Non-abelian cohomology, differential equations, and p-curvatures

One central aspect of algebraic geometry is that the cohomology
of algebraic varieties admit many extra structures as compared with that
of, say, topological spaces. I will discuss some of these structures in the
setting of "non-abelian cohomology", as pioneered by Simpson, and the new
features that arise; we'll see that they are closely related to classical
questions about differential equations such as the Schlesinger system. In
particular, I'll talk about the proof of a conjecture of
Ekedahl--Shepherd-Barron--Taylor's and Bost's in many new cases, which is
about the p-curvatures of differential equations such as Schlesinger's.
This is joint work with Daniel Litt, following ideas of Kisin, Menzies, and
Papaioannou.