Jacob Lurie: A Riemann-Hilbert Correspondence in P-adic Geometry Part 1

Explore the fascinating world of algebraic geometry in this 46-minute lecture by Jacob Lurie at the Hausdorff Center for Mathematics. Delve into the Riemann-Hilbert correspondence and its implications for p-adic geometry. Trace the historical development from Hilbert's 21st problem to the groundbreaking work of Kashiwara and Mebkhout. Examine the challenges of translating this correspondence to non-archimedean fields like Qp. Discover recent advancements in prismatic cohomology and their potential applications. Cover key concepts including Fuchsian systems, monodromy representations, local systems on complex manifolds, algebraic D-modules, and the de Rham complex. Gain insights into the intersection of topology, algebraic differential equations, and complex algebraic varieties in this comprehensive exploration of modern mathematical theory.

Intro
Hilbert's 21st Problem
Fuchsian Systems
The Monodromy Representation
The Riemann-Hilbert Problem
Reformulation
A Solution
Conclusion
Local Systems on Complex Manifolds
Local Systems on Projective Varieties
Local Systems on General Varieties
The Riemann-Hilbert Correspondence for Local Systems
Example: The Gauss-Manin Connection
Direct Image Sheaves
Algebraic D-Modules
Behavior of Flat Sections
The de Rham Complex
The Riemann-Hilbert Functor
Outline