TITLE: Relation between mirror symmetry, homological mirror symmetry, and SYZ conjecture
QUESTION [10 upvotes]: I'm very new to mirror symmetry, and have a hard time establishing a broad overview of the subject. In particular I do not understand the precise relation between the following three conjectures:

Mirror symmetry, as formulated on the first page of these notes
Homological mirror symmetry (HMS)
The SYZ conjecture

A first basic question: when people speak of the "mirror" of a CY variety, do they really always mean a mirror in the sense of point (1) above?
My main question is whether any of these conjectures actually imply each other? For example, HMS predicts an equivalence of categories, which is only applied, in heuristic arguments for SYZ, to skyscraper sheaves. So it seems that SYZ would be at most a (refinement of (skyscraper sheaves correspond to honest Lagrangians, not just any objects in the derived category) a) consequence of HMS. In particular, the two do not seem to imply eachother?

REPLY [3 votes]: Another relation between 2) and 3) is that 2) is an algebraic statement while 3) is a geometric statement. So if one obtains an SYZ mirror by dualizing a Lagrangian torus fibration, then one obtains a pair of manifolds and can then try to prove HMS on that pair.  Specifically, the input needed for SYZ mirror symmetry is a Lagrangian torus fibration structure on the original symplectic manifold, and its mirror complex manifold is the dual fibration. Then one can aim to compute the categories on each side and show they match to prove HMS for that pair. Here is a nice set of notes on SYZ mirror symmetry i.e. T-duality (T for torus): https://math.berkeley.edu/~auroux/papers/slagmirror.pdf