[Turkmath:7973] Istanbul Bilgi University Tuesday Seminars (Nov 1 - Martin Hils)

[Uğur Doğan]

Dear All,

Martin Hils (Paris 7) will give a talk at our departmental  seminar
on Tuesday (Nov 1) at 16:00 in the room D-135. The title and abstract are
below.

Best,

Piotr
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Title: "Generic automorphisms of first order structures"

Abstract:

The first part of the talk will be introductory. For $T$ a model-complete
first order theory, we consider the theory $T_{\sigma}$ whose models are
of the form $(M,\sigma)$ where $M$ is a model of $T$ and $\sigma$ an
automorphism of $M$. If the class of existentially closed models
of $T_{\sigma}$ is axiomatisable, its theory is denoted $TA$, and the
corresponding automorphisms are called generic. We will discuss
some examples, with a particular emphasis on the so-called geometric
axioms for $ACFA$. We note that the model theoretic study of $ACFA$,
paradigmatically the work of Chatzidakis-Hrushovski, has shown to be
extremely useful both in applications (e.g. to algebraic dynamics)
and as a trigger for new advances in pure model theory.

In the second part, we will discuss some necessary and sufficient
conditions (in terms of the initial theory $T$)
for the existence of $TA$. The guiding principle is that this should be
closely related to a good (definable) control of
"multiplicities" in $T$. In various contexts, where multiplicities may
indeed be controlled, on may give a geometric
axiomatisation of $TA$, e.g. for $T$ the theory of differentially closed
fields or Poizat's green fields.
If $T$ is the theory of a group of finite Morley rank, then $TA$ exists if
and only if Morley degree is definable in $T$.
The latter generalises a result of Hasson-Hrushovski and
is joint work with Martin Bays and Misha Gavrilovich.
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