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**Abstract:**

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.