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Explore the intricate connections between p-curvature and non-abelian cohomology in this advanced mathematical lecture from the Institute for Advanced Study's Workshop on Special Cycles and Related Topics. Delve into the cohomological structures of algebraic variety families, examining both Hodge-theoretic and arithmetic aspects through the lens of non-abelian cohomology—specifically the representation spaces of fundamental groups in algebraic variety families. Learn about groundbreaking joint research with Josh Lam that develops analogues of classical cohomological structures for non-abelian settings, extending beyond traditional abelian frameworks. Discover a sophisticated non-abelian generalization of Katz's seminal 1972 proof of the p-curvature conjecture for Gauss-Manin connections, with particular focus on smooth proper morphisms X→S. Understand how vanishing p-curvature of isomonodromy foliations on moduli spaces of flat bundles leads to the fundamental result that π₁(S,s) actions on conjugacy classes of π₁(Xs) representations into GLₙ(ℤ) factor through finite groups. Gain insights into cutting-edge developments in algebraic geometry, representation theory, and arithmetic geometry through this technical presentation suitable for researchers and advanced graduate students in these fields.
