I will explain the key points in the proof of Deligne's main theorem in Weil II, combining Deligne's theory of representations of global and local monodromy groups and Laumon's use of Deligne's $\ell$-adic Fourier transform.


       Here's a tentative plan :


0. Historical sketch

1. $\overline{\mathbf{Q}}_{\ell}$-sheaves, Grothendieck trace formula, Weil sheaves 2. Mixed sheaves, statement of Deligne's main theorem in Weil II 3. The global monodromy theorem, determinantal weights 4. Real sheaves and purity 5. The weight monodromy theorem 6. The curve case implies the main theorem 7. Review of Deligne's $\ell$-adic Fourier transform 8. Proof of the curve case