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.