Felix Klein Lectures 2022

A Riemann-Hilbert Correspondence in p-adic Geometry

Jacob Lurie (Harvard University)

 

Dates: November 14 - December 2, 2022 (Exact dates tbc)

Venue: HIM Lecture Hall, Poppelsdorfer Allee 45, Bonn

 

Abstract:

At the start of the 20th century, David Hilbert asked which representations can arise by studying the monodromy of Fuchsian equations. This question was the starting point for a beautiful circle of ideas relating the topology of a complex algebraic variety X to the study of algebraic differential equations. A central result is the celebrated Riemann-Hilbert correspondence of Kashiwara and Mebkhout, which supplies a fully faithful embedding from the category of perverse sheaves on $X$ to the category of algebraic $\mathscr{D}_{X}$-modules. This embedding is transcendental in nature: that is, it depends essentially on the (archimedean) topology of the field of complex numbers. It is natural to ask if there is some counterpart of the Riemann-Hilbert correspondence over nonarchimedean fields, such as the field $\mathbf{Q}_p$ of $p$-adic rational numbers. In this series of lectures, I will survey some of what is known about this question and describe some recent progress, using tools from the theory of prismatic cohomology (joint work with Bhargav Bhatt).

 

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