I will explain what Koszul duality means, and what it is good for in representation theory, with a focus on the celebrated 1995 paper of Beilinson-Ginzburg-Soergel, and its later generalization to Kac-Moody groups by Bezrukavnikov-Yun. In these papers, which are about perverse sheaves with complex coefficients on (possibly infinite-dimensional) flag varieties, a key role is played by a construction called "logarithm of the monodromy."

I will then discuss new results on Koszul duality with coefficients in a field of positive characteristic. The main difficulty is that logarithm no longer makes sense! This work, which is joint with S. Makisumi, S. Riche, and G. Williamson, is part of our proof of the Riche-Williamson conjecture on characters of tilting modules.