A "Calabi-Yau manifold" refers to a compact Kähler manifold which is Ricci-flat; according to Yau's verification of the Calabi conjecture, such manifolds are known to exist. Mirror symmetry, which is a proposal of physicists beginning in the late 80s, postulates that Calabi-Yau manifolds of complex dimension 3 can be grouped into pairs which share characteristics, such as Euler and Hodge numbers. Based on this conjectural picture, the physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes proposed a formula of enumerative geometry which, given any positive integer , encodes the number of rational curves of degree in a general quintic hypersurface of four-dimensional complex projective space. Bong Lian, Kefeng Liu, and Yau gave a rigorous proof that this formula holds. Alexander Givental had earlier given a proof of the mirror formulas; according to Lian, Liu, and Yau, the details of his proof were only successfully filled in following their own publication.