Delve into the intricacies of p-adic geometry in this 45-minute lecture by Jacob Lurie at the Hausdorff Center for Mathematics. Explore the concept of a Riemann-Hilbert correspondence in non-archimedean fields, particularly focusing on the field of p-adic rational numbers. Trace the historical development from Hilbert's question about Fuchsian equation monodromy to the modern Riemann-Hilbert correspondence of Kashiwara and Mebkhout. Examine the challenges of translating this correspondence to non-archimedean settings and learn about recent progress using prismatic cohomology theory. Follow along as Lurie discusses key topics including constructible sheaves, the Frobenius, étale sheaves, algebraic Frobenius modules, and the relationship with flat connections. Gain insights into the comparisons between different Riemann-Hilbert correspondences and their implications for p-adic geometry.