This page collects the Videos of the talks, the Slides (when available), together with the Posters that have been presented at the conference.
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TITLES AND ABSTRACTS of the talks are also available
HERE
ABSTRACT : The term "Drinfeld¢s lemma" refers to several statements that relate the
geometry of a scheme over Fp to the result of base-extending
to an algebraically closed field, then formally dividing by the "partial
Frobenius" action on the field (fixing the original scheme). One such
statement asserts that for any prime l ¹ p, lisse l-adic
sheaves on the original scheme are the same as on this formal quotient.
This result plays a pivotal role in the approach to the Langlands
correspondence for a reductive group over the function field of a curve
over a finite field, pioneered by Drinfeld for the group
GL(2) and subsequently extended by L. Lafforgue and then V.
Lafforgue.
In this lecture, we describe an analogue of this statement involving
"lisse p-adic sheaves", meaning overconvergent F-isocrystals. The
hope is that this can be used to upgrade Abe¢s proof of the Langlands
correspondence for GL(n) (based on L. Lafforgue¢s method) to
more general reductive groups.
Since overconvergent F-isocrystals are not directly described by
representations of the profinite étale fundamental group, the p-adic
statement is not an immediate corollary of the original Drinfeld¢s
lemma. We deduce it by building up some structural properties of
``F-isocrystals¢¢ (isocrystals with multiple Frobenius structures),
particularly the Newton polygon variation and slope filtration, so that
we can eventually reduce to the prior result.
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Andreas Bode (Ecole Normale Supérieure Lyon, France)
TITLE : Bornological D-modules on rigid analytic spaces.SLIDES
ABSTRACT :
Ardakov-Wadsley introduced p-adic D-cap-modules on rigid analytic spaces in order to study p-adic
representations geometrically, in analogy to the theory of Beilinson-Bernstein localization over
the complex numbers. In this talk, we report on an ongoing project to extend their framework to the
(derived) category of complete bornological D-cap-modules, which allows us to define analogues of
the usual six operations.
We then consider a subcategory playing the role of Dbcoh(D) and prove a number
of stability results.
TITLE : Twisted local Simpson correspondence and crystals on the $q$-crystalline and prismatic sites.SLIDES
ABSTRACT :
We will explain how the twisted Simpson correspondence elaborated these last years in collaboration with B.
Le Stum and A.Quiro\¢os can be interpreted in the language of crystals on the q-crystalline and prismatic
sites recently considered by B. Bhatt and P. Scholze, giving in the same time very explicit examples of these objects.
TITLE : p-adic Tate conjectures and abeloid varieties.SLIDES
ABSTRACT :
We study Tate-type conjectures over p-adic fields and in
particular, a conjecture of Raskind for smooth and
projective varieties over p-adic fields that have totally
degenerate reduction. In the case of abeloid varieties, we
translate Raskind¢s conjecture into a question about
filtered (j,N)-modules and use this to construct
counter-examples. This is joint work with Oliver Gregory.
TITLE : D-modules and irreducibility results for locally analytic representations.SLIDES
ABSTRACT :
Let G be a split reductive p-adic group and let P be a parabolic subgroup of G. Let X be the rigid analytic flag variety of G and Y a P-stable closed subset. After a brief review of the theory of equivariant D-modules on rigid analytic spaces, I will discuss the geometric induction functor, due to K. Ardakov, which relates P-equivariant DX-modules M with support Y to G-equivariant DX-modules ind(M) with support GY. I will then explain how to establish the irreducibility of the induced module ind(M) for M the (push forward of the) structure sheaf on certain smooth Schubert varieties Y. As an application, we reprove geometrically some irreducibility results for locally analytic G-representations in the image of the Orlik-Strauch functor. This is work in progress with Konstantin Ardakov.
TITLE :
Poincaré duality in rigid analytic Hyodo-Kato theory.
ABSTRACT :
Hyodo-Kato theory plays an important role in different parts of arithmetic geometry,
for example in p-adic Hodge theory or the research of p-adic L-fnctions.Especially for the latter it is of advantage to describe explicitly certain cohomology classes, both in the usual
Hyodo-Kato theory and the compactly supported Hyodo-Kato theory.
I will talk about a rigid analytic construction of Hyodo-Kato theory developed together with Kazuki Yamada (Keio University) and its compatibility with Poincaré duality, which is suited for such explicit computations.
TITLE :
Integral p-adic cohomology theories.
SLIDES
ABSTRACT :
Suppose R is a complete local ring with residue field k. The
prospects for a cohomology theory on the category of separated k-schemes
of finite type with values in finitely generated R-modules are not good. This is joint work with Tomoyuki Abe.
TITLE :
Non-Archimedean uniformization and tropicalization: Teichmueller space and the moduli space of curves.
SLIDES
ABSTRACT :
The theories of uniformization for maximally degenerate curves over non-Archimedean curves (by Mumford) and for abelian varieties
(by Raynaud) are one of the big achievements of modern arithmetic algebraic geometry.
In recent years it has become clear that this story also has a tropical aspect.
In fact, one may think of the construction as a two-step process:
first construct a tropical uniformization, then use the combinatorial data of this tropical
uniformization to build the non-Archimedean uniformization.
In this talk, I will illustrate this principle in the case of the moduli space of curves.
In particular I will explain how to build a non-Archimedean uniformization of the moduli space of stable
algebraic curves that is closely connected to the non-Archimedean Schottky space for Mumford curves constructed by Gerritzen and Herrlich.
Our approach will, in particular, exhibit tropical Teichmueller space, a simplicial compactification of Culler-Vogtmann Outer space,
as a strong deformation retract of non-Archimedean Teichmueller space.
The crucial technical ingredient will be the theory of Artin fans which will allow us to lift tropical data to algebraic moduli functors.
TITLE :
Tropical reduction and lifting theorems.
SLIDES
ABSTRACT :
One can often assign to a geometric object over a non-archimedean field
geometric objects over the group of values -- a skeleton or a tropicalization,
and over the residue field -- a reduction. Sometimes one can
combine them in a single object, that we call tropical reduction. In simple cases
it is something well known, like a log variety, but often the definition is still rather ad hoc.
In my talk I will discuss two cases which reveal a surprising similarity: a wildly ramified
cover of curves with minimal wild ramification (joint work with U. Brezner), and a curve with a differential
form when the residue characteristic is zero (joint work with I. Tyomkin). In both cases there is a lifting theorem
indicating that our definition is the correct one and we do not lose any residual/tropical information:
any compatible tropical and residual data can be lifted to an object over the non-archimedean field.
TITLE :
On p-adic vector bundles and local systems.
SLIDES
ABSTRACT :
Correspondences between vector bundles on p-adic varieties and local
systems arise in the context of p-adic Simpson theory for vanishing Higgs field.
I will explain recent contributions to this theory
obtained jointly with Lucas Mann. We use Scholze¢s framework of diamonds
to prove a category equivalence on any proper adic space of finite type over Cp
relating Cp-local systems with integral models to certain modules under the structure sheaf for the v-topology.
TITLE :
K-stability from a non-Archimedean perspective.
SLIDES
ABSTRACT :
K-stability was introduced in complex differential geometry as a conjectural criterion (the YTD conjecture) for the existence
of special Kähler metrics, such as Kähler-Einstein metrics, or constant scalar curvature Kähler metrics. There has been a great
deal of recent activity around K-stability: the YTD conjecture is solved in many cases, a nice picture for K-stable
Fano varieties has emerged, and there is an increased understanding of the notion of K-stability itself.
In my talk, I will explain how K-stability can be viewed through the lens of non-Archimedean geometry.
This is based on joint work with Berman, Blum, Boucksom, and Hisamoto.
TITLE :
Tropical differential equations and their solutions.
SLIDES
ABSTRACT :
Tropical differential equations were introduced in 2015 by Dima
Grigoriev, whose interest was in the computational complexity of
finding the supports of power series solutions of systems of linear
ODEs. The next year, Fuensanta Aroca, Cristhian Garay and Zeinab
Toghani proved a Fundamental Theorem showing that every tropical
solution of a differential ideal lifts to a solution over a field.
I¢ll present this theory and some of the developments Zeinab, I and
others have achieved since, such as extensions to PDEs and connections
to Groebner theory.
TITLE :
Connections and symmetric differentials.
SLIDES
ABSTRACT :
If X is a smooth complex projective variety without differential form, every local system has finite monodromy (Brunebarbe-Klingler-Totaro).
In work in progress with Michael Groechenig, we develop a positive characteristic analog.
We¢ll discuss the rank 2 case and how ideally it relates to isocrystals.
TITLE :
Arithmetic differential operators with overconvergent coefficients.
SLIDES
ABSTRACT :
The ring of differential operators with overconvergent
coefficients plays a central role in Berthelot's theory of
arithmetic D-modules over a formal scheme, as well as
in Caro's theory of overholonomic modules. It is constructed over the complement of a divisor of some smooth formal scheme. We would
like to explain how to do such a construction in the case
where we have a non necessary smooth divisorial compactification of some
smooth formal scheme.
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Masha Vlasenko (Institute of Mathematics of the Polish Academy of Sciences, Warsaw)
TITLE :
Dwork's congruences and p-adic cohomology.
SLIDES
ABSTRACT :
This talk is about some curious p-adic properties of period functions, that were discovered by Bernard Dwork in his work on rationality of zeta functions.
I will demonstrate their generalization and explain a cohomological proof.
Our proof uses an explicit Dwork-style construction of Cartier operation on differential forms on toric hypersurfaces.
This is joint work with Frits Beukers.
TITLE :
A general theory of tropical differential equations.
SLIDES
ABSTRACT :
A few years ago Grigoriev introduced the concept of
tropicalization of differential equations, and then Aroca et
al. proved that there is an analogue of the fundamental
theorem of tropical geometry, where any solution at the
tropical level can be lifted to a classical solution.
Unfortunately, this theory is limited in its applicability,
since it only works with trivial valuations. In this talk I will
describe the work of Stefano Mereta to build a general
framework for tropical differential equations over
non-trivially valued rings, thus allowing potential
applications to p-adic differential equations. This involves
some interesting twists, such as a tropical variant of the
Leibniz rule. We support our theory with a differential
analogue of Payne's inverse limit theorem.
TITLE :
Arithmetic D-modules and rigid cohomologies
ABSTRACT :
Currently, we have at least two cohomological formalisms for p-adic cohomologies: rigid cohomology and cohomology defined using arithmetic D-modules.
It is natural to expect that these coincide, but this is not straightforward.
One of the reasons is that the construction of the functor associating arithmetic D-modules to isocrystals, due to Caro, is very complicated.
We give alternative construction which enables us to compare the cohomology theory.
This is a joint work with Chris Lazda.
TITLE :
Comparison theorems for p-adic analytic varieties.SLIDES
ABSTRACT :
I will discuss comparison theorems for p-adic analytic varieties proved recently in a joint work with Pierre Colmez.
A key input comes from a fine study of Banach-Colmez spaces appearing in cohomology of analytic varieties.
TITLE :
Derived Cartier transform, derived Satake equivalence and cohomology.SLIDES
ABSTRACT :
Derived localization theorem for modules over g=Lie(G)
where G is
a reductive algebraic group over a filed of positive characteristic relates g-modules
to crystalline D-modules on the flag variety G/B. It can be composed
with Cartier transform to relate representations of g to coherent sheaves on the
cotangent bundle T*G/B. A related description of modules over the
Frobenius kernel Gr involves coherent sheaves on a certain derived scheme S
mapping to T*G/B. That derived scheme S plays a role in (local) geometric Langlands
duality which leads to old and possibly new connections between modular representations
and geometric Langlands duality.
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POSTERS
Stefano Mereta (Swansea University, UK & Université Grenoble Alpes, France)
TITLE :
A colimit theorem in tropical differential algebra.POSTER
MAIN GOAL :
Extend the setting of the theory of differential
tropical equations as introduced by D.Grigoriev, and F.Aroca-C.Garay-Z.Toghani
to encompass the non-trivially valued case, introduce
a differential tropicalization functor and
prove an inverse limit theorem in a similar fashion
to S.Payne and J.H.Giansiracusa & N.Giansiracusa.
TITLE :
Fourier-Mukai transform for formal schemes.POSTER
Objective :
In 1981, Mukai constructed the Fourier-Mukai transform for abelian varieties over an algebraically
closed field, which gives an equivalence of categories between quasi-coherent sheaves over A and
the ones over its dual variety. Laumon generalized these results for abelian varieties over a locally
noetherian basis.
One can then ask the following question: can these results be generalized even more? What about
formal abelian varieties? And abelian rigid analytic varieties?
The generalization of the Fourier-Mukai transform's construction is based on a simple idea: make the
classical construction commute with (derived) inverse limit. Even if the idea seems simple, it implies
to clearly understand quasi-coherent sheaves and functors defined over formal varieties.
When the formal Fourier-Mukai transform is constructed and its fundamental results has been proved,
one can then obtain these results over its generic fiber.
