Past Logic Seminar

After a short introduction to the section conjecture, I plan to present a "minimalistic" form of the birational p-adic section conjecture. The result is related to both: Koenigsmann's proof of the birational p-adic section conjecture, and a "minimalistic" Galois characterisation of formally p-adic valuations.

VC dimension and density are properties of a collection of sets which come from probability theory.  It was observed by Laskowski that there is a close tie between these notions and the model-theoretic property called NIP. This tie results in many examples of collections of sets that have finite VC dimension. In general, it is difficult to find upper bounds for the VC dimension, and known bounds are mostly very large. However, the VC density seems to be more accessible. In this talk, I will explain all of the above acronyms, and present a theorem which gives an upper bound (in some cases optimal) on the VC density of formulae in some examples of NIP theories. This represents joint work of myself with M. Aschenbrenner, A. Dolich, D. Macpherson and S.

Starchenko.

I will speak about the CBP (canonical base property) for types of finite SU-rank. This property first appears in a paper by Pillay and Ziegler, who show that it holds for types of finite rank in differentially closed fields of characteristic 0, as well as in existentially closed difference fields. It is unknown whether this property holds for all finite rank types in supersimple theories. I will first recall the definition of a canonical base, and give some natural examples. I will then  talk about a reduction of the problem (which allows one to extend the Pillay-Ziegler result to existentially closed fields of any characteristic), and finally derive some consequences of the CBP, in particular the UCBP, thus answering a question of Moosa and Pillay.  If time permits, I will show an application of these results to difference

fields.

I will talk about two pieces of work on finite covers:

(i) Work of Hrushovski which, for a stable theory, links splitting of certain finite covers with higher amalgamation properties;

(ii) Joint work of myself and Elisabetta Pastori which uses group cohomology to investigate some non-split finite covers of the set of k-sets from a disintegrated set.

In this talk we discuss the main results of my PhD thesis. We begin by giving some background on Moufang polygons. This is followed by a short introduction of the basic model theoretic notions related to the thesis, such as asymptotic classes of finite structures, measurable structures, (superstable) supersimple theories and (finite Morley rank) S_1 rank. We also mention the relation between Moufang polygons and the associated little projective groups.

Moufang polygons have been classified by Tits and Weiss, and a complete list is given in their book `Moufang polygons'.

This work is inspired by a paper of Kramer, Tent and van Maldeghem called "Simple groups of finite Morley rank and Tits buildings". The authors work in a superstable context. They show that Moufang polygons of finite Morley rank are exactly Pappian polygons, i.e., projective planes, symplectic quadrangles and split Cayley hexagons, provided that they arise over algebraically closed fields.

We work under the weaker assumption of supersimplicity. Therefore, we expect more examples. Indeed, apart from those already occuring in the finite Morley rank case, there are four further examples, up to duality, of supersimple Moufang polygons; namely, Hermitian quadrangles in projective dimension 3 and 4, the twisted triality hexagon and the (perfect) Ree-Tits octagon, provided that the underlying field (or `difference' field in the last case) is supersimple.

As a result, we obtain the nice characterization that supersimple Moufang polygons are exactly those Moufang polygons belonging to families which also arise over finite fields.

Examples of supersimple Moufang polygons are constructed via asymptoticity

arguments: every class C of finite Moufang polygons forms an asymptotic class, and every non-principal ultraproduct of C gives rise to a measurable structure, thus supersimple (of finite S_1 rank). For the remaining cases one can proceed as follows: let \Gamma be any Moufang polygon belonging to a family which does not arise over finite fields, and call K its underlying field; then K is

(first-order) definable in \Gamma, and by applying some model theoretic facts this definability is inconsistent with supersimplicity".

This purpose of this talk will be to introduce the idea that the spectrum of nonstandard models of a ``standard''

algebraic object can be used much like a microscope with which one may perceive and codify irrationality invisible within the standard model.

This will be done by examining the following three themes:

\item {\it Algebraic topology of foliated spaces} We define the fundamental germ, a generalization of fundamental group for foliations, and show that the fundamental germ of a foliation that covers a manifold $M$ is detected (as a substructure) by a nonstandard model of the fundamental group of $M$.

\item {\it Real algebraic number theory.} We introduce the group $(r)$ of diophantine approximations of a real number $r$, a subgroup of a nonstandard model of the integers, and show how $(r)$ gives rise to a notion of principal ideal generated by $r$.

The general linear group $GL(2, \mathbb{Z})$ plays here the role of a Galois group, permuting the real ideals of equivalent real numbers.

\item {\it Modular invariants of a Noncommutative Torus.} We use the fundamental germ of the associated Kronecker foliation as a lattice and define the notion of Eisenstein series, Weierstrass function, Weierstrass equation and j-invariant.

A theorem of Kuyk says that every Abelian extension of a

Hilbertian field is Hilbertian.

We conjecture that for an Abelian variety $A$ defined over

a Hilbertian field $K$

every extension $L$ of $K$ in $K(A_\tor)$ is Hilbertian.

We prove our conjecture when $K$ is a number field.

The proofs applies a result of Serre about $l$-torsion of

Abelian varieties, information about $l$-adic analytic

groups, and Haran's diamond theorem.

Let $\mathbb{E}$ be an elliptic curve defined over a number field $k$,

and let $a\in\mathbb{E}(\mathbb{C})$ be a complex point. Among the

possible choices of sequences of division points of $a$, $(a_n)_n$

such that $a_1 = a$ and $na_{nm} = a_m$, we can pick out those which

converge in the complex topology to the identity. We show that the

algebraic content of this effect of the complex topology is very

small, in the sense that any set of division sequences which shares

certain obvious algebraic properties with the set of those which

converge to the identity is conjugated to it by a field automorphism

of $\mathbb{C}$ over $k$.

As stated, this is a result of algebra and number theory. However, in

proving it we are led ineluctably to use model theoretic techniques -

specifically the concept of "excellence" introduced by Shelah for the

analysis of $L_{\omega_1,\omega}$ categoricity, which reduces the

question to that of proving certain unusual versions of the theorems

of Mordell-Weil and Kummer-Bashmakov. I will discuss this and other

aspects of the proof, without assuming any model- or number-theoretic

knowledge on the part of my audience.

Let T be a (one-sorted first order) geometric theory (so T

has infinite models, T eliminates "there exist infinitely many" and

algebraic closure gives a pregeometry). I shall present some results

about T_P, the theory of lovely pairs of models of T as defined by

Berenstein and Vassiliev following earlier work of Ben-Yaacov, Pillay

and Vassiliev, of van den Dries and of Poizat. I shall present

results concerning superrosiness, the independence property and

imaginaries. As far as the independence property is concerned, I

shall discuss the relationship with recent work of Gunaydin and

Hieronymi and of Berenstein, Dolich and Onshuus. I shall also discuss

an application to Belegradek and Zilber's theory of the real field

with a subgroup of the unit circle. As far as imaginaries are

concerned, I shall discuss an application of one of the general

results to imaginaries in pairs of algebraically closed fields,

adding to Pillay's work on that subject.

In 1993 in his paper "A new strongly minimal set" Hrushovski produced a family of counter examples to a conjecture by Zilber. Each one of these counter examples carry a pregeometry. We answer a question by Hrushovski about comparing these pregeometries and their localization to finite sets. We first analyse the pregeometries arising from different variations of the construction before the collapse. Then we compare the pregeometries of the family of new strongly minimal structures obtained after the collapse.

We present a topos-theoretic interpretation of (a categorical generalization of) Fraïssé's construction in Model Theory, with applications to countably categorical theories. The proof of our main theorem represents an instance of exploiting the interplay of syntactic, semantic and geometric ideas in the foundations of Topos Theory.

Given K a separably closed field of finite ( > 1) degree of imperfection, and semiabelian variety A over K, we study the maximal divisible subgroup A^{sharp} of A(K). We show that the {\sharp} functor does not preserve exact sequences and also give an example where A^{\sharp} does not have relative Morley rank. (Joint work with F. Benoist and E. Bouscaren)

Abstract available at: http://people.maths.ox.ac.uk/~kirby/LInnocente.pdf