# Timetable (MAAW01)

## An Introduction to Recent Applications of Model Theory

Tuesday 29th March 2005 to Friday 8th April 2005

 09:00 to 10:00 RegistrationSession: An Introduction to Recent Applications of Model Theory 10:00 to 11:00 Introduction INI 1 11:00 to 11:30 Coffee and posters 11:30 to 12:30 A Pillay ([Illinois])Stability, differential fields, and related structures We will discuss stability theory and the model theory of differential fields, difference fields, separably closed fields, and compact complex manifolds. We will survey the applications to and connections with diophantine geometry, complex geometry, and the arithmetic of differential equations. INI 1 12:30 to 13:30 Lunch at Churchill College 14:00 to 15:00 Stability, differential fields, and related structures We will discuss stability theory and the model theory of differential fields, difference fields, separably closed fields, and compact complex manifolds. We will survey the applications to and connections with diophantine geometry, complex geometry, and the arithmetic of differential equations. INI 1 15:00 to 15:30 Tea and posters 15:30 to 16:30 Model theory of algebraically closed valued fields This tutorial of 5 lectures will be an exposition of a proof of elimination of imaginaries for the theory of algebraically closed valued fields (ACVF), when certain extra sorts from M^eq (coset spaces) are added. The proof will be based on that of [1], though further recent ideas of Hrushovski may be incorporated. The tutorial will begin with a general account of the basic model theory of ACVF and the notion of elimination of imaginaries, and will end with further developments from [2]: in particular, the notion of stable domination. 1. D. Haskell, E. Hrushovski, H.D. Macpherson, Definable sets in algebraically closed valued fields. Part I: elimination of imaginaries', preprint. 2. D. Haskell E. Hrushovski, H.D. Macpherson, Definable sets in algebraically closed valued fields. Part II: stable domination and independence.' INI 1 16:30 to 17:30 Model theory of algebraically closed valued fields This tutorial of 5 lectures will be an exposition of a proof of elimination of imaginaries for the theory of algebraically closed valued fields (ACVF), when certain extra sorts from M^eq (coset spaces) are added. The proof will be based on that of [1], though further recent ideas of Hrushovski may be incorporated. The tutorial will begin with a general account of the basic model theory of ACVF and the notion of elimination of imaginaries, and will end with further developments from [2]: in particular, the notion of stable domination. 1. D. Haskell, E. Hrushovski, H.D. Macpherson, Definable sets in algebraically closed valued fields. Part I: elimination of imaginaries', preprint. 2. D. Haskell E. Hrushovski, H.D. Macpherson, Definable sets in algebraically closed valued fields. Part II: stable domination and independence.' INI 1 17:30 to 18:30 Wine Reception 18:45 to 19:30 Dinner at Wolfson Court (Residents only)
 09:00 to 10:00 A Pillay ([Illinois])Stability, differential fields, and related structuresSession: An Introduction to Recent Applications of Model Theory We will discuss stability theory and the model theory of differential fields, difference fields, separably closed fields, and compact complex manifolds. We will survey the applications to and connections with diophantine geometry, complex geometry, and the arithmetic of differential equations. INI 1 10:00 to 11:00 Stability, differential fields, and related structures We will discuss stability theory and the model theory of differential fields, difference fields, separably closed fields, and compact complex manifolds. We will survey the applications to and connections with diophantine geometry, complex geometry, and the arithmetic of differential equations. INI 1 11:00 to 11:30 Coffee and posters 11:30 to 12:30 Model theory of algebraically closed valued fields This tutorial of 5 lectures will be an exposition of a proof of elimination of imaginaries for the theory of algebraically closed valued fields (ACVF), when certain extra sorts from M^eq (coset spaces) are added. The proof will be based on that of [1], though further recent ideas of Hrushovski may be incorporated. The tutorial will begin with a general account of the basic model theory of ACVF and the notion of elimination of imaginaries, and will end with further developments from [2]: in particular, the notion of stable domination. 1. D. Haskell, E. Hrushovski, H.D. Macpherson, Definable sets in algebraically closed valued fields. Part I: elimination of imaginaries', preprint. 2. D. Haskell E. Hrushovski, H.D. Macpherson, Definable sets in algebraically closed valued fields. Part II: stable domination and independence.' INI 1 12:30 to 13:30 Lunch at Churchill College 14:00 to 15:00 Model theory of algebraically closed valued fields This tutorial of 5 lectures will be an exposition of a proof of elimination of imaginaries for the theory of algebraically closed valued fields (ACVF), when certain extra sorts from M^eq (coset spaces) are added. The proof will be based on that of [1], though further recent ideas of Hrushovski may be incorporated. The tutorial will begin with a general account of the basic model theory of ACVF and the notion of elimination of imaginaries, and will end with further developments from [2]: in particular, the notion of stable domination. 1. D. Haskell, E. Hrushovski, H.D. Macpherson, Definable sets in algebraically closed valued fields. Part I: elimination of imaginaries', preprint. 2. D. Haskell E. Hrushovski, H.D. Macpherson, Definable sets in algebraically closed valued fields. Part II: stable domination and independence.' INI 1 15:00 to 15:30 Tea and posters INI 1 15:30 to 16:30 JB Bost ([Paris])Some problems arising from the Diophantine study of algebraic foliations This survey talk will be devoted to various problems arising in the study of Diophantine properties of algebraic foliations. Hopefully, I will explain (1) how algebraic foliations naturally enters into arithmetic geometry, (2) some known results established notably by means of Diophantine approximation techniques (concerning in particular the Grothendieck-Katz conjecture and its generalizations), and (3) discuss some Diophantine conjectures/problems, and some problems in (differential-)algebraic geometry arising from the use of Diophantine approximation techniques. This last part should present various issues where I expect model theory to be relevant. INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents only)
 09:00 to 10:00 A Pillay ([Illinois])Stability, differential fields, and related structuresSession: An Introduction to Recent Applications of Model Theory We will discuss stability theory and the model theory of differential fields, difference fields, separably closed fields, and compact complex manifolds. We will survey the applications to and connections with diophantine geometry, complex geometry, and the arithmetic of differential equations. INI 1 10:00 to 11:00 Model theory of algebraically closed valued fields This tutorial of 5 lectures will be an exposition of a proof of elimination of imaginaries for the theory of algebraically closed valued fields (ACVF), when certain extra sorts from M^eq (coset spaces) are added. The proof will be based on that of [1], though further recent ideas of Hrushovski may be incorporated. The tutorial will begin with a general account of the basic model theory of ACVF and the notion of elimination of imaginaries, and will end with further developments from [2]: in particular, the notion of stable domination. 1. D. Haskell, E. Hrushovski, H.D. Macpherson, Definable sets in algebraically closed valued fields. Part I: elimination of imaginaries', preprint. 2. D. Haskell E. Hrushovski, H.D. Macpherson, Definable sets in algebraically closed valued fields. Part II: stable domination and independence.' INI 1 11:00 to 11:30 Coffee and posters 11:30 to 12:30 A local Andre-Oort conjecture via ACFA INI 1 12:30 to 13:30 Lunch at Churchill College 14:00 to 15:00 I Fesenko ([Nottingham])Two dimensional arithmetic geometry and nonstandard mathematics This is review of a variety of potential and actual applications of nonstandard mathematics to arithmetic geometry; for the text see www.maths.nott.ac.uk/personal/ibf/rem.pdf Related Links http://www.maths.nott.ac.uk/personal/ibf/rem.pdf INI 1 15:00 to 15:30 Tea and posters 15:30 to 16:30 On a theorem of Simpson INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents only)
 09:00 to 10:00 Zariski-type structuresSession: An Introduction to Recent Applications of Model Theory INI 1 10:00 to 11:00 Zariski-type structures INI 1 11:00 to 11:30 Coffee and posters 11:30 to 12:30 CW Henson & A Berenstein ([Illinois])Model theory for metric structures A metric structure is a many-sorted structure with each sort a metric space, which for convenience is assumed to have finite diameter. Additionally, there are functions (of several variables) between sorts, assumed to be uniformly continuous. Examples include metric spaces themselves, measure algebras (with the metric d(A,B) = m(A*B), where * is symmetric difference), and structures based on Banach spaces (where one interprets the sorts as balls), including Banach lattices, C*-algebras, etc. The usual first-order logic does not work very well for such structures, and several good alternatives have been developed. One is the logic of positive bounded formulas with an approximate semantics (see [3]). Another is the setting of compact abstract theories (cats) (see [1]). A recent development, which will be emphasized in this course, is the convergence of these two points of view and the realization that they are also connected to the [0,1]-valued continuous logic that was studied extensively in the 1960s and then dropped (see [2]). This leads to a new formalism for the logic of metric structures in which the operations of sup and inf play a central role. The development of this continuous logic for metric structures is an ongoing project being carried out as a collaboration among Itay Ben-Yaacov (Wisconsin), Alex Berenstein (Illinois), Ward Henson (Illinois), and Alex Usvyatsov (Jerusalem). The analogy between this logic for metric space structures and the usual first order logic for ordinary structures is far reaching. Continuous logic satisfies the compactness theorem, Lowenheim-Skolem theorems, diagram arguments, existence of saturated and homogeneous models, characterizations of quantifier elimination and model completeness, Beth's definability theorem, Craig's interpolation theorem, the omitting types theorem, fundamental results of stability theory, and appropriate analogues of essentially all results in basic model theory of first order logic. The theory of definable sets and functions in this continuous logic for metric structures has novel aspects and there are numerous open problems of apparent interest. Extending important concepts from first order model theory to this continuous logic often presents a challenge. The purpose of this tutorial is to present this logic for metric structures, and to show a few of its application areas, with emphasis on probability spaces and certain linear spaces from functional analysis. References: [1] I. Ben-Yaacov, Positive model theory and compact abstract theories, J. Math. Logic 3 (2003), 85--118. [2] C. C. Chang and H. J. Keisler, Continuous Model Theory, Princeton Univ. Press (1966). [3] C. W. Henson and J. Iovino, Ultraproducts in Analysis, in Analysis and Logic, London Mathematical Society Lecture Notes 262, Cambridge University Press (2002), 1--113. INI 1 12:30 to 13:30 Lunch at Churchill College 14:00 to 15:00 CW Henson & A Berenstein ([Illinois])Model theory for metric structures A metric structure is a many-sorted structure with each sort a metric space, which for convenience is assumed to have finite diameter. Additionally, there are functions (of several variables) between sorts, assumed to be uniformly continuous. Examples include metric spaces themselves, measure algebras (with the metric d(A,B) = m(A*B), where * is symmetric difference), and structures based on Banach spaces (where one interprets the sorts as balls), including Banach lattices, C*-algebras, etc. The usual first-order logic does not work very well for such structures, and several good alternatives have been developed. One is the logic of positive bounded formulas with an approximate semantics (see [3]). Another is the setting of compact abstract theories (cats) (see [1]). A recent development, which will be emphasized in this course, is the convergence of these two points of view and the realization that they are also connected to the [0,1]-valued continuous logic that was studied extensively in the 1960s and then dropped (see [2]). This leads to a new formalism for the logic of metric structures in which the operations of sup and inf play a central role. The development of this continuous logic for metric structures is an ongoing project being carried out as a collaboration among Itay Ben-Yaacov (Wisconsin), Alex Berenstein (Illinois), Ward Henson (Illinois), and Alex Usvyatsov (Jerusalem). The analogy between this logic for metric space structures and the usual first order logic for ordinary structures is far reaching. Continuous logic satisfies the compactness theorem, Lowenheim-Skolem theorems, diagram arguments, existence of saturated and homogeneous models, characterizations of quantifier elimination and model completeness, Beth's definability theorem, Craig's interpolation theorem, the omitting types theorem, fundamental results of stability theory, and appropriate analogues of essentially all results in basic model theory of first order logic. The theory of definable sets and functions in this continuous logic for metric structures has novel aspects and there are numerous open problems of apparent interest. Extending important concepts from first order model theory to this continuous logic often presents a challenge. The purpose of this tutorial is to present this logic for metric structures, and to show a few of its application areas, with emphasis on probability spaces and certain linear spaces from functional analysis. References: [1] I. Ben-Yaacov, Positive model theory and compact abstract theories, J. Math. Logic 3 (2003), 85--118. [2] C. C. Chang and H. J. Keisler, Continuous Model Theory, Princeton Univ. Press (1966). [3] C. W. Henson and J. Iovino, Ultraproducts in Analysis, in Analysis and Logic, London Mathematical Society Lecture Notes 262, Cambridge University Press (2002), 1--113. INI 1 15:00 to 15:30 Tea and posters 15:30 to 16:30 Closed trajectories of plane systems of ODE's, moments, iterated integrals, and compositions - with the stress on some "formal" aspects Recently some new connections have been found between the structure of the closed trajectories of plane systems of ODE's and certain questions in classical analysis and algebra. In particular, this concerns the vanishing problem of certain moment-like expressions and of iterated integrals on one side, and the structure of the composition factorization of analytic functions on the other. These connections provide a useful information on the Center conditions (for all the trajectories around a critical point to be closed) and on the distribution of the isolated closed trajectories (limit cycles). Some of the above effects are rather "structural" or "formal" in their nature, and, as the preliminary considerations show, they possibly can be considered in the framework of the Model Theory. INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents only)
 09:00 to 10:00 Zariski-type structuresSession: An Introduction to Recent Applications of Model Theory INI 1 10:00 to 11:00 Zariski-type structures INI 1 11:00 to 11:30 Coffee and posters 11:30 to 12:30 CW Henson & A Berenstein ([Illinois])Model theory for metric structures A metric structure is a many-sorted structure with each sort a metric space, which for convenience is assumed to have finite diameter. Additionally, there are functions (of several variables) between sorts, assumed to be uniformly continuous. Examples include metric spaces themselves, measure algebras (with the metric d(A,B) = m(A*B), where * is symmetric difference), and structures based on Banach spaces (where one interprets the sorts as balls), including Banach lattices, C*-algebras, etc. The usual first-order logic does not work very well for such structures, and several good alternatives have been developed. One is the logic of positive bounded formulas with an approximate semantics (see [3]). Another is the setting of compact abstract theories (cats) (see [1]). A recent development, which will be emphasized in this course, is the convergence of these two points of view and the realization that they are also connected to the [0,1]-valued continuous logic that was studied extensively in the 1960s and then dropped (see [2]). This leads to a new formalism for the logic of metric structures in which the operations of sup and inf play a central role. The development of this continuous logic for metric structures is an ongoing project being carried out as a collaboration among Itay Ben-Yaacov (Wisconsin), Alex Berenstein (Illinois), Ward Henson (Illinois), and Alex Usvyatsov (Jerusalem). The analogy between this logic for metric space structures and the usual first order logic for ordinary structures is far reaching. Continuous logic satisfies the compactness theorem, Lowenheim-Skolem theorems, diagram arguments, existence of saturated and homogeneous models, characterizations of quantifier elimination and model completeness, Beth's definability theorem, Craig's interpolation theorem, the omitting types theorem, fundamental results of stability theory, and appropriate analogues of essentially all results in basic model theory of first order logic. The theory of definable sets and functions in this continuous logic for metric structures has novel aspects and there are numerous open problems of apparent interest. Extending important concepts from first order model theory to this continuous logic often presents a challenge. The purpose of this tutorial is to present this logic for metric structures, and to show a few of its application areas, with emphasis on probability spaces and certain linear spaces from functional analysis. References: [1] I. Ben-Yaacov, Positive model theory and compact abstract theories, J. Math. Logic 3 (2003), 85--118. [2] C. C. Chang and H. J. Keisler, Continuous Model Theory, Princeton Univ. Press (1966). [3] C. W. Henson and J. Iovino, Ultraproducts in Analysis, in Analysis and Logic, London Mathematical Society Lecture Notes 262, Cambridge University Press (2002), 1--113. INI 1 12:30 to 13:30 Lunch at Churchill College 14:00 to 15:00 CW Henson & A Berenstein ([Illinois])Model theory for metric structures A metric structure is a many-sorted structure with each sort a metric space, which for convenience is assumed to have finite diameter. Additionally, there are functions (of several variables) between sorts, assumed to be uniformly continuous. Examples include metric spaces themselves, measure algebras (with the metric d(A,B) = m(A*B), where * is symmetric difference), and structures based on Banach spaces (where one interprets the sorts as balls), including Banach lattices, C*-algebras, etc. The usual first-order logic does not work very well for such structures, and several good alternatives have been developed. One is the logic of positive bounded formulas with an approximate semantics (see [3]). Another is the setting of compact abstract theories (cats) (see [1]). A recent development, which will be emphasized in this course, is the convergence of these two points of view and the realization that they are also connected to the [0,1]-valued continuous logic that was studied extensively in the 1960s and then dropped (see [2]). This leads to a new formalism for the logic of metric structures in which the operations of sup and inf play a central role. The development of this continuous logic for metric structures is an ongoing project being carried out as a collaboration among Itay Ben-Yaacov (Wisconsin), Alex Berenstein (Illinois), Ward Henson (Illinois), and Alex Usvyatsov (Jerusalem). The analogy between this logic for metric space structures and the usual first order logic for ordinary structures is far reaching. Continuous logic satisfies the compactness theorem, Lowenheim-Skolem theorems, diagram arguments, existence of saturated and homogeneous models, characterizations of quantifier elimination and model completeness, Beth's definability theorem, Craig's interpolation theorem, the omitting types theorem, fundamental results of stability theory, and appropriate analogues of essentially all results in basic model theory of first order logic. The theory of definable sets and functions in this continuous logic for metric structures has novel aspects and there are numerous open problems of apparent interest. Extending important concepts from first order model theory to this continuous logic often presents a challenge. The purpose of this tutorial is to present this logic for metric structures, and to show a few of its application areas, with emphasis on probability spaces and certain linear spaces from functional analysis. References: [1] I. Ben-Yaacov, Positive model theory and compact abstract theories, J. Math. Logic 3 (2003), 85--118. [2] C. C. Chang and H. J. Keisler, Continuous Model Theory, Princeton Univ. Press (1966). [3] C. W. Henson and J. Iovino, Ultraproducts in Analysis, in Analysis and Logic, London Mathematical Society Lecture Notes 262, Cambridge University Press (2002), 1--113. INI 1 15:00 to 15:30 Tea and posters 15:30 to 16:30 Y Raynaud ([Paris])Ultrapowers and ultraroots in Banach spaces theory INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents only)
 09:00 to 10:00 Zariski-type structuresSession: An Introduction to Recent Applications of Model Theory INI 1 10:00 to 11:00 CW Henson & A Berenstein ([Illinois])Model theory for metric structures A metric structure is a many-sorted structure with each sort a metric space, which for convenience is assumed to have finite diameter. Additionally, there are functions (of several variables) between sorts, assumed to be uniformly continuous. Examples include metric spaces themselves, measure algebras (with the metric d(A,B) = m(A*B), where * is symmetric difference), and structures based on Banach spaces (where one interprets the sorts as balls), including Banach lattices, C*-algebras, etc. The usual first-order logic does not work very well for such structures, and several good alternatives have been developed. One is the logic of positive bounded formulas with an approximate semantics (see [3]). Another is the setting of compact abstract theories (cats) (see [1]). A recent development, which will be emphasized in this course, is the convergence of these two points of view and the realization that they are also connected to the [0,1]-valued continuous logic that was studied extensively in the 1960s and then dropped (see [2]). This leads to a new formalism for the logic of metric structures in which the operations of sup and inf play a central role. The development of this continuous logic for metric structures is an ongoing project being carried out as a collaboration among Itay Ben-Yaacov (Wisconsin), Alex Berenstein (Illinois), Ward Henson (Illinois), and Alex Usvyatsov (Jerusalem). The analogy between this logic for metric space structures and the usual first order logic for ordinary structures is far reaching. Continuous logic satisfies the compactness theorem, Lowenheim-Skolem theorems, diagram arguments, existence of saturated and homogeneous models, characterizations of quantifier elimination and model completeness, Beth's definability theorem, Craig's interpolation theorem, the omitting types theorem, fundamental results of stability theory, and appropriate analogues of essentially all results in basic model theory of first order logic. The theory of definable sets and functions in this continuous logic for metric structures has novel aspects and there are numerous open problems of apparent interest. Extending important concepts from first order model theory to this continuous logic often presents a challenge. The purpose of this tutorial is to present this logic for metric structures, and to show a few of its application areas, with emphasis on probability spaces and certain linear spaces from functional analysis. References: [1] I. Ben-Yaacov, Positive model theory and compact abstract theories, J. Math. Logic 3 (2003), 85--118. [2] C. C. Chang and H. J. Keisler, Continuous Model Theory, Princeton Univ. Press (1966). [3] C. W. Henson and J. Iovino, Ultraproducts in Analysis, in Analysis and Logic, London Mathematical Society Lecture Notes 262, Cambridge University Press (2002), 1--113. INI 1 11:00 to 11:30 Coffee and posters 11:30 to 12:30 F Loeser ([ENS, Paris])Operations on constructible functions (I) Constructible functions appear under various guises in many different setting. Starting from real and o-minimal geometry, we shall journey to the p-adic and motivic setting. We plan to focus mainly on the construction of direct image (pushforward) but we intend also taking some time to discuss other operations, like the Fourier transformation. INI 1 12:30 to 13:30 Lunch at Churchill College 14:00 to 15:00 F Loeser ([ENS, Paris])Operations on constructible functions (II) Constructible functions appear under various guises in many different setting. Starting from real and o-minimal geometry, we shall journey to the p-adic and motivic setting. We plan to focus mainly on the construction of direct image (pushforward) but we intend also taking some time to discuss other operations, like the Fourier transformation. INI 1 15:00 to 15:30 Tea and posters 15:30 to 16:30 The theorem of the complement for nested sub-Pfaffian sets We show that the Pfaffian closure of an o-minimal structure with analytic cell decomposition is model complete. This is achieved by proving a theorem of the complement for nested sub-Pfaffian sets over the o-minimal structure in question. INI 1 16:30 to 17:30 Nonstandard 1-dimensional tori are locally modular In earlier work we showed how a uniform family of biholomorphisms of 1-dimensional complex tori and algebraic cubics is definable in $R_{an,exp}$, covering in this way all smooth cubics, but not all tori. As a corollary, one obtains in elementary extensions of $R_{an,exp}$ some nonstandard'' 1-dimensional tori. I will discuss the induced analytic structure on these tori and show that these nonstandard tori are strongly minimal and locally modular. INI 1 20:00 to 18:00 Conference Dinner at Magdalene College (Dining Hall)
 09:00 to 10:00 Isotriviality criteria for families of non-algebraic compact K\" ahler manifolds, and model-theoretic nonmultidimensionality of the class C.Session: An Introduction to Recent Applications of Model Theory A question raised by A.Pillay is whether the class $\calC$ of compact complex manifolds $F$ bimeromorphic to some compact K\" ahler manifold $F'$ (depending on $F$) is nonmultidimensional in the model theoretic sense. Specialised to the case of {\it simple} manifolds $F$ (those which are not covered by proper compact analytic subsets, and of complex dimension at least $2$), this means that if $f:X\to S$ is a surjective holomorphic map with $X$ in $\calC$, and general smooth fibre $X_s$ simple, then $f$ is {\it isotrivial}, which means that any two such fibres are isomorphic. We show that this is indeed the case for (most of) the known simple manifolds: the non-projective hyperk\" ahler manifolds, and the general complex tori. The talk is intended for non-specialists in complex geometry. INI 1 10:00 to 11:00 Analogues of Hilbert's tenth problem This presentation is part of the tutorial on Hilbert's Tenth Problem, which will be presented jointly with Thanases Pheidas. The aim is to give a comprehensive overview of results concerning decidability questions related to solving diophantine equations. For more information I refer to the abstract submitted by Thanases Pheidas. INI 1 11:00 to 11:30 Coffee and posters 11:30 to 12:30 Analogues of Hilbert's tenth problem This presentation is part of the tutorial on Hilbert's Tenth Problem, which will be presented jointly with Thanases Pheidas. The aim is to give a comprehensive overview of results concerning decidability questions related to solving diophantine equations. For more information I refer to the abstract submitted by Thanases Pheidas. INI 1 12:30 to 13:30 Lunch at Churchill College 14:00 to 15:00 Hilbert's tenth problem for function fields We will discuss how elliptic curves of rank one can be used to prove undecidability of Hilbert's Tenth Problem for function fields of characteristic zero, such as R(t) and C(t_1,...,t_n) (n at least 2). The approach for function fields of positive characteristic is very different, and we will sketch some of the methods used there. INI 1 15:00 to 15:30 Tea and posters 15:30 to 16:30 F Loeser ([ENS, Paris])Operations on constructible functions (III) Constructible functions appear under various guises in many different setting. Starting from real and o-minimal geometry, we shall journey to the p-adic and motivic setting. We plan to focus mainly on the construction of direct image (pushforward) but we intend also taking some time to discuss other operations, like the Fourier transformation. INI 1 16:30 to 17:30 Real integration of oscillating functions We present new results on real integration of oscillating functions, for example, Fourier transforms of subanalytic functions; it is joint work with Aschenbrenner and Rolin. We begin to control very good the transcendental functions we have to add, (namely, oscillating versions of basic Abelian integrals), in order to describe these parameterized integrals. The method is in principle algorithmic, with a similar algorithm as to compute motivic oscillating integrals. Yet, conjectures linking real and motivic integrals remain unsolvable. INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents only)
 09:00 to 09:30 Analytic Zariski geometries and the Hrushovski CollapseSession: An Introduction to Recent Applications of Model Theory In recent years much progress has been made along the 'Zilber Programme' intended to explain Hrushovski-type constructions in terms of analytic geometry. These results, however, have only dealt with Hrushovski-type structures of infinite rank. we will show that considering appropriate expansions of the infinite rank ab initio structrue, Hrushovski's strongly minimal sets can be obtained as infinitesimal neighborhoods (in terms of specializations) of carefully enough chosen points. We will also show that these results can be extended to other Hrushovski-type structres, most notably to ones supporting a structure of an algebraically closed field. If time allows we will discuss briefly: (a) The obstacles in generalizing these results to more complicated structures (e.g. the fusion of two strongly minimal Zariski geometries). (b) The nature of a possible analytic prototype in which our interpretation of the collapse could be described in analytic terms. INI 1 09:30 to 10:00 S Barbina ([Turin])Reconstruction of homogeneous relational structures Reconstruction results give conditions under which the automorphism group of a structure determines the structure up to bi-interpretability or bi-definability. Here we examine a large class of omega-categorical combinatorial structures, which was isolated by Herwig and contains K_n-free graps, k-hypergraphs and Henson digraphs. Using a Baire category approach we show how to obtain reconstruction for this class, proving that a reconstruction condition, developed by M. Rubin, holds. The method rests on the existence of a generic pair of automorphisms. INI 1 10:00 to 10:30 Arc-properties of functions definable in o-minimal structures There are two topics at the base of this talk: arc-analytic functions (mostly studied in the subanalytic setting) and the relationships between o-minimal structures and Hardy fields. They lead us to study the following problem: for a function definable in some o-minimal structure (over the field of reals), what kind of property may be detected in restriction to some "small" space of definable arcs. We shall prove for instance that for most classical polynomially bounded o-minimal structures, continuity is equivalent to continuity on restriction to polynomial arcs. We will finish with problems of arc-definability for which we will give several open questions. INI 1 10:30 to 11:00 The theory of a Hilbert space with a generic automorphism I will present an axiomatization for the continuous theory of a generic unitary representation of the group Z (integers), i.e. the theory of a Hilbert space with a generic automorphism. This is also the theory of the regular unitary representation of Z. I will describe the properties of this theory (e.g. superstable, non-multidimensional, non-omega stable) and present a full characterization of types using the spectral decomposition theorem. This is a joint work with Itay Ben-Yaacov and Moshe Zadka. INI 1 11:00 to 11:30 Coffee and posters 11:30 to 12:00 T Mellor ([Regensburg])An Euler characteristic for real closed valued fields I shall consider a real closed valued field together with certain extensions. There are four classes of definable set: considering both structures, both as fields and valued fileds. The relationships between these classes allow one to prove the existance of an Euler characteristic for the (valued field) definable sets of the origional structure. Related Links http://math.berkeley.edu/~scanlon/papers/blc29sep00ii.pdf - From the webpage of Thomas Scanlon describing the theory of Euler Characteristics INI 1 12:30 to 13:30 Lunch at Churchill College INI 1 15:00 to 15:30 Tea and posters INI 1 15:30 to 16:30 Application of arc spaces to the model theory of fields with commuting derivations INI 1 16:30 to 17:30 Model theoretic topics in valued fields I INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents only)
 09:00 to 10:00 On the quasi-minimality of certain expansions of the complex fieldSession: An Introduction to Recent Applications of Model Theory I shall consider various natural pre-geometries on expansions of the complex field inspired by the work of Peterzil and Starchenko on the development of complex analysis within an o-minimal structure. Unfortunately, I still cannot realise my original aim of using such methods to show that the complex exponential field is quasi-minimal (ie every definable subset of the complex numbers is either countable or co-countable) but I can at least show that we do have quasi-minimality if we only allow the operations of raising to real powers. (I should point out, however, that if one assumes postive answers to certain conjectures from diophantine geometry and transcendence theory then Zilber has already shown this,and more. See 'Raising to powers in algebraically closed fields', J Math Logic vol 3(2), 2003, 217-238.) Another aspect of the talk is that it gives some sort of answer to a question of Hrushovski (private communication a couple of years ago) which asks whether elimation of quantifiers for algebraically closed fields may be naturally deduced from elimination of quantifiers for real closed ordered fields. I show that even though the definable closure operator on an o-minimal structure (expanding a real closed field) does not satisfy the modular law, it may nevertheless be linearised and thereby induce a pregeometry on an expansion of its algebraic closure. The Cauchy-Riemann equations play a role here so that, for example, in the pure field case this pregeometry IS algebraic closure (and not,say, "algebraic closure of the set of real and imaginary parts"). INI 1 10:00 to 10:30 Grothendieck-Katz conjecture for elliptic curves We will consider a special case of Grothendieck-Katz conjecture given by the logarithmic derivative. INI 1 10:30 to 11:00 Schanuel conditions for Weierstrass differential equations I will discuss a version of Schanuel's conjecture for Weierstrass equations in differential fields. This gives a necessary and sufficient condition for a system of Weierstrass differential equations to have a solution. The necessity part builds on work by James Ax, who proved the equivalent statement for the exponential equation, and by Brownawell and Kubota who proved an analogue for complex power series. The sufficiency part builds on work of Cecily Crampin. I hope also to show connections to the theory of the complex Weierstrass p-functions and to structures constructed via Hrushovski's amalgamation technique. INI 1 11:00 to 11:30 Coffee and posters 11:30 to 12:00 Mordell-Lang theorem for Drinfeld modules and minimal groups in the theory of separably closed fields We study the rings of quasi-endomorphisms of certain minimal groups in the theory of separably closed fields. These groups are associated to Drinfeld modules of finite characteristic. Based on our results, we are able to prove a Mordell-Lang statement for Drinfeld modules of finite characteristic. INI 1 12:00 to 12:30 Classes of (Lie) Differential Fields without Model Companions A Lie differential field is a field F given with some Lie algebra L acting on F as derivations. If we fix L we get the class of L-differential fields, which has amalgamation when the characteristic is 0. If in addition L is finite dimensional over F then the above class has a model companion (and hence a model completion). However if L isn't finitely presented (at least locally), i.e. if there is a finitely generated sub Lie algebra of L without a finite presentation, then the class of L-differential fields does NOT have a model companion. I will describe how this result is proved by producing a non-eliminable quantifier, using a system of linear PDEs. The question of a tighter connection between companionability and finite presentability remains open. INI 1 12:30 to 13:30 Lunch at Churchill College 14:00 to 15:00 Model theoretic topics in valued fields II INI 1 15:00 to 15:30 Tea and posters 15:30 to 16:30 Model theory of elliptic functions: model completeness, uniformity, decidability INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents only)