  We study the notion of formal duality introduced by Cohn, Kumar, and
Sch\"urmann in their computational study of energy-minimizing particle
configurations in Euclidean space. In particular, using the Poisson summation
formula we reformulate formal duality as a combinatorial phenomenon in finite
abelian groups. We give new examples related to Gauss sums and make some
progress towards classifying formally dual configurations.
