The program consists of 7 mini-courses of 4 to 6 hours.

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

The workshop on model theory of friday, July 2 is directly related to the conference's topic.

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.

----

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).

----

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.

----

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 Â 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).

----

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.

----

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.

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

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 innite 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.

----

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.

----

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.

----

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.

----

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 innite 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.

----

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.

----

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.

----

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.

----

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.

----

Johannes NICAISE: Berkovich spaces and limit mixed Hodge structures.

----

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.

----

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.

----

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.

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.

----

Johannes NICAISE: Berkovich spaces and limit mixed Hodge structures.

----

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.

----

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.

----

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.