Twentieth-century physics was built on two fundamental hypotheses: that spacetime is a manifold, and that all matter is made of zero-dimensional elementary particles. I will explain how these ideas are ultimately incompatible, and together lead to irreparable infinities in calculations of physical processes. I will then introduce a resolution of this problem: a theory of quantum gravity, furnished by string theory. Using string theory to model quantum gravity in our universe involves the study of six-manifolds with vanishing Ricci curvature, obtained as hypersurfaces in algebraic varieties. If time permits, I will outline how our work on the computational geometry of string theory connects to various areas of mathematics.