Past Advanced Class Logic

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.

(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.