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

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ZOE CHATZIDAKIS, Paris 7

BRADD HART, McMaster/Fields Institute
To be announced

FRANZ-VIKTOR 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 t-adic 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 Ax-Kochen and Ershov for the p-adics 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 Z-group 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 non-Archimedean 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; 153-190] 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 L-equivalence, taking the place of the usual notion of isomorphism of L-structures. 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 so-called model categories will be developed. An example of the connections is the fact that L-equivalence, with a suitable L, for simplicial Kan-complexes 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 t-invariant 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 local-global 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 K-theoretic 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 real-closed 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 non-trivial line bundles.

I will show that for the defining ideal of an affine curve C Ì A_Kⁿ, such a uniform bound does exist, except possibly when C is locally generated by the least possible minimal number of generators (namely n-1). 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 n-1 or n generators.

The proof uses a non-standard argument together with the Forster-Swan Theorem and (the positive solution of) the EE-Conjecture.

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^CÈA) for every one-to-finite 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 type-definable in C^eq over Æ. If p is Q-internal via p-regular generic parameter (i.e. every forking extension of the type of the generic parameter is orthogonal to p) then Aut(p/Q) is direct-limit-definable 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 Wisconsin-Madison, 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 P⁰₁ classes) in place of Boolean spaces. They showed that for every P⁰₁ 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 torsion-free abelian and nilpotent groups.

KATRIN TENT, Unviersität Würzburg, Am Hubland, 97074  Wüerzburg, Germany
BN-pairs of finite Morley rank

We report on recent progress on the Cherlin-Zil'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' BN-pair. The following is a partial analog of the famous paper by Fong and Seitz on finite groups with split BN-pairs:

Theorem [T]. Let G be a simple group with a definable BN-pair 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 PSL₃(K) for some algebraically closed field K.
(ii) If U is nilpotent and |W| ¹ 16, then G is definably isomorphic to PSL₃(K), PSp₄(K) or G₂(K) for some algebraically closed field K.
(iii) If Z(U) contains a B-minimal subgroup A with RM(A) ³ RM(P_i/B) for both parabolic subgroups P₁ and P₂, then G is as in (ii).

References

[T ] K. Tent, Split BN-pairs 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 h-formulas 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 (h-formulas) defined by E. A. Palyutin in Categorical Horn classes, I. Algebra and Logic 19(1980), 377-400.

ROBERT E. WOODROW, University of Calgary, Calgary, Alberta
Absorbing sets in arc-coloured 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.