Feb 03, 2014
Monday

09:15 AM  09:30 AM


Welcome

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements



09:30 AM  10:30 AM


Specialities for nonspecialists
Antoine ChambertLoir (Université ParisSud (Orsay))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The goal of this 3hour 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 ManinMumford and MordellLang (which became theorems in the last 30 years under the hands of Raynaud, Hindry, Faltings, McQuillan, Vojta, Hrushovski...), the more recent conjectures of BombieriMasserZannier on anomalous intersections, the conjecture of AndréOort, and their extensions by ZilberPink. 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'' MumfordTate 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.
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


An Introduction to StabilityTheoretic Techniques
Pierre Simon (Centre national de la recherche scientifique (CNRS))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The aim of this tutorial is to present the modeltheoretic 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
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


A modeltheorist's view of algebraically closed valued fields
Deirdre Haskell (McMaster University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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 modeltheoretic approach to Berkovich space
 Supplements


03:00 PM  04:00 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



04:00 PM  05:00 PM


MSRI/Evans Lecture: Around Approximate Subgroups (Evans Hall, UC Berkeley)
Ehud Hrushovski (Hebrew University)

 Location
 UC Berkeley, 60 Evans Hall
 Video


 Abstract
 
 Supplements



Feb 04, 2014
Tuesday

09:30 AM  10:30 AM


Specialities for nonspecialists
Antoine ChambertLoir (Université ParisSud (Orsay))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The goal of this 3hour 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 ManinMumford and MordellLang (which became theorems in the last 30 years under the hands of Raynaud, Hindry, Faltings, McQuillan, Vojta, Hrushovski...), the more recent conjectures of BombieriMasserZannier on anomalous intersections, the conjecture of AndréOort, and their extensions by ZilberPink. 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'' MumfordTate 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.
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


An Introduction to StabilityTheoretic Techniques
Pierre Simon (Centre national de la recherche scientifique (CNRS))

 Location
 MSRI: Atrium
 Video

 Abstract
The aim of this tutorial is to present the modeltheoretic 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.
 Supplements


12:00 PM  01:30 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



01:30 PM  02:30 PM


Some applications of Hrushovski's theorem about the Frobenius map to algebraic dynamics
Ekaterina Amerik (Higher School of Economics)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I aim to survey some results in algebraic dynamics which rely on Hrushovski's theory of the Frobenius map: density of periodic points (by Fakhruddin) and of nonpreperiodic algebraic points (by myself) and, if time permits, some related work.
 Supplements


02:30 PM  03:30 PM


A Model Theoretic Approach to Berkovich Spaces
Martin Hils (Université Paris Diderot  Paris VII)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
If K is a (complete) eld with respect to a nonarchimedean 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 complexanalytic case. One approach to overcome this problem is due to Berkovich in the late 80's. He develops a theory of Kanalytic spaces, adding points to the set of naive points. In particular, for an algebraic variety V dened over K he constructs what is now called its Berkovich analytication 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)denable space bV which is a close analogue of V an, and they establish strong topological tame ness properties in this denable setting, combining ominimality 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 analytication of an algebraic variety. I will then explain how the model theory of ACVF, in particular the theory of stable domination (due to HaskellHrushovskiMacpherson 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 denability, I will sketch HrushovskiLoeser's construction of a strong de formation retraction of bV onto a piecewise linear subspace (in the denable category).
 Supplements


03:30 PM  04:00 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



04:00 PM  05:00 PM


Model theory and multiplicative combinatorics
Lou van den Dries (University of Illinois at UrbanaChampaign)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 
 Supplements


05:00 PM  07:00 PM


Reception

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements




Feb 05, 2014
Wednesday

09:30 AM  10:30 AM


Specialities for nonspecialists
Antoine ChambertLoir (Université ParisSud (Orsay))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The goal of this 3hour 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 ManinMumford and MordellLang (which became theorems in the last 30 years under the hands of Raynaud, Hindry, Faltings, McQuillan, Vojta, Hrushovski...), the more recent conjectures of BombieriMasserZannier on anomalous intersections, the conjecture of AndréOort, and their extensions by ZilberPink. 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'' MumfordTate 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.
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


An Introduction to StabilityTheoretic Techniques
Pierre Simon (Centre national de la recherche scientifique (CNRS))

 Location
 MSRI:
 Video

 Abstract
The aim of this tutorial is to present the modeltheoretic 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.
 Supplements


12:00 PM  01:00 PM


The PilaWilkie Theorem and variations
Margaret Thomas (Universität Konstanz)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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 ominimal expansions of the real field. This result has brought about a lively interaction in recent years between ominimality 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 PilaWilkie Theorem, indicating the main ingredients involved in the proof. In particular, we will focus on the key step known as the PilaWilkie Reparameterization Theorem. This is a model theoretic statement about the geometry of sets definable in ominimal 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 PilaWilkie 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 PilaWilkie notion of parameterization could play an important role in the pursuit of this conjecture.
 Supplements



Feb 06, 2014
Thursday

09:30 AM  10:30 AM


A Model Theoretic Approach to Berkovich Spaces
Martin Hils (Université Paris Diderot  Paris VII)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
If K is a (complete) eld with respect to a nonarchimedean 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 complexanalytic case. One approach to overcome this problem is due to Berkovich in the late 80's. He develops a theory of Kanalytic spaces, adding points to the set of naive points. In particular, for an algebraic variety V dened over K he constructs what is now called its Berkovich analytication 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)denable space bV which is a close analogue of V an, and they establish strong topological tame ness properties in this denable setting, combining ominimality 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 analytication of an algebraic variety. I will then explain how the model theory of ACVF, in particular the theory of stable domination (due to HaskellHrushovskiMacpherson 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 denability, I will sketch HrushovskiLoeser's construction of a strong de formation retraction of bV onto a piecewise linear subspace (in the denable category).
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Stability theory and Diophantine geometry
Anand Pillay (University of Notre Dame)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will discuss various aspects of the model theoretic approach to ManinMumford and functional MordellLang, 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 MordellLang to function field ManinMumford, and possibly the recent work of Roessler and Corpet
 Supplements


12:00 PM  01:30 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



01:30 PM  02:30 PM


OMinimal Ingredients in Proofs of Arithmetical Conjectures Such as ManinMumford and AndreOort
Ya'acov Peterzil (University of Haifa)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The goal of these talks is to describe a method which was rst proposed by Pila and Zannier in their new proof of the ManinMumford conjecture ([4]), and since then implemented to various other arithmetical questions. Most notably, it was used by Pila to prove the AndreOort 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 (apriori 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 denable sets in ominimal structures. That theorem itself has a similar form to the above conjectures: Assume that X Rn is a denable set in an ominimal structure. If X \ Qn is \large" then X must contain an innite semialgebraic set. More precisely, if one removes from X all innite semialgebraic sets then the remaining intersection with Qn is \small". The application of PilaWilkie to the arithmetic problems goes via three steps: 1. Find a map , denable in an ominimal structure M which transforms the variety V to an Mdenable set eV Cn. Let (X) = eX . These maps are usually dened via classical transcendental maps such as the complex exponentials or the Jinvariant. 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 PilaWilkie. 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 PilaWilkie, 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 PilaWilkie one obtains a nontrivial semialgebraic subset eS eX . Replacing S by such subset of maximal dimension, one tries to show that
 Supplements


02:30 PM  03:30 PM


Model theory and multiplicative combinatorics
Lou van den Dries (University of Illinois at UrbanaChampaign)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 
 Supplements


03:30 PM  04:00 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



04:00 PM  05:00 PM


A regularity lemma for definable sets over finite fields, and expanding polynomials
Terence Tao (University of California, Los Angeles)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Szemeredi's regularity lemma provides a structural description of arbitrary large dense graphs; roughly speaking, it decomposes any such graphs into a bounded number of pieces, most of which behave "pseudorandomly". While this lemma applies to arbitrary graphs, the dependence on constants can be terrible, and there can also exist "bad" components which are not pseudorandom. However, these defects can sometimes be removed if further assumptions are placed on the original graph. We present an "algebraic regularity lemma" which covers the case when the graph is definable (with bounded complexity) over a finite field of large characteristic. By combining the modeltheoretic classification of definable sets in this setting with an ultraproduct argument (to transfer to the characteristic zero setting) combined with algebraic geometry tools (such as the etale fundamental group) and the LangWeil inequality, we are able to get much better control on the pseudorandomness and definability properties of the components, and also can exclude the existence of bad components. As an application of this lemma, we classify the polynomials over finite fields of large characteristic which are "expanding" in a combinatorial sense.
 Supplements



Feb 07, 2014
Friday

09:30 AM  10:30 AM


A Model Theoretic Approach to Berkovich Spaces
Martin Hils (Université Paris Diderot  Paris VII)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
If K is a (complete) eld with respect to a nonarchimedean 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 complexanalytic case. One approach to overcome this problem is due to Berkovich in the late 80's. He develops a theory of Kanalytic spaces, adding points to the set of naive points. In particular, for an algebraic variety V dened over K he constructs what is now called its Berkovich analytication 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)denable space bV which is a close analogue of V an, and they establish strong topological tame ness properties in this denable setting, combining ominimality 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 analytication of an algebraic variety. I will then explain how the model theory of ACVF, in particular the theory of stable domination (due to HaskellHrushovskiMacpherson 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 denability, I will sketch HrushovskiLoeser's construction of a strong de formation retraction of bV onto a piecewise linear subspace (in the denable category).
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Stability theory and Diophantine geometry
Anand Pillay (University of Notre Dame)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will discuss various aspects of the model theoretic approach to ManinMumford and functional MordellLang, 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 MordellLang to function field ManinMumford, and possibly the recent work of Roessler and Corpet
 Supplements


12:00 PM  01:30 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



01:30 PM  02:30 PM


OMinimal Ingredients in Proofs of Arithmetical Conjectures Such as ManinMumford and AndreOort
Ya'acov Peterzil (University of Haifa)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The goal of these talks is to describe a method which was rst proposed by Pila and Zannier in their new proof of the ManinMumford conjecture ([4]), and since then implemented to various other arithmetical questions. Most notably, it was used by Pila to prove the AndreOort 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 (apriori 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 denable sets in ominimal structures. That theorem itself has a similar form to the above conjectures: Assume that X Rn is a denable set in an ominimal structure. If X \ Qn is \large" then X must contain an innite semialgebraic set. More precisely, if one removes from X all innite semialgebraic sets then the remaining intersection with Qn is \small". The application of PilaWilkie to the arithmetic problems goes via three steps: 1. Find a map , denable in an ominimal structure M which transforms the variety V to an Mdenable set eV Cn. Let (X) = eX . These maps are usually dened via classical transcendental maps such as the complex exponentials or the Jinvariant. 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 PilaWilkie. 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 PilaWilkie, 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 PilaWilkie one obtains a nontrivial semialgebraic subset eS eX . Replacing S by such subset of maximal dimension, one tries to show that
 Supplements


02:30 PM  03:30 PM


Algebraic Dynamics and the Model Theory of Difference Fields
Alice Medvedev (City College, CUNY)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
An algebraic dynamical system is a variety $X$ with a selfmorphism $F : X \rightarrow X$. 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 modeltheoretic 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 coordinatewise polynomial dynamics.
 Supplements


03:30 PM  04:00 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



04:00 PM  05:00 PM


Model theory and multiplicative combinatorics
Lou van den Dries (University of Illinois at UrbanaChampaign)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 
 Supplements


