For even some of the smallest and most well-understood finite groups, classifying indecomposable representations over a field of positive characteristic is impossible. Since the development of support varieties in the 1980s, one rougher attempt to understand these categories of representations is to classify indecomposable objects up to a suitable equivalence; formally, this goal amounts to classifying the thick ideals of the category, and a full classification for finite groups was given by Benson—Carlson—Rickard. Tensor-triangular geometry, initiated in the early 2000s by Paul Balmer, gives a vast generalization of these techniques, and produces a topological space, the Balmer spectrum, to any tensor triangulated category; these categories have a tensor product which behaves in a similar way to the tensor product of vector spaces, and the Balmer spectrum is analogous to the prime spectrum of a commutative ring, where the tensor product plays the role of multiplication. We will discuss some recent progress in extending the Benson—Carlson—Rickard theorem to representation categories of finite-dimensional Hopf algebras, which is joint with Nakano and Yakimov.