Homological mirror symmetry
Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory.
In an address to the 1994 International Congress of Mathematicians in Zurich, Kontsevich speculated that mirror symmetry for a pair of Calabi-Yau manifolds X and Y could be explained as an equivalence of a triangulated category constructed from the algebraic geometry of X and another triangulated category constructed from the symplectic geometry of Y. Kontsevich commented that the conjecture could be proved in the case of elliptic curves using theta functions.
Following this route, Alexander Polishchuk and Eric Zaslow provided a proof of a version of the conjecture for elliptic curves. Kenji Fukaya was able to establish elements of the conjecture for abelian varieties. Later, Kontsevich and Yan Soibelman provided a proof of the majority of the conjecture for nonsingular torus bundles over affine manifolds using ideas from the SYZ conjecture. Recently, Paul Seidel proved the conjecture in the case of the quartic surface.