Calabi-Yau manifolds have become a topic of study in both mathematics and physics, dissolving the boundaries between the two subjects.
A manifold is a type of geometrical space where each small region looks like normal Euclidean space. For example, an ant on the surface of the Earth sees its world as flat, rather than the curved surface of the sphere. Calabi-Yau manifolds are complex manifolds, that is, they can be disassembled into patches which look like flat complex space. What makes them so special is that these patches can only be joined together by the complex analogue of a rotation.
Proving a conjecture of Eugenio Calabi, Shing-Tung Yau has shown that Calabi-Yau manifolds have a property which is very interesting to physics. Einstein's equations show that spacetime curves according to the distribution of energy and momentum. But what if space is all empty? By Yau's theorem, not only is flat space a solution but so are Calabi-Yau manifolds. Furthermore, for this reason, Calabi-Yau spaces are possible candidates for the shape of extra spatial dimensions in String Theory.