Abstract: Given a graded Hopf algebra $A$, one wants to compute the stable representation ring $Stab(A)$. By work of Margolis, computing all possible Adams spectral sequence $E_2$-terms for finite module spectra over certain commutative ring spectra amounts to computing the cohomology of A with coefficients in each generator for Stab(A), when is a subalgebra of the Steenrod algebra. However, actually computing $Stab(A)$ is (in Margolis' words) "a very difficult problem in general."

In this talk we describe this relationship between Stab(A) and Adams spectral sequences, and we describe a new approach to the computation of Stab(A) which uses a twisted version of the deformation theory of modules. While untwisted first-order deformations of an A-module M are classified by the Hochschild cohomology group $HH^1(A, End(M))$, our twisted deformations instead are classified by a nonabelian (that is, with coefficients in a nonsymmetric module) version of the "higher-order Hochschild cohomology" of Pirashvili. We discuss existence and uniqueness results for these nonabelian higher-order Hochschild cohomologies, and the relative difficulty of actually making these computations (in particular, when they do and do not run up against of the unsolvability of the word problem!).