




Model Theoretic Algebra / Algèbre en théorie des modèles (Bradd Hart, F.V. Kuhlmann and S. Kuhlmann, Organizers)
 ZOE CHATZIDAKIS, Paris 7

 BRADD HART, McMaster/Fields Institute
To be announced

 FRANZVIKTOR KUHLMANN, Department of Mathematics, University of Saskatchewan,
Saskatoon, Saskatchewan S7N 5E6
Additive polynomials and the model theory of valued fields in
positive characteristic

Let F_{p} ((t)) denote the field of formal Laurent series over
the field with p elements; it carries the tadic valuation v_{t}. A
well known open problem in model theoretic algebra is to find a
complete recursive axiom system for the elementary theory of
F_{p} ((t)). This would yield that it is decidable. A similar
result was shown by AxKochen and Ershov for the padics
Q_{p},v_{p}. An adaptation of their axiom system to the case of
F_{p}((t)) is: ``henselian defectless valued field of
characteristic p with value group a Zgroup and residue
field F_{p}''.
We showed in 1989 that this axiom system is not complete. In 1998 we
developed an elementary axiom scheme which holds in maximal valued
fields of positive characteristic and describes valuation theoretic
properties of additive polynomials on such fields. This axiom scheme is
independent of the above axioms. This work is to appear in JSL. In
1999, in joint work with L. van den Dries, we developed a particularly
simple version of this axiom scheme for F_{p} ((t)), using the
fact that this field is locally compact. This work is to appear in
Canad. Math. Bull.
 SALMA KUHLMANN, Saskatchewan, 106 Wiggins Road, Saskatoon,
Saskatchewan S7N 5E6
Lexicographic chains

Lexicographic chains appear in several contexts in Set Theory and
Logic. They also appear naturally in Model Theory. For example, they
play a crucial role when constructing nonArchimedean models of real
exponentiation. They have an Arithmetic that is fascinating in its own
right, and about which relatively little is known. In this talk, I
will discuss several aspects of this Arithmetic, present some results,
and open questions.
 JIM LOVEYS, McGill
To be confirmed

 MICHAEL MAKKAI, Department of Mathematics and Statistics, McGill University,
Montreal, Quebec H3A 2K6
Logic and D. G. Quillen's homotopical algebra

In [Logic Colloquium '95, Lecture Notes of Logic 11, Springer,
1998; 153190] the author introduced First Order Logic with Dependent
Sorts (FOLDS) for certain foundational purposes. The main semantical
feature of FOLDS is the presence, associated with any given FOLDS
signature L, of a concept called Lequivalence, taking the
place of the usual notion of isomorphism of Lstructures. In this talk,
modeltheoretical results for FOLDS, of both of a general and of an
applied nature, will be presented. In particular, connections with
Quillen's classical approach to homotopy theory via the socalled model
categories will be developed. An example of the connections is the fact
that Lequivalence, with a suitable L, for simplicial Kancomplexes
is the same as homotopy equivalence.
 MURRAY MARSHALL, University of Saskatchewan
Open questions in the theory of spaces of orderings

Every formally real field F has associated to it a reduced special
group G = F^{*}/SF^{*2} (terminology of M. Dickmann) and a space
of orderings (X,G) (my terminology) where X is the topological
space of orderings of F. The two concepts are known to be essentially
equivalent. There are a number of hard open problems in the theory of
spaces of orderings (e.g., L. Bröcker's tinvariant problem;
M. Coste's problem about mod 2^{k} representation in the reduced Witt
ring of F) which would be answered positively if it were known that
positive primitive formulas in the language of special groups which
hold true on every finite quotient of G, hold on G as well. There
is some evidence for this conjecture. The statement that a quadratic
form over F is weakly isotropic is such a pp formula (by the
localglobal principle for weak isotropy due to L. Bröcker and
A. Prestel). The characterization of the powers of the fundamental
ideal of the reduced Witt ring of F in terms of signatures can also
be reduced to such a pp formula (by the solution of the signature
conjecture by Dickmann and Miraglia, using Ktheoretic results of
Voevodsky). Thus it would appear that even if the conjecture is false,
it is still important to determine for which sorts of pp formulas and
which sorts of special groups it is true.
 CHRIS MILLER, Ohio State University, Columbus, Ohio 43210, USA
Hausdorff dimension, analytic sets and transcendence

Every analytic (in the sense of descriptive set theory) set of real
numbers having positive Hausdorff dimension contains a transcendence
base. Equivalently, every analytic proper realclosed subfield of the
reals has Hausdorff dimension zero. (Joint work with G. A. Edgar.)
 TONIANN PITASSI, Toronto
To be announced

 HANS SCHOUTENS, Rutgers University, New Brunswick, New Jersey, USA
Determining the number of equations of an affine curve

It is in general a very hard problem to find the minimal number of
generators of an ideal in a finitely generated algebra over a field. In
fact, there can be no computer algebra system that calculates this
number exactly (I will briefly discuss a counterexample due to
SCHMIDT). The obstruction lies in the possible unboundedness
of the degrees of a minimal set of generators and is also related to
the presence of nontrivial line bundles.
I will show that for the defining ideal of an affine curve
C Ì A_{K}^{n}, such a uniform bound does exist, except
possibly when C is locally generated by the least possible minimal
number of generators (namely n1). This answers a question raised by
VAN DEN DRIES for affine curves. Consequently, there exists an
algorithm calculating the exact number of generators of the ideal of an
affine reduced curve C (barring the exceptional case) provided we
take the arithmetic of the field as an oracle. In the exceptional case
(which includes the smooth case!), we know at least that the ideal of
C requires either n1 or n generators.
The proof uses a nonstandard argument together with the ForsterSwan
Theorem and (the positive solution of) the EEConjecture.
 ZIV SHAMI, McMaster University and Fields Institute
Towards a binding group theorem for simple theories

Let T be simple. Let p Î S(Æ) be internal in Q Î L and
suppose the global algebraic closure is weakly transitive on
Q^{C}, that is DCL(acl_{R}(Q)) Í ACL(Q^{C}ÈA) for every onetofinite definable relation
R defined over A (where ACL(S) (DCL(S)) for a set
S denotes those elements (in C^{eq}) which have finite
orbit (an orbit of size 1 resp.) with respect to the action of
Aut(C^{eq}/Q^{C}). acl_{R}(Q)={b   R(b,[`(c)]) for some tuple [`(c)] from Q^{C}}). Then
Aut(p/Q)={s p^{C}  s Î Aut(C/Q^{C}) } with its action on
p^{C} is typedefinable in C^{eq} over
Æ. If p is Qinternal via pregular generic parameter
(i.e. every forking extension of the type of the generic
parameter is orthogonal to p) then Aut(p/Q) is
directlimitdefinable in C^{eq} over Æ and its
action on p^{C} is definable. This is a joint work with
B. Hart.
 REED SOLOMON, Department of Mathematics, University of WisconsinMadison,
Madison, Wisconsin 53706
The effective content of spaces of orders on groups and fields

For a formally real field F, the space of orders X(F) has a natural
topology which makes it a Boolean space (compact, Hausdorff, and totally
disconnected). Craven proved that given any Boolean space B, there is a
formally real field F such that B is homeomorphic to X(F). Metakides
and Nerode examined the computable content of this correspondence by
considering effectively closed subspaces of the Cantor space (called
P^{0}_{1} classes) in place of Boolean spaces. They showed that
for every P^{0}_{1} class C, there is a computable field F such that
C is homeomorphic to X(F) by a Turing degree preserving homeomorphism.
Downey and Kurtz asked whether a similar correspondence holds for
computable orderable groups. In this talk, we will discuss these
results as well as give a negative answer to the Downey and Kurtz
question for the classes of torsionfree abelian and nilpotent groups.
 KATRIN TENT, Unviersität Würzburg, Am Hubland, 97074 Wüerzburg,
Germany
BNpairs of finite Morley rank

We report on recent progress on the CherlinZil'ber Conjecture, viz.
that a simple group of finite Morley rank is an algebraic group over an
algebraically closed field. The conjecture implies in particular that
any such group must have a definable `split' BNpair. The following is
a partial analog of the famous paper by Fong and Seitz on finite groups
with split BNpairs:
Theorem [T]. Let G be a simple group with a definable BNpair of rank 2 where B=U·T for T=BÇN and a normal subgroup U of B with Z(U) ¹ 1. It was shown in [TVM2] that the Weyl group W=N/BÇN has cardinality 2n with n=3,4,6,8 or 12. If G has finite Morley rank then furthermore the following holds:
(i) If n=3, then G is definably isomorphic to PSL_{3}(K) for some algebraically closed field K.
(ii) If U is nilpotent and W ¹ 16, then G is definably isomorphic to PSL_{3}(K), PSp_{4}(K) or G_{2}(K) for some algebraically closed field K.
(iii) If Z(U) contains a Bminimal subgroup A with RM(A) ³ RM(P_{i}/B) for both parabolic subgroups P_{1} and P_{2}, then G is as in (ii).
References
[T ] K. Tent, Split BNpairs of finite Morley rank.
preprint.
[TVM1 ] K. Tent and H. Van Maldeghem, Cherlin's Conjecture
for buildings with panels of Morley rank 1. preprint.
[TVM2 ] , On irreducible (B,N)pairs of rank
2. Math. Forum, to appear.
 ROSS WILLARD, Department of Pure Mathematics, Waterloo, Ontario N2L 3G1
Palyutin's hformulas and a problem from universal algebra

In this talk we describe a nice syntactic characterization of a
structural property of general algebras (the strict refinement
property) in terms of certain formulas (hformulas) defined by
E. A. Palyutin in Categorical Horn classes, I. Algebra and Logic
19(1980), 377400.
 ROBERT E. WOODROW, University of Calgary, Calgary, Alberta
Absorbing sets in arccoloured tournaments

Let T be a tounament whose arcs are coloured with k colours. Call
a subset X of the vertices of T absorbing if from each vertex of
T no in X there is a monochromatic directed path to some vertex in
X. We consider the question of the minimum size of absorbing sets,
extending known results and using new approaches based on notions from
the Theory of Relations. Most of the work deals with finite
tournaments, but extensions to the infinite are also discussed.

