Past Advanced Class Logic

Abstract: We will consider a model theoretic approach to Gelfand-Naimark duality, from the point of view of (generalized) Zariski structures. In particular we will show quantifier elimination for compact Hausdorff spaces in the natural Zariski language. Moreover we may see a slightly unusual construction and tweak to the language, which improves stability properties of the structures.
 

In this talk I will present some answers to the question when every specialisation from a \kappa-saturated extension of 
a Zariski structure is \kappa-universal? I will show that for algebraically closed fields, all specialisations from a \kappa-
saturated extension is \kappa-universal. More importantly, I will consider this question for finite and infinite covers of
Zariski structures. In these cases I will present a counterexample to show that there are covers of Zariski structures 
which have specialisations from a \kappa-saturated extension that are not \kappa-universal. I will present some natural 
conditions on the fibres under which all specialisations from a \kappa-saturated extension of a cover is \kappa-universal. 
I will explain how this work points towards a prospective Ladder Theorem for Specialisations and explain difficulties and 
further works that needs to be considered.
 

This will be a little potpourri containing some of the recent developments on the model theory of F_p((t)) and of algebraic extensions of Q_p.

 "We give a diophantine criterion for a polynomial with rational coefficients not to have any
rational zero, i.e. an existential formula in terms of the coefficients expressing this property. This can be seen as a kind of restricted
model-completeness for Q and answers a question of Koenigsmann."

following the joint paper with L.Shaheen http://people.maths.ox.ac.uk/zilber/wLb.pdf

I will explain how algebraic spaces can be presented as Zariski geometries and prove some classical facts about algebraic spaces using the theory of Zariski geometries.

I will report on some joint work with Jacob Tsimerman
concerning multiplicative relations among singular moduli.
Our results rely on the ``Ax-Schanuel'' theorem for the j-function
recently proved by us, which I will describe.

 This is joint work with Angus Macintyre. We prove a general model-completeness theorem for Henselian valued fields
stating that a Henselian valued field of characteristic zero with value group a Z-group and with finite ramification is model-complete in the language of rings provided that its residue field is model-complete. We apply this to extensions of p-adic fields showing that any finite or infinite extension of p-adics with finite ramification is model-complete in the language of rings.

Motivated by an open conjecture in anabelian geometry, we investigate which arithmetic properties of the rationals are encoded in the absolute Galois group G_Q. We give a model-theoretic framework for studying absolute Galois groups and discuss in what respect orderings and valuations of the field are known to their first-order theory. Some questions regarding local-global principles and the transfer to elementary extensions of Q are raised.

Everything measurable about a quantum system, as modelled by a noncommutative operator algebra, is captured by its commutative subalgebras. We briefly survey this programme, and zoom in one specific incarnation: any bilinear associative function on the set of n-by-n matrices over a field of characteristic not two, that makes the same vectors orthogonal as ordinary matrix multiplication and gives the same trace as ordinary matrix multiplication, must in fact be ordinary matrix multiplication (or its opposite). Model-theoretic questions about the hypotheses and scope of this theorem are raised.

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.