Jacob Lurie: A Riemann-Hilbert Correspondence in p-adic Geometry

Explore the fifth installment of Jacob Lurie's lecture series on the Riemann-Hilbert correspondence in p-adic geometry. Delve into advanced mathematical concepts including local systems, Galois representations, Hodge theory, étale cohomology, and p-adic Hodge theory. Examine the cyclotomic character, Hodge-Tate decomposition, and the fundamental calculation leading to the Riemann-Hilbert correspondence. Investigate the comparison conjecture, period sheaves, and de Rham local systems. Gain insights into the classical story and its p-adic analogue, bridging geometric and algebraic perspectives in this 53-minute lecture from the Hausdorff Center for Mathematics.

Intro
Setting the Stage
Local Systems on a Point
Analogue over K?
The Cyclotomic Character
Galois Representations from Geometry
Classical Hodge Theory
Hodge-Tate Representations
The Hodge-Tate Decomposition
Digression
Coherent Description of Étale Cohomology
Rigid-Analytic Description
Fundamental Calculation
The Riemann-Hilbert Correspondence of Lecture 2
The Functor RHC
p-adic Hodge Theory
The Classical Story
Example
The Comparison Conjecture
Sketch of Proof
Period Sheaves
Comparison with Coefficients
de Rham Local Systems