# Felix Klein Lectures 2022

## A Riemann-Hilbert Correspondence in p-adic Geometry

## Jacob Lurie (Institute for Advanced Studies, Princeton)

Dates:

Lecture 1: MPI, Tuesday, Nov 15, 12:00--13:00 video pdf

Lecture 2: MPI, Thursday, Nov 17, 14:00--15:00 video pdf

Lecture 3: MPI, Tuesday, Nov 22, 16:30--17:30 video pdf

Lecture 4: MPI, Thursday, Nov 24, 14:00--15:00 video pdf

Lecture 5: MPI, Tuesday, Nov 29, 16:30--17:30 video pdf

Lecture 6: MPI, Thursday, Dec 1, 14:00--15:00 video pdf

Venue: MPIM - Max-Planck-Institute for Mathematics, Vivatsgasse 7, 53111 Bonn

### Abstract:

At the start of the 20^{th} 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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