Jacob Lurie: A Riemann-Hilbert Correspondence in P-adic Geometry Part 2

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.

Intro
The Classical Riemann-Hilbert Correpondence
Constructible Sheaves
The Frobenius
Overview
Étale Sheaves on a Point
Finiteness
Algebraic Frobenius Modules
Katz's Theorem
A Generalization
Some Analogies
Analogy with the de Rham Complex
Computing Cohomology with the Artin-Schreier Sequenc
Explicit Description
Relationship with the de Rham Functor
Properties of the Riemann-Hilbert Functor
An Example
Unit Frobenius Modules
Relationship with Flat Connections
The Riemann-Hilbert Correspondence of Emerton-Kisin
Comparison of Riemann-Hilbert Correspondences