Frobenius exact symmetric tensor categories

I will report on a joint work in progress with K. Coulembier and V. Ostrik. We show that a symmetric tensor category in characteristic p>0 admits a fiber functor to the Verlinde category (semisimplification of Rep(Z/p)) if and only if it has moderate growth and its Frobenius functor (an analog of the classical Frobenius in the representation theory of algebraic group) is exact. For example, for p=2 and 3 this implies that any such category is (super)-Tannakian. We also give a characterization of super-Tannakian categories for p>3. This generalizes Deligne's theorem that any symmetric tensor category over C of moderate growth is super-Tannakian to characteristic p. At the end I'll discuss applications of this result to modular representation theory.