Number/Representation Theory

In 1972, Katz proved the Grothendieck-Katz $p$-curvature conjecture for linear differential equations arising from algebraic geometry—that is, Gauss-Manin connections. His proof made use of the structures and properties of the cohomology of a family of varieties: for example, the Hodge and conjugate filtrations, the Hodge index theorem, etc. I'll explain analogues of these structures and properties for non-abelian cohomology (that is, the moduli of representations of $\pi_1$) and how to use them to prove a version of the Grothendieck-Katz $p$-curvature conjecture in the non-abelian setting.