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Let G be a finite p-group and k be a field of characteristic
p. Motivated by the generating hypothesis -- a deep conjecture in stable
homotopy theory, we study maps between finite dimensional kG-modules
that are invisible to Tate cohomology --- such maps are called ghosts.
A natural question is to find the smallest integer n such that the
composition of any n ghosts is stably trivial, i.e., factors through a
projective. I will talk about these new integer invariants for kG with
a pleasant mix of techniques from  group theory, representation theory,
triangulated category theory and constructions motivated from homotopy
theory. This is joint work with Jan Minac and Dan Christensen.
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