Kontsevich's homological mirror symmetry conjecture

Let $X$ and $Y$ be mirror varieties, let $\mathcal{F}uk(X)$ denote the Fukaya category of $X$, and let $D^b\operatorname{Coh}(Y)$ denote the bounded derived category of coherent sheaves on $Y$. Kontsevich's homological mirror symmetry conjecture. There exists an equivalence

$$\mathcal{F}uk(X)\cong D^b\operatorname{Coh}(Y)$$

and this homological mirror symmetry implies enumerative mirror symmetry. This is proposed as a fundamental explanation for the correspondence between symplectic and algebraic invariants; the stated source reports genus-zero results for Calabi–Yau varieties under suitable assumptions, while the general claim remains open.