Past Advanced Class Logic

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.

 We consider two decision problems for linear recurrence sequences (LRS) 
over the integers, namely the Positivity Problem (are all terms of a given 
LRS positive?) and the Ultimate Positivity Problem (are all but finitely 
many terms of a given LRS positive?). We show decidability of both 
problems for LRS of order 5 or less, and for simple LRS (i.e. whose 
characteristic polynomial has no repeated roots) of order 9 or less. Our 
results rely on on tools from Diophantine approximation, including Baker's 
Theorem on linear forms in logarithms of algebraic numbers. By way of 
hardness, we show that extending the decidability of either problem to LRS 
of order 6 would entail major breakthroughs on Diophantine approximation 
of transcendental numbers.

This is joint with work with Joel Ouaknine and Matt Daws.

 A classical question in the model theory of fields is to find out which fields are model complete in the language of rings. It turns out that all well-known examples of model complete fields are quite rigid when it comes to henselianity. We discuss some first results which indicate that in residue characteristic zero, definable henselian valuations prevent model completeness.

A topological space is called rigid if its only autohomeomorphism is the identity map. Using the Axiom of Choice it is easy to construct rigid subsets of the real line R, but sets constructed in this way always have size continuum. I will explore the question of whether it is possible to have rigid subsets of R that are small, meaning that their cardinality is smaller than that of the continuum. On the one hand, we will see that forcing can be used to produce models of ZFC in which such small rigid sets abound. On the other hand, I will introduce a combinatorial axiom that can be used to show the consistency with ZFC of the statement "CH fails but every rigid subset of R has size continuum". Only a working knowledge of basic set theory (roughly what one might remember from C1.2b) and topology will be assumed.

 Following Prestel and Ziegler, we will explore what it means for a field
to be t-henselian, i.e. elementarily equivalent (in the language of
rings) to some non-trivially henselian valued field. We will discuss
well-known as well as some new properties of t-henselian fields.

This is joint work with Angus Macintyre. We study model-theoretic properties of 
the ring of adeles, the hyperring of adele classes (studied by Connes-Consani), 
and residual hyperfields of valued fields (in the sense of Krasner).

I'll sketch some context for future and past research around valued fields
and motivic integration, from a model theoretic viewpoint, leaving out technical details. 
The talk will be partly conjectural.

I shall present a geometric property valid in many Hrushovski
amalgamation constructions, relative CM-triviality, and derive
consequences on definable groups: modulo their centre they are already
products of groups interpretable in the initial theories used for the
construction. For the bad field constructed in this way, I shall
moreover classify all interpretable groups up to isogeny.

This is joint work with Uri Onn. We use motivic integration to get the growth rate of the sequence consisting of the number of conjugacy classes in quotients of G(O) by congruence subgroups, where $G$ is suitable algebraic group over the rationals and $O$ the ring of integers of a number field.

The proof uses tools from the work of Nir Avni on representation growth of arithmetic groups and results of Cluckers and Loeser on motivic rationality and motivic specialization.