  I will discuss results of three different types in geometry and topology. (1)
General vanishing and rigidity theorems of elliptic genera proved by using
modular forms, Kac-Moody algebras and vertex operator algebras. (2) The
computations of intersection numbers of the moduli spaces of flat connections
on a Riemann surface by using heat kernels. (3) The mirror principle about
counting curves in Calabi-Yau and general projective manifolds by using
hypergeometric series.
