Program

KARL SCHWEDE: INTRODUCTION TO POSITIVE CHARACTERISTIC GEOMETRY

LINQUAN MA / HANLIN CAI / KEVIN TUCKER: PRISMATIC COHOMOLOGY AND APPLICATIONS TO COMMUTATIVE ALGEBRA

We give an introduction to prismatic cohomology. We will focus on its connections to perfectoid rings, the almost purity theorem, and its applications to mixed characteristic commutative algebra via the perfectoidization functor.

JAKUB WITASZEK: RIEMANN-HILBERT CORRESPONDENCES AND APPLICATIONS TO SINGULARITIES

We will provide a brief introduction to the theory of perverse sheaves as well as Riemann-Hilbert correspondences in positive and mixed characteristics. We will focus on applications to commutative algebra, particularly to the theory of splinters.

WIESLAWA NIZIOL: HIDDEN STRUCTURES ON THE DE RHAM COHOMOLOGY OF P-ADIC ANALYTIC VARIETIES

I will survey what is known about extra structures (Hodge filtration, Frobenius, monodromy) appearing on de Rham cohomology of analytic varieties over local fields of mixed characteristic.

WALTER GUBLER: ON THE NON-ARCHIMEDIAN MONGE-AMPERE EQUATION IN MIXED CHARACTERISTIC

Monge-Ampère equations are important PDE's in Analysis. In a celebrated result in the seventies, Yau solved the complex Monge-Ampère equation to prove a long standing conjecture of Calabi about the curvature on Kähler manifolds. In this talk, we explain that there is an analogue of the Monge-Ampère equation for projective varieties over non-archimedean fields as the p-adic numbers and that we can use test ideals in mixed characteristic to solve them assuming resolution of singularities. This is joint work with Yanbo Fang and Klaus Künnemann.

ARTHUR-CESAR LE BRAS: DE RHAM FARGUES-FONTAINE STACKS

Rodriguez Camargo recently introduced in analytic geometry a variant of Simpson’s de Rham stack, whose theory of quasicoherent sheaves relates to the category of analytic D-modules. Perhaps surprisingly, the theory is very « topological » in nature; the analytic de Rham stack can be defined even for spaces with no good theory of differential forms. In particular, one can form the analytic de Rham stacks of Fargues-Fontaine curves. I will explain what this means and how it should allow to revisit some recent results in p-adic Hodge theory. Joint work very much in progress with Anschütz, Bosco and Rodriguez Camargo.

Charles Vial: On proper splinters in positive characteristic

A commutative ring is called a splinter if any finite-module ring extension splits. By the direct summand conjecture, now a theorem due to André, every regular ring is a splinter. The notion of splinter can naturally be extended to schemes. In that context, every normal scheme in characteristic zero is a splinter. In contrast, Bhatt observed in his thesis that the splinter property for proper schemes in positive characteristic imposes strong constraints on the global geometry; for instance, the structure sheaf of a proper splinter in positive characteristic has vanishing positive-degree cohomology.

I will report on joint work with Johannes Krah where we describe further restrictions on the global geometry of proper splinters in positive characteristic. I will also address the derived-invariance of the (derived-)splinter property both in positive and mixed characteristics.