Vladimir BERKOVICH (Weizmann Institute) : F1 Geometry

Antoine DUCROS (Université Paris 6) : Etale cohomology

Mattias JONSSON (University of Michigan) : Dynamics on Berkovich spaces

François LOESER (Ecole Normale Supérieure, Paris) : Model theory and analytic geometry

Jérome POINEAU (Université de Strasbourg) : Berkovich spaces over Z

Michael TEMKIN (IAS and University of Pennsylvania) : Introduction to Berkovich analytic spaces

Bertrand REMY et Amaury THUILLIER (Université de Lyon) : Bruhat-Tits buildings and analytic geometry

Vladimir BERKOVICH (Weizmann Institute) : F1 Geometry

Abstract:

The minicourse is an introduction to work in progress on foundations of
algebraic and analytic geometry over the field of one element. This work
originates in non-Archimedean analytic geometry as a result of a search
for appropriate framework for so called skeletons of analytic spaces and
formal schemes, and is related to logarithmic and tropical geometry.

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Antoine DUCROS (University Paris 6) : Etale cohomology

Abstract:

Étale cohomology was introduced in the scheme-theoretic context by
Grothendieck in the 50's and 60's in order to provide a purely algebraic
cohomology theory, satisfying the same fundamental properties as the
singular cohomology of complex varieties, which was needed for proving the
Weil conjectures.

For other deep arithmetic reasons (that times, related to Langlands
program) it appeared later that it should also be worthwhile developing
such a theory in the p-adic analytic context ; this was done by Berkovich
in the early 90's.

In  this serie of lectures, I plan, after having given some general
motivations, to spend some times about the notion of a Grothendieck
topology and its associated cohomology theory; then I will explain the
basic ideas and properties of both scheme-theoretic and
Berkovich-theoretic étale cohomology theories (which are closely related
to each other), and the fundamental results like various comparison
theorems, Poincaré duality, purity and so on.

My purpose won't be to give detailed proofs, which are for most of them
highly technically involved; I will rather insist on examples, trying to
show how étale cohomology can at the same time be quite closed to the
classical topological intuition, and deal in a completely natural manner
with deep field arithmetic phenomena (as an example, it encodes Galois
theory).... which allows sometimes to think to the latter in a purely
geometrical way.

References:

The foundations of scheme-theoretic étale cohomology (definition of étale morphisms, the fundamental group, the notion of a Grothendieck topology and the étale cohomology itself) have been written in SGA 1 (LNM 224, new edition by the SMF now available) and SGA IV (LNM 269, 270 and 305), but both those books are highly technical, and not easily readable by somebody who feels not very comfortable with scheme theory. The book SGA 4 1/2 (LNM 569) which presents the results without proofs, is more accessible ; it has the advantage to give some motivations and comparisons with the classical (i.e. topological) situation.

I would however suggest to the reader who is a little bit afraid by the SGA's to refer to Milne's book Étale cohomology , which is certainly simpler -- though it also requires some familiarity with schemes.

I mention a beautiful historical text about étale cohomology by Illusie, Grothendieck et la cohomologie étale.

As far as Berkovich's étale cohomology is concerned, the only reference I know at the moment is Berkovich's foundational article Étale cohomology for non-Archimedean analytic spaces
 in IHES 78 (1993).

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Mattias JONSSON (University of Michigan) : Dynamics on Berkovich spaces

Abstract:

In these lectures I will present two instances where dynamics on
Berkovich spaces appear naturally.

The first case is in the context of iterations of selfmaps of the (standard) projective line over a non-archimedean field such as the p-adic numbers. When trying to extend results from
the archimedean setting (over the complex numbers), it turns out be both natural and fruitful
to study the induced dynamics on the associated Berkovich projective line.

The second case concerns iterations of (germs of) holomorphic selfmaps
f:C2->C2 fixing the origin, f(0)=0. When the fixed point is superattracting, that is, the differential df(0) is identically zero,
the dynamics can be analyzed by studying the induced action on the Berkovich affine plane over the field C equipped with the trivial valuation.

Beyond the subject of dynamics, these lectures will provide a
"hands-on" introduction to Berkovich spaces in relatively concrete settings, where the topological structure is essentially
that of an R-tree. In studying the second instance above, we will
also have the opportunity to explore the link between Berkovich
spaces and the algebro-geometric study of valuations, going back to Zariski.

If time permits, I will also other dynamical situations, such as
the behavior at infinity of iterates of two-dimensional polynomial selfmaps. I may also briefly discuss the higher-dimensional case.
 
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François LOESER (Ecole normale supérieure) : Model theory and analytic geometry

Abstract:

Lecture 1 : A quick review of Model Theory.

1) Basic notions: Languages, structures, definable sets, types.
2) An example : o-minimal structures as a paradigm for tame topology.

Lecture 2 : Model Theory of valued fields.

1) An application of Model Theory in the 60's:  the Ax-Kochen-Ersov theorem.
2) Quantifier elimination for algebraically closed valued fields.
3) Elimination of imaginaries.

Lecture 3 : The basic object of study.

Given an algebraic variety U over a non-archimedean valued field, we introduce
the space  of  stably dominated types on it and prove that it is strictly pro-definable.
We also explain how it compares with U^an.

Lecture 4 : Some topological properties of Û

1) Definable compactness and its relation with properness
2) Γ-internal subsets of Û are topologically tame
3) More advanced material (if times allows).

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Jérome POINEAU (Strasbourg's university) : Berkovich spaces over Z

Abstract:

We shall split the course into two parts.
First we shall recall Berkovich's general construction of
analytic spaces over Z with a particular emphasis on the affine line. It
is a remarkable fact that in many respects this space behaves like usual
analytic spaces: it is for instance Hausdorff, locally compact and locally
path connected, and its local rings are Henselian, Noetherian, and regular.


Second, we shall turn our attention to the study of Stein subsets of the
affine line over Z. These subsets are defined in terms of the vanishing of
coherent cohomology. We shall derive some applications of this study to
the construction of convergent power series with integral coefficients
having prescribed poles, and to the inverse Galois problem.

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Michael TEMKIN (Penn state university) : Introduction to Berkovich analytic spaces

Abstract:

In this mini-course we will introduce Berkovich analytic spaces over a non- archimedean field
and will study their basic properties. A familiarity with algebraic geometry and commutative algebra
is the main prerequisite for the course. Some familiarity with field valuations and formal schemes
may also be helpful, though I will mention briefly the facts we will need about them. In order to cover
the large amount of material we will concentrate on describing definitions and constructions and
formulating the main results of the theory, although in some cases main ideas of the proofs will be outlined.
The course can be divided to five parts as follows: §1 valuations, non-archimedean fields and Banach algebras,
§2 affinoid algebras and spaces, §3 analytic spaces, §4 connection to other categories: analytification of algebraic
varieties and generic fiber of formal schemes, §5 analytic curves.

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Bertrand REMY et Amaury THUILLIER (Université de Lyon) : Bruhat-Tits buildings and analytic geometry

Abstract:

Let G be a reductive algebraic group defined over a non-Archimedean local field k.

During the 60ies and 70ies, F. Bruhat and J. Tits have been working on a fine description
of groups of rational points like G(k). The achievement of this work is a combinatorial
description that can be stated in geometric terms, i.e., using the Euclidean building of G over k.
The latter space, which is both a complete metric space and a simplicial complex, can
be seen in many ways as a (singular) analogue of the Riemannian symmetric space of a
semisimple real Lie group.

During the 80ies, V. Berkovich developed an approach to analytic geometry
over non-Archimedean complete fields, thus enriching the classical theory
due to Tate-Raynaud. From the beginning he mentionned the possibility to
combine his theory with Bruhat-Tits' one.

In these talks, we intend to present a joint work with A. Werner, in which
we develop and extend Berkovich's ideas. We show in particular that they
enable one to define and describe natural compactifications of the Bruhat-Tits
building of a group G over k as above. These compactifications can also be obtained
- again by means of Berkovich geometry - by procedures which very much look like
Satake's initial ideas for symmetric spaces.

The necessary material concerning Bruhat-Tits theory will be presented during the first lecture.

Matthew BAKER: Complex dynamics and adelic potential theory.

We will discuss the proof of the following theorem: For any xed complex
numbers a and b, and any integer d at least 2, the set of complex numbers c for which
both a and b are preperiodic for zd + c is innite if and only if ad = bd. This provides
an affirmative answer to a question of Zannier. The main ingredients in the proof are a
complex-analytic study of certain generalized Mandelbrot sets and an adelic equidistribution
theorem for preperiodic points over number elds and function elds. Somewhat
surprisingly, non-Archimedean Berkovich spaces play an essential role in the arguments
even though the theorem is purely about complex dynamics.
This is joint work with Laura DeMarco.

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Laurent FARGUES: p-adic periods and local Langlands correspondences.

I will speak about the realization of local Langlands correspondences in the
l-adic étale cohomology of rigid analytic spaces contructed by Rapoport and Zink. Those are
p-adic local analogs of Shimura varieties and hermitian symmetric spaces.

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Walter GUBLER: Canonical measures on subvarieties of abelian varieties.

Chambert-Loir has introduced a measure on the Berkovich space which is a
p-adic analogue of a top-dimensional wedge product of first Chern forms.
If we consider a subvariety X of an abelian variety and a symmetric ample
line bundle endowed with a canonical metric, then we get a positive
measure on the Berkovich space associated to X which we call a canonical
measure. In the talk, we present a very explicit description of this
canonical measure in terms of a semistable alteration and convex geometry.
This has applications for the Bogomolov conjecture over function fields.

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Kiran KEDLAYA: Nonarchimedean geometry of Witt vectors.

The construction of Witt vectors provides a functorial way to convert a
perfect ring R of characteristic p into a p-adically separated and
complete ring W(R) for which W(R)/(p) is isomorphic to R. Each Witt
vector can be written as a "power series in p" whose coefficients are
elements of a distinguished set of representatives of R in W(R) (the
Teichmuller lifts). This suggests that one might be able to relate the
geometry of the Gelfand spectra of R (with the trivial norm) and W(R)
(with the p-adic norm) by analogy with the comparison between R (with
some norm) and R[T] (with the Gauss norm). That is, one expects the
spectrum of W(R) to look like a fibration in discs over the spectrum of
R. I'll indicate some ways in which this turns out to be correct, and
some ways in which this is still conjectural. For example, we prove that
the spectrum of W(R) is a strong deformation retract of the spectrum of
R; hence the fibres of the projection from the spectrum of W(R) onto the
spectrum of R are contractible.

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Emmanuel LEPAGE: Tempered fundamental group.

The tempered fundamental group of a Berkovich space classifies étale
coverings that become topological coverings after pullback by some finite
étale covering. Even in geometric cases (over C_p for example), the
tempered fundamental group depends much more of the variety itself than
the profinite fudamental group of Grothendieck or the topological
fundamental group of complex analytic varieties. For example, according to
Mochizuki, one can recover the graph of the stable reduction of a
hyperbolic curve from its tempered fundamental group. For Mumford curves,
one can even recover the width of the nodes of the stable reduction.

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Johannes NICAISE: Berkovich spaces and limit mixed Hodge structures.

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Frédéric PAUGAM: spectral symmetries of zeta functions and global analytic geometry.

We will define a symplectic pairing on Connes/Meyer's spectral interpretation
of the zeroes of Riemann's zeta function. This symplectic pairing was defined
by Sarnak in a letter to Bombieri under the Riemann hypothesis. Our construction
is unconditionnal. We will also define independently a sheafified version of Weil's
local spectral interpretation using Berkovich's global analytic geometry and describe
the difficulties that appear when trying to combine those two results.

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Sam PAYNE: Lifting tropical intersections.

I will discuss tropicalizations of embedded subvarieties of toric varieties,
which are piecewise linear images of their nonarchimedean analytifications.
The tropicalization of the intersection of two subvarieties is contained in
the intersection of their tropicalizations, but this containment is
sometimes strict.  I will discuss joint work with Brian Osserman giving
sufficient conditions for lifting points in the tropical intersection to
algebraic points in the intersection of the subvarieties.

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Ilya TYOMKIN: Berkovich spaces and Tropical geometry.

In my talk I will describe the connection between skeletons of Berkovich spaces and
tropical varieties. I will show that the canonical tropicalization of an algebraic curve
with marked points is nothing but the skeleton of the corresponding Berkovich analytification.
As an application of tropical curves (=skeletons of analytic curves), I will prove Zariski's
theorem in arbitrary characteristic.