Superstring theory is the leading candidate for the ultimate unified theory of all the matters and forces in nature. It postulates that fundamental building blocks are not point particles but strings, and it is defined in (9+1)-dimensional spacetime. The rich structure in particle physics in our (3+1) dimensions, such as quark and lepton flavors, gauge forces, and the Higgs boson to break the gauge symmetry, is expected to emerge from the geometry of the extra 6 (= 9 - 3) dimensions, called a Calabi-Yau space. We want to derive quantitative predictions on our universe from the geometry of the Calabi-Yau space. However, the Calabi-Yau space is so complicated that we do not even know an expression for its metric, namely how to measure the distance between two points on the space. Given this limitation, what can we do? In this talk, I will describe techniques we have developed to overcome this difficulty and applications of these techniques to problems in physics and mathematics.