Given a symmetric tensor functor between two symmetric tensor categories, Tannakian formalism allows us to write the first category as the representations of an affine group scheme in the second category. This gives a template for classifying all symmetric tensor categories of moderate growth: find all of the "incompressible" categories over which everything else fibres, such as vector spaces and super vector spaces, and then study their affine group schemes. In positive characteristic, this task is still far from complete, but significant progress has been made in the semisimple setting. It was shown by Victor Ostrik in 2015 that all fusion categories fibre over an unusual incompressible category called the semisimple Verlinde category, and in 2022, Siddharth Venkatesh developed a theory of affine group schemes in this category. In this talk I will give an overview of these concepts and how they apply to the more general construction of Verlinde categories of simple algebraic groups. I will then discuss some joint work with Kevin Coulembier and Pavel Etingof which uses this theory to classify symmetric tensor categories generated by objects with constrained symmetric and exterior powers.