Workshop

The goal of this 3-hour tutorial is to motivate and explain in some detail the statements of various conjectures/theorems which have attracted the attention of many number theorists in recent years: from the classic conjectures of Manin-Mumford and Mordell-Lang (which became theorems in the last 30 years under the hands of Raynaud, Hindry, Faltings, McQuillan, Vojta, Hrushovski...), the more recent conjectures of Bombieri-Masser-Zannier on anomalous intersections, the conjecture of André-Oort, and their extensions by Zilber-Pink. Most of these conjectures take place in an ambient variety with a natural notion of special points and of special subvarieties and ask whether subvarieties which possess a dense subset of special points must be special. The specific notion varies among the context. The ambient variety may be an abelian variety (or more generally a commutative algebraic group), in which case special points are torsion points, or the points of a subgroup of finite rank, and special subvarieties are build up from translates of connected algebraic subgroups and ``constant'' subvarieties. The ambient variety may also be a Shimura variety, then special points and special varieties are those associated with ``maximal'' Mumford-Tate groups. The maximal generalization is due to Pink and lies within the framework of Shimura mixed varieties. The main goal of these lectures is to describe the statements of these conjectures/results. Time permitting, I will give a rough sketch of some of the proofs, so as both to make the connexion with the other lectures and to introduce the audience to the vast litterature on the topic.

The aim of this tutorial is to present the model-theoretic tools used in applications to other areas of mathematics. I will focus on notions which will be employed in the other tutorials such as pseudofinite structures, stability, NIP, measures

I will start with the definition of an algebraically closed valued field, and observe the role played by the families of open and closed balls, and how the residue field and value group can be seen in different subfamilies. We will look at the definable sets and definable equivalence classes in one variable, and discuss how to generalise to more variables. This will lead to looking at examples of stably dominated types, which play a fundamental role in the model-theoretic approach to Berkovich space

The goal of this 3-hour tutorial is to motivate and explain in some detail the statements of various conjectures/theorems which have attracted the attention of many number theorists in recent years: from the classic conjectures of Manin-Mumford and Mordell-Lang (which became theorems in the last 30 years under the hands of Raynaud, Hindry, Faltings, McQuillan, Vojta, Hrushovski...), the more recent conjectures of Bombieri-Masser-Zannier on anomalous intersections, the conjecture of André-Oort, and their extensions by Zilber-Pink. Most of these conjectures take place in an ambient variety with a natural notion of special points and of special subvarieties and ask whether subvarieties which possess a dense subset of special points must be special. The specific notion varies among the context. The ambient variety may be an abelian variety (or more generally a commutative algebraic group), in which case special points are torsion points, or the points of a subgroup of finite rank, and special subvarieties are build up from translates of connected algebraic subgroups and ``constant'' subvarieties. The ambient variety may also be a Shimura variety, then special points and special varieties are those associated with ``maximal'' Mumford-Tate groups. The maximal generalization is due to Pink and lies within the framework of Shimura mixed varieties. The main goal of these lectures is to describe the statements of these conjectures/results. Time permitting, I will give a rough sketch of some of the proofs, so as both to make the connexion with the other lectures and to introduce the audience to the vast litterature on the topic.

The aim of this tutorial is to present the model-theoretic tools used in applications to other areas of mathematics. I will focus on notions which will be employed in the other tutorials such as pseudofinite structures, stability, NIP, measures.

Pila and Wilkie's influential counting theorem provides a bound on the density of rational points of bounded height lying on the 'transcendental parts' of sets definable in o-minimal expansions of the real field. This result has brought about a lively interaction in recent years between o-minimality and diophantine geometry, including several important applications to arithmetical conjectures which will be explored further in Ya'acov Peterzil's tutorial. As a prelude to this, we will provide an introduction to the Pila-Wilkie Theorem, indicating the main ingredients involved in the proof. In particular, we will focus on the key step known as the Pila-Wilkie Reparameterization Theorem. This is a model theoretic statement about the geometry of sets definable in o-minimal expansions of real closed fields - namely that they can be covered by the images of finitely many sufficiently differentiable functions with bounded derivatives. Following the Pila-Wilkie Theorem, subsequent work carried out by a number of authors, including Pila, Besson, Boxall, Butler, Jones, Masser and myself, has focussed on establishing that a sharper bound holds in certain situations. One important goal is a conjecture of Wilkie concerning sets definable in the real exponential field. We shall explore some of the cases of this conjecture already established and the methods involved, indicating how a suitable modification of the Pila-Wilkie notion of parameterization could play an important role in the pursuit of this conjecture.

If K is a (complete) eld with respect to a non-archimedean ab- solute value, then K is totally disconnected as a topological eld. This is a serious obstacle when one wants to develop analytic geometry over K as this is done in the complex-analytic case. One approach to overcome this problem is due to Berkovich in the late 80's. He develops a theory of K-analytic spaces, adding points to the set of naive points. In particular, for an algebraic variety V dened over K he constructs what is now called its Berkovich analytication V an which contains V (K) (with its natural topology) as a dense subspace and which is locally compact and locally arcwise connected. Recently, using the geometric model theory of algebraically closed valued elds (ACVF), Hrushovski and Loeser constructed a (pro-)denable space bV which is a close analogue of V an, and they establish strong topological tame- ness properties in this denable setting, combining o-minimality with tools from stability theory. These properties are then shown to transfer to the ac- tual Berkovich setting. In the tutorial, I will rst introduce the notion of a Berkovich space, em- phasising the analytication of an algebraic variety. I will then explain how the model theory of ACVF, in particular the theory of stable domination (due to Haskell-Hrushovski-Macpherson and which will be discussed in Deirdre Haskell's talk), is used in the work of Hrushovski and Loeser to construct the space bV . Finally, in the case where V is an algebraic curve, stressing the role of denability, I will sketch Hrushovski-Loeser's construction of a strong de- formation retraction of bV onto a piecewise linear subspace (in the denable category).

I will discuss various aspects of the model theoretic approach to Manin-Mumford and functional Mordell-Lang, originating in Hrushovski's work in the 1990's. The underlying structural results were dichotomy theorems for minimal sets in difference closed fields and differentially closed fields. I will also touch on later developments: the canonical base property, the possibility of reducing function field Mordell-Lang to function field Manin-Mumford, and possibly the recent work of Roessler and Corpet

The goal of these talks is to describe a method which was rst proposed by Pila and Zannier in their new proof of the Manin-Mumford conjecture ([4]), and since then implemented to various other arithmetical questions. Most notably, it was used by Pila to prove the Andre-Oort conjecture for Cn, [3]. A common feature to the problems which t this method is their general form: Name a collection of \special" complex algebraic varieties (e.g. Abelian varieties, Shimura varieties, mixed Shimura varieties). Within one such special variety V one isolates a set of \special points" (e.g. the set of torsion points in the group of an abelian variety). Consider an algebraic subvariety X of V (a-priori not necessarily special itself) and assume that X contains \many" (e.g. A Zariski dense set of) special points. The conjectures claim that under these assumptions X itself must be a special variety, or at least contain a nontrivial special subvariety. The novelty in the method of Pila and Zannier is the application of a Theorem of Pila and Wilkie, [8] about rational points on denable sets in o-minimal structures. That theorem itself has a similar form to the above conjectures: Assume that X  Rn is a denable set in an o-minimal structure. If X \ Qn is \large" then X must contain an innite semialgebraic set. More precisely, if one removes from X all innite semialgebraic sets then the remaining intersection with Qn is \small". The application of Pila-Wilkie to the arithmetic problems goes via three steps: 1. Find a map , denable in an o-minimal structure M which transforms the variety V to an M-denable set eV  Cn. Let (X) = eX . These maps are usually dened via classical transcendental maps such as the complex exponentials or the J-invariant. In all cases thus far it is sucient to consider the structure = Ran;exp. 2. Show that if X contains \many" special points in the arithmetical sense then eX contains a \large" set of rational or algebraic points, in the sense of Pila-Wilkie. This step is number theoretic in nature. It requires rst to show that  sends the special points in X to rational (or more generally algebraic) points in eX . Mainly, in order to apply Pila-Wilkie, one is required to obtain precise lower bounds on the number of the rational (or algebraic) points, in terms of their height. 3. After applying Pila-Wilkie one obtains a nontrivial semialgebraic subset eS  eX . Replacing S by such subset of maximal dimension, one tries to show that 

An algebraic dynamical system is a variety $X$ with a self-morphism $F : X \rightarrow X$. Difference equations are a natural tool for understanding such discrete dynamical systems, just as differential equations are natural toop for understanding continuous dymanical systems.  "Model theory of difference fields" (1999) by Chatzidakis and Hrushovski is a beautiful application of geometric stability theory (one of the shiniest parts of modern model theory!) to algebraic difference equations. Several results in arithmetic dynamics have now been obtained through this model-theoretic way of thinking about difference equations. I will describe several of these, including the work on Chatzidakis and Hrushovski on descent in algebraic dynamics and my work with Scanlon on coordinate-wise polynomial dynamics.