Besides providing a beautiful geometric explanation of mirror symmetry, the SYZ conjecture [153] has been exerting its long-lasting effect on many related areas of mathematics as well. Let us briefly describe several examples of applications in this regard.

For example, given a Lagrangian section of a special Lagrangian torus fibration $\mu:X\to B$, its intersection point with a fiber $L$ of $\mu$ determines a flat $U(1)$-connection on the dual torus $L^{\vee}$. Patching these flat $U(1)$-connections together should give a holomorphic line bundle over the total space of the dual fibration, which is the mirror $\check{X}$. This simple idea, first envisioned by Gross [66, 67], was explored by Arinkin and Polishchuk [7] and Leung, Yau and Zaslow [112] to construct SYZ transforms, which were applied to prove and understand the HMS conjecture in the semi-flat Calabi-Yau case. Later, the same idea was also employed to study the HMS conjecture for toric varieties [1, 2, 42, 44, 43, 15, 23, 36].

In some more recent works [17, 25, 24], SYZ transforms were applied to construct geometric Fourier–type functors (on the objects level) which realize the HMS categorial equivalences for certain examples of local Calabi-Yau such as resolutions of the $A_{n}$-singularities and the smoothed conifold, where one encounters SYZ fibrations with singular fibers and hence nontrivial quantum corrections. On the other hand, work in progress by K.-L. Chan, Leung and Ma [27, 26] have shown that SYZ transforms can also used to construct the HMS equivalences on the morphism level, at least in the semi-flat case. The ultimate goal is to construct a canonical geometric Fourier-type functor associated to any given SYZ fibration, which realizes the equivalences of categories asserted by the HMS conjecture, thereby enriching our understanding of the geometry of the HMS conjecture, and also mirror symmetry as a whole.

In general, such a qualitative description is still hard to obtain because we do not have nontrivial examples of SYZ fibrations on compact Calabi-Yau manifolds, and open Gromov-Witten theory is not well-understood. However, recent pioneering works of Gaiotto, Moore and Neitzke [56, 57] have shed new light on the hyperkähler case. They proposed a new (partially conjectural) construction of hyperkähler metrics on the total spaces of complex integrable systems, the simplest example of which reproduces the well-known Ooguri-Vafa metric [138].

Let us describe their construction in a bit more details. Given a complex integrable system $\psi:M\to B$, i.e. $M$ is holomorphic symplectic and the fibers of $\psi$ are complex Lagrangian submanifolds, they tried to first construct the twistor space of the desired hyperkähler metric by writing down a $\mathbb{C}^{\times}$-family of holomorphic Darboux coordinates on $M$ which satisfy the hypotheses of a theorem of Hitchin et al. [87]. In particular, they required the coordinates to satisfy certain wall-crossing formulas which describe the discontinuity of the coordinates across the so-called BPS rays, where the (virtual) counts of BPS states jump.

These wall-crossing formulas turn out to be of the same kind as those used by Kontsevich and Soibelman [106] and Gross and Siebert [78] in their constructions of toric degenerations of Calabi-Yau manifolds (and on the other hand they are the same as wall-crossing formulas in motivic Donaldson-Thomas theory [96, 104]). In view of this and the SYZ conjecture, it is natural to expect that the hyperkähler metrics constructed by Gaiotto, Moore and Neitzke can be expressed in terms of holomorphic disks.

This was done for the simplest example - the Ooguri-Vafa metric in [16]. More recent works of W. Lu [124, 125] have demonstrated that the construction of hyperkähler metrics in [56, 57] in the case of Hitchin systems produced the same data which are required to run the Gross-Siebert program [78], hence showing that there must be some (perhaps implicit) relations between the metrics and tropical disks counting. In his PhD thesis [119], Y.-S. Lin considered elliptic $K3$ surfaces and tropical disk counting invariants. He proved that his invariants satisfy the same wall-crossing formulas as appeared in [56, 57]. This again shows that the hyperkähler metrics are cloely related to disk counting. There are also recent works by Stoppa [151, 45] demonstrating the intimate relations between the wall-crossing formulas in motivic Donaldson-Thomas theory and the construction of Gaiotto, Moore and Neitzke.

In their recent joint project [71], Gross, Hacking and Keel constructed mirror families to log Calabi-Yau surfaces, i.e. pairs $(Y,D)$ where $Y$ is a nonsingular projective rational surface and $D\in|-K_{Y}|$ is a cycle of rational curves, by extending the construction in [78] to allow the affine manifolds to have more general (i.e. worse) singularity types. Amazingly, their results could be applied to give a proof of a 30-year-old conjecture of Looijenga [123] concerning smoothability of cusp singularities.

In an even more recent preprint [72], they applied their construction again to prove a Torelli theorem for log Calabi-Yau surfaces, which was originally conjectured in 1984 by Friedman [48]. On the other hand, their construction is also closely connected with the theory of cluster varieties, and they have suggested a vast generalisation of the Fock-Goncharov dual bases. For a nice exposition of these exciting new results and developments, we refer the reader to the nice survey article by Gross and Siebert [74].

In another unexpected direction, the SYZ construction has recently been applied to construct new knot invariants. For a knot $K$ in $S^{3}$, its conormal bundle $N^{*}K$ is canonically a Lagrangian cycle in the cotangent bundle $T^{*}S^{3}$. In [39], Diaconescu, Shende and Vafa constructed a corresponding Lagrangian cycle $L_{K}$ in the resolved conifold $X:=\mathcal{O}_{\mathbb{P}^{1}}(-1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-1)$, which is roughly speaking done by lifting the conormal bundle $N^{*}K$ off the zero section and letting $T^{*}S^{3}$ undergo the conifold transition. Their construction was motivated by a mysterious phenomenon called large $N$ duality in physics.

In [6], Aganagic and Vafa defined a new knot invariant by a generalized SYZ construction applied to the pair $(X,L_{K})$. Roughly speaking, their invariant is the generating series of open Gromov-Witten invariants for $(X,L_{K})$. It turned out that the resulting invariant is always a polynomial and they conjectured that it should be a deformation of the classical A-polynomial in knot theory [38]. Furthermore, an interesting relation between their invariant and augmentations of the contact homology algebra of $K$ [134] was suggested. Substantial evidences for this relation was obtained in a very recent preprint [5].

These two new applications of the SYZ conjecture, together with many more which are yet to come, open up new directions in mirror symmetry and other branches of mathematics and physics,⁶⁶ 6 One interesting story that we have not mentioned is the application of the SYZ picture to $G_{2}$ manifolds by Gukov, Yau and Zaslow [84] which was aimed at explaining the duality between M-theory and heterotic string theory. and they are all pointing out to further research works for the future.