Speakers

Confirmed Speakers:

****Tomoyuki Abe (Kavli Inst. Tokyo)

****Yves André (Inst. de Math. Jussieu-Paris Rive Gauche)

Finite covers and the fpqc topology: from perfectoid theory to singularitytheory

Abstract:  The basic question of whether (and when) finite covers of schemes are fpqc covers has long been known - in commutative algebra, if not in algebraic geometry-  to have decisive implications on singularity theory. Perfectoid theory has shed light on this question.

****Francesco Baldassarri (Univ. di Padova)

Duality for linear topologies and integrtion pairing on locally analytic groups

****Vladimir Berkovich (Weizmann Inst.)

Comparison theorem for de Rham cohomology of non-Archimedean analytic spaces

Abstract: In a work in progress, I defined integral "etale" cohomology groups for certain non-Archimedean analytic spaces over the field of Laurent power series with complex coefficients. They are finitely generated abelian groups that give rise to the l-adic etale cohomology groups and the de Rham cohomology groups of such a space X. The latter comparison isomorphism depends on the choice of a uniformizing element of the power series ring or, equivalently, on the choice of a universal covering of the punctured complex plane. One of the reasons for this is that the construction of the integral etale cohomology groups depends on such a choice. In this talk I’ll give a description of the comparison result in the form an isomorphism between two local systems on a “classifying space” of universal coverings of the punctured complex plane. The first one is a complexified local system of the integral etale cohomology groups of X, and the second one is the local system of horizontal sections of a vector bundle with an integrable connection associated to the de Rham cohomology groups of X.

****Richard Crew (Univ. of Florida-Gainesville)

Nilpotent arithmetic D-modules

I will discuss the ideal theory of Berthelot's rings of arithmetic D-modules. A certain completion of the ring of level m arithmetic differential operators gives a natural interpretation of the category of topologically nilpotent D-modules of level m. I will discuss the problem of determining the global dimension of these rings.

****Valentina Di Proietto (Univ. of Exeter)

****Veronika Ertl (Univ. Regensburg)

Rigid analytic reconstruction of Hyodo--Kato theory

Abstract: I will give a new and very intuitive construction of Hyodo-Kato cohomology and the Hyodo-Kato map.It is based on log rigid cohomology and independent of the choice of a uniformiser.I will explain its dependence on the choice of a branch of the p-adic logarithm and show how it is related to the classical construction.This is joint work with Kazuki Yamada (Keio University).

****Hélène Esnault (Freie Univ. Berlin)

Formal $\ell$-adic Lie groups of multiplicative type and applications.

Abstract: Joint with Moritz Kerz. We prove some property which in particular implies Hard Lefschetz in rank $1$ in positive characteristic.

****Michel Gros (Univ. de Rennes1)

Around Simpson's correspondences

Abstract : I will give an overview of results obtained these last years as well as some more recent advances  and questions raised by them (joint work with B. Le Stum and A. Quiros).

****Christine Huyghe (Univ. de Strasbourg)

Intermediate extensions and crystalline distribution algebras

Abstract:This is joint work with Tobias Schmidt and Matthias Strauch. Let G be a complex reductive algebraic group, and g its Lie algebra. There is an equivalence of categories between the category of D-modules over the flag variety of G and g-modules with central character. The D-module corresponding to the simple quotient of a Verma module with trivial central character is obtained as an intermediate extension of the constant sheaf of a Bruhat cell. I will explain a p-adic analogue of this result, using recent results of Abe-Caro on arithmetic D-modules. On the way, I will also introduce an arithmetic p-adic category analogous to the classical O-category appearing in classical representation theory of Lie algebras.

****Kiran Kedlaya ( UC-San Diego)

Etale and crystalline companions
Abstract: Let X be a smooth variety over a finite field of characteristic p. In his "Weil 2" paper, Deligne conjectured that for any prime l different from p, any l-adic etale local system on X which is irreducible with determinant of finite order belongs to a compatible system. This means that on one hand, for every prime l' other than p, one has a "companion" in the category of l'-adic etale local systems; this has been established by Drinfeld using the Langlands correspondence for GL(n) for function fields. On the other hand, there should also be a companion for l' = p, which we now know should be in the category of overconvergent F-isocrystals.
Using recent developments in p-adic cohomology (particularly Abe's extension of the Langlands correspondence to include p-adic coefficients), one can now prove a more symmetric version of this statement, where both l and l' can equal p. In fact it is critical to prove the stronger version: one first proves the case where l=p (this was done independently by Abe-Esnault), and then uses this to handle the case where l'=p.

****Christopher Lazda (Univ. of Amsterdam-UVA/Warwick)

Trace morphisms, compactly supported rigid cohomology, and arithmetic D-modules
Abstract: Inspired by Le Stum's theory of constructible isocrystals, I will describe a new construction of Caro's specialisation functor sp_+ from overconvergent isocrystals to overholonomic D-modules. The utility of the construction is that it enables more direct proofs of comparison theorems between rigid cohomology and D-module cohomology with coefficients. I will explain how to use analytic incarnations of the trace morphism to prove such a comparison theorem for compactly supported cohomology. This is joint work with Tomoyuki Abe.

slides

****Atsushi Shiho (Univ of Tokyo)

On integral p-adic cohomology

Abstract: In this talk, we discuss several approaches to define integral p-adic cohomology of varieties which is compatible  with (log) crystalline cohomology and rigid cohomology. This is a joint work (partly in progress) with Veronika Ertl and Johannes Sprang.

****Daxin Xu (Caltech)

Bessel F-isocrystals for reductive groups