Titles and Abstracts

Martin Bays (Universität Münster, Germany)
"Definability in the infinitesimal subgroup of a simple compact Lie group"

Abstract:
Let $G$ be a simple compact Lie group, for example $G=SO_3(\mathbb{R})$. The infinitesimal subgroup $G^{00}$ in an $\aleph_0$-saturated elementary extension of the real field is the inverse image of the identity under the standard part map, so $G^{00}$ is definable in the corresponding valued field. We show that the pure group structure on $G^{00}$ recovers the valued field, making this a bi-interpretation. Hence the definable sets in the group are as rich as possible. Joint work with Kobi Peterzil.

Olivier Benoist (Université de Strasbourg, France)
"Algebraic curves on real algebraic varieties"

Abstract:
Consider a loop drawn on the real locus of a real algebraic variety. Can it be approximated by a real algebraic curve lying on the variety? This is a classical question in real algebraic geometry. We will describe new positive and negative results. We will also discuss the corresponding problem over arbitrary real closed fields.
This is joint work with Olivier Wittenberg.

Alessandro Berarducci (Università di Pisa, Italy)
"On the dimension of bounded hyperdefinable sets"

Abstract:
Let $X$ be a definable set in an o-minimal structure and let $E$ be a type-definable equivalence relation of bounded index on $X$. Under suitable assumptions, including a form of compact domination, we show that the o-minimal dimension of $X$ equals the dimension of $X/E$, as a space with the logic topology. In particular we obtain a new proof of Pillay's conjecture $\dim(G)=\dim(G/G^{00})$ where $G$ is a definably compact group. The proof depends on a homotopy comparison result between $X$ and $X/E$. Joint work with Alessandro Achille.

Raf Cluckers (Université de Lille I, France)
"Expansion minimality"

Abstract:
Together with I. Halupczok and S. Rideau, we develop a notion of tame geometry for the definable sets in henselian valued fields of characteristic zero, called expansion minimality. We will present this notion and some properties.

Pablo Cubides (Université de Caen, France)
"Definables types and elimination of imaginaries in closed ordered differential fields"

Abstract:
We show that for every definable set $X$ in a closed ordered differential field $K$, there is a definable type $p$ in $X$ which is definable over the code of $X$.
As an application we give another proof of elimination of imaginaries of CODF. Using a fibered dimension function on definable subsets of models of CODF, we further show that such a type $p$ may be chosen as having the same dimension as $X$.
This is a joint work with Quentin Brouette and Françoise Point.

Paola D'Aquino (Università della Campania, Italy)
"Zero sets of exponential polynomials"

Abstract:
I will show the existence of generic solutions for certain exponential polynomials over the complex field (joint work with A. Fornasiero and G.Terzo).

Philip Dittmann (University of Oxford, UK)
"Uniformity in the Model Theory of Algebraic Fields"

Abstract:
We discuss the first-order theory of algebraic fields of characteristic zero, showing in particular that every model of this theory is dense in all its real and p-adic closures. This ties in with the theory of PRC and PpC fields, and has consequences for the property of having embedded residue. This is joint work with Sylvy Anscombe and Arno Fehm.

Ayhan Günaydın (Boğaziçi University, Turkey)
"Topological Study of Pairs of Algebraically Closed Fields"

Abstract:
The study of an algebraically closed field $K$ with a distinguished algebraically closed field $L$ goes back to Keisler's work in 1964, where he proves a completeness result. Since then there had been many developments around such pairs and more general kinds of pairs of stable structures.
In this talk, we equip each $K^n$ with a topology refining Zariski topology in a way that sets definable in the pair $(K,L)$ are precisely the constructible sets of this topology.
We shall also mention the relations of this topology with Morley rank and another notion of dimension arising from a pregeometry.

Tobias Kaiser (Universität Passau, Germany)
"Asymptotics of parameterized exponential integrals given by Brownian motion on globally subanalytic sets"

Abstract:
(Joint work with Julia Ruppert.) Understanding integration in the o-minimal setting is an important and difficult task. By the work of Comte, Lion and Rolin, succeeded by the work of Cluckers and Miller, parameterized integrals of globally subanalytic functions are very well analyzed. But very little is known when the exponential function comes into the game. We consider certain parameterized exponential integrals which come from considering the Brownian motion on globally subanalytic sets. We are able to show nice asymptotic expansions of these integrals.

Eva Leenknegt (KU Leuven, Belgium)
"Cell Decomposition for P-minimal structures: a story"

Abstract:
P-minimality is a concept that was developed by Haskell and Macpherson as a p-adic equivalent for o-minimality. For o-minimality, the cell decomposition theorem is probably one of the most powerful tools, so it is quite a natural question to ask for a p-adic equivalent of this.
In this talk I would like to give an overview of the development of cell decomposition in the p-adic context, with an emphasis on how questions regarding the existence of definable skolem functions have complicated things. The idea of p-adic cell decomposition was first developed by Denef, for p-adic semi-algebraic structures, as a tool to answer certain questions regarding quantifier elimination, rationality and p-adic integration. This first version eventually resulted in a cell decomposition theorem for P-minimal structures. This theorem, proven by Mourgues, was however dependent on the existence of definable Skolem functions. The second part of the talk will focus a bit more on Skolem functions, and sketch a generalized version of the Denef-Mourgues theorem that does not rely on the existence of such functions, by introducing a notion of clustered cells. We will explain the notion, give an informal sketch of the proof, and compare with other versions of cell decomposition.

François Loeser (Université Paris VI, France)
"Non-archimedean integrals as limits of complex integrals"

Abstract:
A meromorphic family of complex projective varieties on the punctured unit disc gives rise at the limit to a Berkovich space. Chambert-Loir and Ducros have recently developed a theory of real $(p,q)$-forms and currents on Berkovich spaces which is parallel to the theory of differential forms on complex spaces. We will present the following result which is part of a joint project with A. Ducros and E. Hrushovski: the integral of such a $(n,n)$-form on the limit Berkovich space of dimension $n$ can always be expressed in a natural way as a limit of complex integrals on the original family.

Angus Macintyre (University of London, UK)
"Issues of definability and decidability concerning $\widehat{\mathbb{Z}}$ and the finite adeles over $\mathbb{Q}$"

Abstract:
We are concerned with variations and generalizations of the classical Feferman-Vaught method in the context of products of henselian domains.Various part of the analysis are joint with respectively Derakhshan, D'Aquino and D'Aquino-Otero. Some topics to be considered are:
1. $Spec(\widehat{\mathbb{Z}})$, ideals of of $\widehat{\mathbb{Z}}$, quotients of $\widehat{\mathbb{Z}}$, with emphasis on some well-behaved special cases.
2. Axioms for $M/nM$, where $M$ is a nonstandard model of PA, and $n$ is a highly composite element of $M$.
3. Definitions in such M/nM.

Kobi Peterzil (University of Haifa, Israel)
"The topological closure of algebraic and o-minimal flows in complex tori"

Abstract:
(Joint work with Sergei Starchenko). Consider the covering map $p:\mathbb{C}^n \to T$ of a compact complex torus and assume that $X$ is an irreducible algebraic subvariety of $\mathbb{C}^n$. It follows from a theorem of Ax and Lindemann that the Zariski closure of $p(X)$ is a coset of a complex Lie subgroup of $T$.
We answer a similar question, of Ullmo and Yafaev, about the topological closure of $p(X)$ and prove a modified version of their conjecture by showing that the topological closure of $p(X)$ is the union of $p(X)$ with finitely many families of real sub-tori of $T$ of positive dimension. The original conjecture was proven by them when $\mathrm{dim} X=1$, but turns out to be false in general
We also prove an analogous result for the covering map of a compact real torus and for a set $X$ in $\mathbb{R}^n$ that is definable in an o-minimal structure.
The proofs use the theory of stabilizers of mu-types in o-minimal structures developed earlier by the authors, as well as the theory of tame pairs in algebraically closed valued fields and o-minimal structures.

Françoise Point (Université Paris VII, France)
"Topological large fields and their generic expansion to differential fields"

Abstract:
In his thesis, W. Johnson showed that if $\mathcal{K}$ is an expansion of a field, $K$ infinite, and $dp$-minimal (a stronger condition than NIP) but not strongly minimal, then $\mathcal{K}$ can be endowed with a non-discrete definable field topology, namely $\mathcal{K}$ has a uniformly definable basis of neighbourhoods of zero compatible with the field operations. Assuming that $\mathcal{K}$ has characteristic $0$, we consider the ${\it generic}$ expansion of $\mathcal{K}$ with a derivation $D$, namely we put no a priori continuity assumptions on $D$. In certain classes of existentially closed
differential expansions $\mathcal{K}_D$, we relate the properties of the definable subsets, definable functions and then definable subgroups of $\mathcal{K}_D$ with those in the reduct $\mathcal{K}$.
We will give two applications; one on the density of definable types and the other on the dual VC-density of a formula in these differential expansions.

Alex Wilkie (University of Manchester, UK)
"A valuation inequality for complex structure"

Abstract:
For valued fields of equicharacteristic 0 the valuation inequality asserts that the transcendence degree of the field (assumed finite here) is an upper bound for the sum of the rank of the value group and the transcendence degree of the residue field. This was subsequently generalized to the case of polynomially bounded o-minimal structures where transcendence degree is replaced by the usual o-minimal dimension defined via the pregeometry of Skolem closure. (The corresponding inequality was then applied to showing that the real exponential field has a model complete and o-minimal theory.)
In this talk I present a version of the valuation inequality for the complex field associated to an o-minimal structure where the notion of Skolem closure is defined via holomorphic definable functions. I then suggest applications of this inequality to the problem of showing that certain expansions of the complex field are quasi-minimal and report on one small success.

Boris Zilber (University of Oxford, UK)
"Positive model theory and approximation by finite structures"

Abstract:
I will discuss a notion of structural approximation and its relation to positive and continuous model theory.
The most interesting class of examples that allow an approximation by finite structures allows also a natural topology, dimension and measure theory based on measure in pseudo-finite structures. The key example is a model of quantum mechanics.