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

Explore the fourth part of Jacob Lurie's lecture series on the Riemann-Hilbert correspondence in p-adic geometry. Delve into advanced mathematical concepts such as perfectoid Riemann-Hilbert functors, perfected Hodge-Tate crystals, and perfectoidization of schemes. Examine the properties of perfectoidization, spectral methods, and the computation of perfectoidization. Investigate the module structure on RH, affineness of perfectoidization, and properties of the Riemann-Hilbert functor in both characteristic p and mixed characteristic settings. Learn about finiteness theorems, globalization, rigid geometry, and Zavyalov's theorem. Conclude with an exploration of the primitive comparison theorem, exactness, duality, and applications of these complex mathematical ideas.

Intro
Last Time: Perfectoid Riemann-Hilbert Functors
Perfected Hodge-Tate Crystals
Crystals from the Riemann-Hilbert Functor
The Perfectoid Case
Perfectoidization of Schemes
Properties of the Perfectoidization
A More Concrete Riemann-Hilbert Functor
The Module Structure on RH
Spectral Methods
Computing the Perfectoidization
Affineness of Perfectoidization
Properties of the Riemann-Hilbert Functor
Example: Characteristic p
Mixed Characteristic
Finiteness Theorem
Globalization
Some Rigid Geometry
A Formula for RH
Zavyalov's Theorem
The Primitive Comparison Theorem
Exactness
Duality
Application