I will define Fourier-Mukai numbers following S. Mukai, and show that
they can be computed using elementary group theory and the theory of
lattices. I will give an explicit formula for these numbers in terms
of the Picard lattice of a K3 surface (which I will define), and then
explain some of its applications. One of them relates Fourier-Mukai
numbers to class numbers of real quadratic fields of prime
discriminants. Another application is a solution to an old problem of
T. Shioda about abelian surfaces. This is joint with S. Hosono, K.
Oguiso, and S.T. Yau.