I believe this is the case for e.g. monotone symplectic manifolds.
Best Answer
V. I. Arnol'd, June 12, 1937 - June 3, 2010.
The very sad news of his death is reported today here.
After Floer, the main difficulty in solving the weak Arnol'd conjecture on a compact symplectic manifold $M$ lies in defining a Floer chain complex generated by 1-periodic orbits of an arbitrary non-degenerate Hamiltonian $H: S^1\times M \to \mathbb{R}$, in such a way that the homology is independent of $H$. Once one has that, the remaining step (proving an isomorphism with Morse homology) can be done either by a computation with small autonomous Hamiltonians, or by a "PSS" isomorphism.
When $M$ is monotone, the crucial compactness theorems for solutions to Floer's equation (used to define the candidate-differential on the Floer complex, to prove that it squares to zero, and, in a variant, to prove the invariance of the theory) can be proved using index considerations. When $M$ is Calabi-Yau, compactness needs an additional idea, that holomorphic spheres generically don't hit cylinders solving Floer's equation. This is beautifully worked out in
In general, where there may be holomorphic spheres with small negative Chern number, one has little choice but to allow "stable trajectories" consisting of broken Floer trajectories with holomorphic bubble-trees attached. Transversality is proved by introducing multi-valued perturbations to the equations, and this forces one to use rational coefficients. References:
[Edit: both these references offer proofs of the weak Arnol'd conjecture with rational coefficients.] For a detailed introduction to these "virtual transversality" methods, see
The technical complications of virtual transversality theory are notorious, and one could wish for a fully detailed textbook account.
What's left?
So far as I know, there is no proof for general manifolds that the number $h$ of 1-periodic orbits of a non-degenerate Hamiltonian is at least the sum of the mod $p$ Betti numbers. The strong Arnol'd conjecture for non-degenerate Hamiltonians, that $h$ is at least the minimum number of critical points of a Morse function, is wide open.