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Résumés

Franck Benoist

Model theory of Hasse fields and application to algebraic geometry

Abstract: We will give some properties of existentially closed fields with a Hasse derivation, in arbitrary characteristic. We will focus on possible applications to algebraic geometry, by studying two different ways of putting additional structure on algebraic varieties using the Hasse derivation. Our aim is to give a criterion concerning the field of definition of an abelian variety over a function field of positive characteristic, in the spirit of the work done around Mordell-Lang conjecture.

Alexandre Borovik

Linear and permutation groups of finite Morley rank.

Zoé Chatzidakis

Algebraic dynamics over function fields and model theory of difference varieties.

Problèmes de dynamique sur des corps de fonctions, théorie des modèles des corps de différence.

Résumé (pdf)

Gregory Cherlin

Genericity, Generosity, and Tori

Abstract: We show that if a connected group of finite Morley rank G generically satisfies the condition x^n=1, where n is a power of 2, then G is a 2-group of exponent at most n. The proof makes use of results of a general character inspired by work on the structure of the simple groups of finite Morley rank. An important part of this is the understanding of the properties of generic elements, and their relationship with tori which arise as the definable hulls of torsion subgroups. The same ideas can also be usefully brought to bear in the theory of definably primitive permutation groups of finite Morley rank, yielding a bound on the rank of such a group in terms of the rank of the set on which it acts.

Adrien Deloro

How to recognize PSL2(K) (among other groups of finite Morley rank)

Abstract: Making assumptions on the structure of the Sylow 2-subgroup(s) is common in the study of groups of finite Morley rank. We shall deal with the case of a simple group of finite Morley rank (and minimal in some sense), whose Sylow 2-subgroup is a finite extension of the group of complex numbers of order some power of 2. These assumptions do characterize PSL2 over some algebraically closed field. The method we will discuss requires a "0-characteristic theory of unipotence" for groups of finite Morley rank, which makes it still valid in case a bad field should appear during the conference.

David Evans

Abstract:The notion of an angular function has been introduced by Zilber in his paper `Non-commutative geometry and new stable structures' as one possible way of connecting non-commutative geometry with two `counterexamples' from model theory: the non-classical Zariski curves of Hrushovski and Zilber, and Poizat's field with green points. In this talk I will discuss some questions from Zilber's paper relating to existentially closed structures in the class of algebraically closed fields with an angular function.

Olivier Frécon

dans les groupes de rang de Morley fini.

Abstract: If G is a group of finite Morley rank, we say that a

subgroup C is a Carter subgroup of G if it is definable, connected,

nilpotent and of finite index in its normalizer in G. These subgroups are

analogous to Cartan subgroups , that is to the centralizers of maximal tori in

connected algebraic groups. In joint work with E. Jaligot we proved their

existence in every group of finite Morley rank. Henceforth the conjugacy of

Carter subgroups is the most important open problem

concerning these subgroups.

For the Cherlin-Zil'ber Conjecture, the study of Carter subgroups is especially

interesting in minimal connected simple groups of finite Morley rank, which are

the infinite simple groups all of whose proper connected definable

subgroups are solvable.

We prove the following result.

Theorem: In any minimal connected simple group, Carter subgroups are conjugate.

(This talk will be in French with transparencies in English.)

Assaf Hasson

On Zilber's Dichotomy in 1-dimensional reducts of o-minimal structures.

Abstract: We discuss several aspects of the modular/non-modular dichotomy in 1-dimensional reducts of o-minimal structures. In particular, we prove that if a 1-dimensional reduct of an o-minimal expansion of a field is stable it must be locally modular.

Joint work with A. Onshuus and Y. Peterzil.

Martin Hils

Mauvais corps en caractéristique 0

Zusammenfassung: Die GrÜnen existieren auch.

Abstract: A bad field is an algebraically closed field K together with a proper infinite subgroup T of the multiplicative group, such that the structure (K,+, . ,0,1,T) is of finite Morley rank.

We show that bad fields exist in characteristic 0. More precisely, we construct such a bad field by collapsing Poizat's green fields in characteristic 0.

Joint work with A. Baudisch, A. Martin-Pizarro and F. Wagner

(The talk will be in French, with slides in English.)

Ehud Hrushovski

On Grothendieck rings and interpretations

Abstract: Grothendieck rings of definable sets can be viewed as intrinsic cardinality structures on a first order theory. Comparing rings belonging to different theories can reveal significant geometric information, and leads immediately to some model theoretic questions. I will discuss some of these, both abstractly and for specific theories around algebraic geometry: ACF, PF, ACFA, ACVFA.

Guillaume Malod

Complete problems in Valiant's theory.

Abstract: Valiant's theory is a simple framework to study computations by circuits. One of the nice results of this theory relates the possible equality of two key classes to the feasability of expressing any permanent as a determinant of reasonable size. This is a consequence of the completeness of the determinant and permanent polynomials for the relevant classes. However, very few other complete problems are known, and for other classes we do not know any natural complete problems. We will describe different attempts to find such complete problems, the difficulties involved, and the results.

Christian Michaux

Deux ou trois choses que je ne sais pas sur (IN, +,V_2)

Résumé (pdf)

Anand Pillay

Return of generix

Abstract: We discuss attempts to find a common generalization of "genericity" in (i) compact groups, (ii) stable groups, (iii) "definably compact" groups in o-minimal structures and p-adically closed fields.

Zlil Sela

Free and hyperbolic groups are stable.

Abstract: We apply the iterative procedure that was used for proving quantifier elimination over a free group, and the structure of definable sets obtained by this procedure, to prove stability for free (and hyperbolic) groups. Equationality and other related questions are considered as well.

Alex Usvyatsov

Categoricity for uncountable continuous theories.

Abstract: We suggest a notion of primeness in the continuous superstable context and as an application prove the categoriciy theorem for uncountable continuous theories.

John Wilson

Characterizations of solubility. for finite groups.

Martin Ziegler

Red Fields

Abstract: They exist. (Joint work with Baudisch and Pizarro).

Boris Zilber

Quantum Zariski geometries and groups

Abstract: We show how to construct a Zariski geometry V(A) out of the category of A-modules for many (conjecturally all) quantum algebras A at roots of unity. Typically the Zariski geometry turns out to be non-classical, that is not definable in an algebraically closed field.

When A is a Hopf algebra (i.e. a quantum group) we find a group configuration in V(A) and so deduce that any quantum group at roots of unity is in a canonical correspondence with an algebraic group.