When Yau proved the famous Calabi conjecture [168, 169] in 1976, most mathematicians, probably Yau himself included, would not have imagined that Calabi-Yau manifolds were going to play such a pivotal and indispensable role [12, 152] in string theory - a modern physical theory which is aimed at unifying general relativity and quantum field theory. Nor would people speculate that the resulting interaction between geometry and physics would eventually lead to the important discovery of mirror symmetry - a subject that has been drastically influencing the development of many branches of mathematics for more than two decades.

The story began in the late 1980’s when Dixon [40] and Lerche, Vafa and Warner [109] suggested that theoretically two different Calabi-Yau manifolds could give rise to identical physics. This surprising prediction was soon verified by Greene and Plesser [63] and Candelas, Lynker and Schimmrigk [13] when they independently constructed pairs of Calabi-Yau manifolds which exhibit an interchange of Hodge numbers. We call these mirror pairs. In their remarkable 1991 paper [11], Candelas, de la Ossa, Green and Parkes carried out an even more astonishing calculation which led to a prediction about the numbers of rational curves on the quintic 3-fold in $\mathbb{P}^{4}$. Mathematicians were particularly intrigued by their prediction because it went way beyond what algebraic geometers could achieve at that time.

This has triggered the development of many important subjects such as Gromov-Witten theory, and finally culminated in the proofs of mirror theorems by Givental [59, 60] and Lian-Liu-Yau [115, 116, 117, 118] independently, which in particular verified the predictions by Candelas et al. for the quintic 3-fold. This is certainly a magnificent achievement. However, all the proofs rely on the geometry of the ambient toric varieties which contain the Calabi-Yau manifolds, and in particular, they do not provide an intrinsic way to understand the geometry of mirror symmetry.

The first intrinsic mathematical formulation of mirror symmetry was Kontsevich’s Homological Mirror Symmetry (HMS) conjecture, proposed in his 1994 ICM address [102]. In string theory, a Calabi-Yau manifold $X$ determines two topological string theories: the A-model and B-model, which depends on the symplectic and complex geometry of $X$ respectively [160, 166]. From this perspective, mirror symmetry predicts that if $X$ and $\check{X}$ are a mirror pair of Calabi-Yau manifolds, then there is an isomorphism between the A-model of $X$ and the B-model of $\check{X}$, and vice versa. The above enumerative predictions about the quintic 3-fold is one of the many interesting manifestations of this bigger picture.

Kontsevich’s HMS conjecture formulates mirror symmetry succinctly as an equivalence between the Fukaya category of Lagrangian submanifolds in $X$ (A-model) and the derived category of coherent sheaves on the mirror $\check{X}$ (B-model). His conjecture is both deep and elegant, and is expected to imply the enumerative predictions by mirror symmetry. Nevertheless, it does not indicate how such and equivalence can be found, nor does it tell us how to construct the mirror of a given Calabi-Yau manifold.