In the first part of today's talk, we discuss the existence of an almost complex structure on a complex manifold. Subsequently, we state the Newlander-Nirenberg theorem, which asserts that the vanishing of the Nijenhuis tensor implies the integrability of an almost complex structure.

In the second part, we set the stage for proving Weinstein's Lagrangian neighborhood theorem: Any sufficiently small neighborhood of a Lagrangian submanifold $L$ in any symplectic manifold $M$ can be symplectomorphically mapped onto a neighborhood of the zero section of the cotangent bundle $T^*L$, extending the identity map of $L$.