• What is a handle-slide? Oszvath-Szabo has a nice article here.
• How do we see if the two Lagrangian torus are equivalent or not?

His 2011 talk slide on the big pictures.

We are building 4d TFT, which should assign a number to a closed 4-manifold. a vector space to a closed 3-manifold, and a category to a closed 2-manifold. Moreover, for a cobordism of 3-manifolds $Y_0^3 \leadsto Y_1^3$ by $W^4$, we have a homomorphism from $Z(W): Z(Y_0) \to Z(Y_1)$. For a cobordism $Y: \Sigma_1 \leadsto \Sigma_2$, we have a functor $Z(Y): Z(\Sigma_1) \to Z(\Sigma_2)$.

(hmm, what is a 4d TFT? Is the category of abelian group a monoidal category? Yes, there is tensor product defined. So, to the empty 2-manifold, we should assign the coefficient category, Ab. And to a closed 3-manifold $Y$, we should assign a functor from Ab to Ab, which I think should be the monoidal unit, $1$. In the cat of Ab, it is $\Z$. See also Kapustin's 2010 ICM talk notes

What is Heegaard-Floer theory? Do we just do the usual Floer homology of two torus in a symmetric product of surfaces? OK. That's sort of easy. But, how to think about symmetric product of Lagrangians? Aha, and Ozsvath-Szabo has that solved already, for branched double cover.