Surgery Problems, Geography Problems and the L-space conjecture

Heegaard Floer homology is a powerful invariant of 3-manifolds which takes value in the category of vector spaces. It is natural to ask
which vector spaces arise as the Heegaard Floer homology of some 3-manifold. In this talk I will discuss some work in progress with
Antonio Alfieri on various special cases of this question as well as some other related problems.