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I won't claim this means ''understanding mirror symmetry'', but if you are familiar with the derived category of coherent sheaves, then there is a consequence of Kontsevich's homological mirror symmetry that can be understood, without knowing anything about the Fukaya category:

For every symplectic diffeomorphism of the mirror $\hat X$, there is a autoequivalence of $\mathrm D^b(X)$.

Examples:

1. If $X$ is a an elliptic curve, then $\mathrm{SL}_2(\mathbb Z)$ acts as a group of symplectic diffeomorphisms on the mirror $\hat X = \mathbb{R}^2/\mathbb{Z}^2$. There is a corresponding action of (a central extension of) $\mathrm{SL}_2(\mathbb Z)$ generated by Fourier-Mukai transform induced by the Poincare line bundle on $X \times X$, and by tensoring by a line bundle of degree one (and shifts).

2. A Dehn twist corresponds to the ''spherical twist'' $\mathrm{ST}_E$ at an spherical object $E$ (see arXiv:math.AG/0001043).

3. More examples have been studied by Horja, see arXiv:0103.5231.

1

I won't claim this means ''understanding mirror symmetry'', but if you are familiar with the derived category of coherent sheaves, then there is a consequence of Kontsevich's homological mirror symmetry that can be understood, without knowing anything about the Fukaya:

For every symplectic diffeomorphism of the mirror $\hat X$, there is a autoequivalence of $\mathrm D^b(X)$.

Examples:

1. If $X$ is a an elliptic curve, then $\mathrm{SL}_2(\mathbb Z)$ acts as a group of symplectic diffeomorphisms on the mirror $\hat X = \mathbb{R}^2/\mathbb{Z}^2$. There is a corresponding action of (a central extension of) $\mathrm{SL}_2(\mathbb Z)$ generated by Fourier-Mukai transform induced by the Poincare line bundle on $X \times X$, and by tensoring by a line bundle of degree one (and shifts).

2. A Dehn twist corresponds to the ''spherical twist'' $\mathrm{ST}_E$ at an spherical object $E$ (see arXiv:math.AG/0001043).

3. More examples have been studied by Horja, see arXiv:0103.5231.