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

Explore the intricacies of p-adic geometry in this advanced mathematics lecture on the Riemann-Hilbert correspondence. Delve into topics such as computing the Riemann-Hilbert functor, perfectoid rings, perfectoidization with compact supports, and prismatic cohomology. Examine the properties and simplifications of the Riemann-Hilbert functor, analyze sheaves on special fibers, and investigate perfectoid spaces. Compare different Riemann-Hilbert functors and gain insights into characteristic p examples and the almost category. This in-depth talk provides a comprehensive exploration of cutting-edge concepts in p-adic geometry for mathematicians and researchers in related fields.

Syllabus

Intro
Last Time
Computing the Riemann-Hilbert Functor
Beyond Characteristic p?
Perfection
Perfectoid Rings
Existence of Perfectoidizations
Perfectoidization with Compact Supports
Relationship with the Riemann-Hilbert Functor
Perfected Prismatic Cohomology with Compact Supports
Extension to Derived Categories
Properties of the Riemann-Hilbert Functor
A Simplification
Example: Characteristic p
Anatomy of Sheaves
Sheaves on the Special Fiber
The Almost Category
Perfectoid Spaces
Recollections
Digression
Perfectoid Analogue
Comparison of Riemann-Hilbert Functors
Variant