Editing Mirror symmetry (string theory) (section)

===Enumerative geometry===
{{main article|Enumerative geometry}}

[[File:Apollonius8ColorMultiplyV2.svg|thumb|right|alt=Three black circles in the plane and eight additional overlapping circles tangent to these three.|[[Problem of Apollonius|Circles of Apollonius]]: Eight colored circles are tangent to the three black circles. ]]

Many of the important mathematical applications of mirror symmetry belong to the branch of mathematics called enumerative geometry. In enumerative geometry, one is interested in counting the number of solutions to geometric questions, typically using the techniques of [[algebraic geometry]]. One of the earliest problems of enumerative geometry was posed around the year 200 [[BCE]] by the ancient Greek mathematician [[Apollonius of Perga|Apollonius]], who asked how many circles in the plane are tangent to three given circles. In general, the solution to the [[problem of Apollonius]] is that there are eight such circles.<ref name=autogenerated8>{{harvnb|Yau|Nadis|2010|page=166}}.</ref>

[[File:Clebsch Cubic.png|thumb|right|alt=A complex mathematical surface in three dimensions.|The [[Clebsch cubic]] ]]
Enumerative problems in mathematics often concern a class of geometric objects called [[algebraic varieties]] which are defined by the vanishing of [[polynomial]]s. For example, the [[Clebsch cubic]] (see the illustration) is defined using a certain polynomial of [[degree of a polynomial|degree]] three in four variables. A celebrated result of nineteenth-century mathematicians [[Arthur Cayley]] and [[George Salmon]] states that there are exactly 27 straight lines that lie entirely on such a surface.<ref>{{harvnb|Yau|Nadis|2010|page=167}}.</ref>

Generalizing this problem, one can ask how many lines can be drawn on a quintic Calabi–Yau manifold, such as the one illustrated above, which is defined by a polynomial of degree five. This problem was solved by the nineteenth-century German mathematician [[Hermann Schubert]], who found that there are exactly 2,875 such lines. In 1986, geometer [[Sheldon Katz]] proved that the number of curves, such as circles, that are defined by polynomials of degree two and lie entirely in the quintic is 609,250.<ref name=autogenerated8 />

By the year 1991, most of the classical problems of enumerative geometry had been solved and interest in enumerative geometry had begun to diminish. According to mathematician [[Mark Gross (mathematician)|Mark Gross]], "As the old problems had been solved, people went back to check Schubert's numbers with modern techniques, but that was getting pretty stale."<ref name=autogenerated6>{{harvnb|Yau|Nadis|2010|page=169}}.</ref> The field was reinvigorated in May 1991 when physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes showed that mirror symmetry could be used to count the number of degree three curves on a quintic Calabi–Yau. Candelas and his collaborators found that these six-dimensional Calabi–Yau manifolds can contain exactly 317,206,375 curves of degree three.<ref name=autogenerated6 />

In addition to counting degree-three curves on a quintic three-fold, Candelas and his collaborators obtained a number of more general results for counting rational curves which went far beyond the results obtained by mathematicians.<ref>{{harvnb|Yau|Nadis|2010|page=171}}.</ref> Although the methods used in this work were based on physical intuition, mathematicians have gone on to [[mathematical proof|prove rigorously]] some of the predictions of mirror symmetry. In particular, the enumerative predictions of mirror symmetry have now been rigorously proven.<ref name=autogenerated13 />