Model theory, Feb 2002, Birmingham

Generic variations of models of T

The free roots of the random graph

(Joint work with Frank Wagner). We study the theory of the graphs obtained by iterating the extraction of square roots from the random graph in a free way. This theory does not eliminate hyperimaginaries and it does not have the strict order property. All the previous known examples of theories without elimination of hyperimaginaries did have the strict order property. Since the theory it is not simple, the existence of a simple theory without elimination of hyperimaginaries remains open.

Groups definable in generic difference fields

A difference field is a field K with a distinguished automorphism, and a generic difference field is an existentially closed difference field (we work in the language of rings augmented by a unary function symbol s for the automorphism). There has been quite a lot of work on the model-theoretic properties of these fields, and this work has led to applications to diophantine geometry. One of the main results in the area is a dichotomy result: if a type of finite rank is not modular, then it is non-orthogonal to a type containing a formula s(x)=x, or s^n(x)=x^{p^m} if the characteristic is positive. In characteristic 0, one can show that modular types are stable and stably embedded, but in positive characteristic this is no longer the case.

Model theory and quadratic forms

Geometry of 1-based minimal types

(Joint work with Tristram de Piro). We study interesting features of 1-based rank-1 types. Negatively, a non-affine example of the type having locally modular geometry is constructed. Positively, we prove that, in general, the geometry of the type is only a subgeometry of projective space over some division ring, but it is full projective geometry when we consider 2-fold and 3-fold of the geometry altogether. This shows that the type has strongly minimal (relativised) reduct in its eq. Using the results we can show that in any $\omega$-categorical non-trivial 1-based theory, a projective space over a finite field is definably recovered.

On the Arithmetic of Lexicographic Exponentiation

Problems related to automorphisms groups

Countable models and profinite structures

I have less and less hope for proving Vaught's conjecture for example for superstable theories. Still some partial results are possible. I mean here describing some structural properties of countable models of some superstable theories, or some "pieces" of them. For this purpose small profinite structures may be useful. They appear naturally for example as the sets of generic types of a definable group. It is my impression that in order to describe even as well-behaved object as a locally modular superstable group forking dependence, unlike in the case of a totally transcendental or superstable theory of finite rank, does not suffice. In effect some more refined notions of dependence may be studied, and here again profinite structures may be helpful. It turns out, that still there are some similarities between the treatment of forking dependence and of these more refined dependence relations (induced by isolation, of course).

On certain complete theories of modules over skew polynomial rings

In collaboration with P. Dellunde and F. Delon, we axiomatized the theory of modules of separably closed fields of fixed characteristic and imperfection degree. These theories of modules are instances of complete theories of modules over skew polynomial rings. Presenting our former results in a more general setting, will allow us to give a description of the theories of modules of ultraproducts of separably closed fields of fixed (finite) imperfection degree.

Uniformity in the characteristic p isotrivial Mordell-Lang conjecture

A few years ago I observed that a version of the Mordell-Lang conjecture for isotrivial semiabelian varieties in characteristic p follows from very easy model-theoretic considerations. Working jointly with Rahim Moosa I show that the induced structure on the points of an isotrivial semiabelian variety over a finitely generated field is more complicated than what one sees in characteristic zero, it is still stable. As a consequence, we obtain some uniform Diophantine bounds.

The use of non-standard Frobenius in tight closure theory

Tight closure theory is a fairly new development in commutative algebra initiated by Hochster and Huneke at the end of the '80's. It uses properties of the Frobenius in positive characteristic and then lifts these results to characteristic zero using reduction techniques and Artin Approximation. This yields an extremely powerful method which allows one to prove many deep theorems with relative ease, at least in positive characteristic; the zero characteristic case requires typically some additional work. I will explain how one can often circumvent this by using the non-standard Frobenius in characteristic zero--essentially this is obtained by taking an ultraproduct of rings of positive characteristic in conjunction with certain definability and uniformity results. This works especially well for finitely generated algebras over the complex numbers as I will illustrate with some examples: the Briancon-Skoda Theorem, the Hochster-Roberts Theorem and Boutot's Theorem on rational quotient singularities.

Non abelian C-minimal groups

The notion of C-minimality, introduced by D. Macpherson and C. Steinhorn, provides a natural setting for the study of algebraically closed valued fields and some valued groups. Here we study the structure of non-abelian C-minimal valued groups and prove that they are nilpotent-by-finite.

TBA

The group configuration in simple theories and its applications

I shall present recent results by Itay Ben-Yaacov, Ivan Tomasic and myself about the group configuration theorem for simple theories, and its applications to locally modular types, k-linearity, and the binding group.

Some remarks about universal graphs

We will begin with an exposition of certain results of Cherlin-Shelah-Shi concerning universal graphs omitting a given subgraph. Further information and details about an explicit example will be provided, including results due to Wesleyan student Rehana Patel.

Excellent classes and field arithmetic