11:00
Axiomatizing Q by "G_Q + ε"
Abstract
we discuss various conjectures about the absolute Galois group G_Q of the field Q of rational numbers and to what extent it encodes the elementary theory of Q.
Forthcoming events in this series
we discuss various conjectures about the absolute Galois group G_Q of the field Q of rational numbers and to what extent it encodes the elementary theory of Q.
In this talk, I will introduce an internal, structural
characterisation of certain convergence properties (Fréchet-Urysohn, or
more generally, radiality) and apply this structure to understand when
Stone spaces have these properties. This work can be generalised to
certain Zariski topologies and perhaps to larger classes of spaces
obtained from other structures.
"We consider certain distinguished extensions of the field F_p((t)) of formal Laurent series over F_p, and look at questions about their model theory and Galois theory, with a particular focus on decidability."
Algebraically closed fields, and in general varieties are among the first examples
of Zariski Geometries.
I will consider specialisations of algebraically closed fields and varieties.
In the case of an algebraically closed field K, I will show that a specialisation
is essentially a residue map, res from K to a residue field k.
In both cases I will show universality of the specialisation is controlled by the
transcendence degree of K over k.
Given a field K and an ordered abelian group G, we can form the field K((G)) of generalised formal power series with coefficients in K and indices in G. When is this field decidable? In certain cases, decidability reduces to that of K and G. We survey some results in the area, particularly in the case char K > 0, where much is still unknown.
Given a field K and an ordered abelian group G, we can form the field K((G)) of generalised formal power series with coefficients in K and indices in G. When is this field decidable? In certain cases, decidability reduces to that of K and G. We survey some results in the area, particularly in the case char K > 0, where much is still unknown.
(Joint work with Jochen Koenigsmann) Admitting a p-henselian
valuation is a weaker assumption on a field than admitting a henselian
valuation. Unlike henselianity, p-henselianity is an elementary property
in the language of rings. We are interested in the question when a field
admits a non-trivial 0-definable p-henselian valuation (in the language
of rings). They often then give rise to 0-definable henselian
valuations. In this talk, we will give a classification of elementary
classes of fields in which the canonical p-henselian valuation is
uniformly 0-definable. This leads to the new phenomenon of p-adically
(pre-)Euclidean fields.
In an o-minimal expansion of the real field, while few holomorphic functions are globally definable, many may be locally definable. Wilkie conjectured that a few basic operations suffice to obtain all of them from the basic functions in the language, and proved the conjecture at generic points. However, it is false in general. Using Ax's theorem, I will explain one counterexample. However, this is not the end of the story.
This is joint work with Jones and Servi.
Hrushovski-Martin-Rideau have proved rationality of Poincare series counting
numbers of equivalence classes of a definable equivalence relation on the p-adic field (in connection to a problem on counting representations of groups). For this they have proved
uniform p-adic elimination of imaginaries. Their work implies that these Poincare series are
motivic. I will talk about their work.
This is joint work with Angus Macintyre. I will discuss new developments in
our work on the model theory of adeles concerning model theoretic
properties of adeles and related issues on adelic geometry and number theory.