Model theory, Feb 2002, Birmingham
This conference follows a regional
meeting of the London Mathematical Society and takes place
from Thursday Feb 28th 2002 to Saturday March 2nd 2002.
The following is a DRAFT programme. Please note that some
details may change. However, the end of the meeting will
be at 4.30 on the Saturday, to give plenty of time for people
to travel home, or for further discussion in the pub,
as is traditional.
Talks on the Thursday and Friday will be in Lecture room B
(first floor, Watson Building) whereas talks on the Saturday will
be in Lecture room A (ground floor)
This meeting is supported by the
London Mathematical
Society. A limited amount of money may be available for
travel or other expenses for participants. Please enquire
if you are interested in attending.
Invited speakers, titles, and short abstracts
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 modeltheoretic
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 nonorthogonal
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 1based minimal types
(Joint work with Tristram de Piro).
We study interesting features
of 1based rank1 types. Negatively, a nonaffine 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 2fold and 3fold 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
nontrivial 1based theory, a projective space over a finite field is
definably recovered.
On the Arithmetic of Lexicographic Exponentiation
Hausdorff developed several arithmetic operations on totally
ordered sets, generalizing Cantor's ordinal arithmetic.
Many open questions arise naturally, that we have been studying in the
last few years. This talk will give an overview of our main results.
Please see the full abstract (in PDF format)
for more details and references.
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 wellbehaved 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 MordellLang
conjecture
A few years ago I observed that a version of the MordellLang
conjecture for isotrivial semiabelian varieties in characteristic p
follows from very easy modeltheoretic 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 nonstandard 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
nonstandard Frobenius in characteristic zeroessentially 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 BrianconSkoda Theorem, the HochsterRoberts Theorem
and Boutot's Theorem on rational quotient singularities.
Non abelian Cminimal groups
The notion of Cminimality,
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 nonabelian
Cminimal valued groups and prove that they are nilpotentbyfinite.
TBA
The group configuration in simple theories and its applications
I shall present recent
results by Itay BenYaacov, Ivan Tomasic
and myself about the group configuration theorem for simple theories, and
its applications to locally modular types, klinearity, and the binding
group.
Some Diophantine properties of curves definable in ominimal expansions
of the real field
Let X be a 1dimensional set definable in some ominimal
expansion of the real field.Suppose that the intersection of
X with any 1dimensional,semialgebraic set is bounded. Then
the set of points on X with integer coordinates is,in a sense
to be made precise, very sparse.
Some remarks about universal graphs
We will begin with an exposition of certain results
of CherlinShelahShi 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
Richard Kaye
Email: R.W.Kaye@bham.ac.uk
Home page: http://web.mat.bham.ac.uk/R.W.Kaye/
LMS regional meeting and Models 2002:
http://web.mat.bham.ac.uk/R.W.Kaye/models2002/
