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Logic / Logique
(Bradd Hart and Claude Laflamme, Organizers)

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LUC BÉLAIR, UQAM, Montreal, Quebec  H3C 3P8
Delta-rings and axioms for Frobenius on Witt vectors

We point out how the structure of delta-ring is relevant to the model theory of the Frobenius map on Witt vectors.

MAX BURKE, University of Prince Edward Island
Borel measurability of separately continuous functions

Lebesgue proved that every separately continuous function f: R×R ® R is a pointwise limit of continuous functions. W. Rudin extended this by showing that if X is a metric space, then for any topological space Y, every separately continuous function f: X×Y® R is a pointwise limit of continuous functions. This statement can fail if we take for X an arbitrary linearly ordered space, even if X is separable. However, we show that if X = X₁×¼×X_n, where X₁,...,X_n are linearly ordered spaces which are either all separable or all countably compact, and Y is any topological space, then every separately continuous function f: X×Y® R is Borel measurable. We also give, under a cardinal arithmetic assumption, an example of a linearly ordered space X and a separately continuous function f: X×X® R which is not Borel measurable.

ILIJAS FARAH, Department of Mathematics, University of Rutgers, Piscataway, New Jersey  08854, USA
Layered Ramseyan structures (joint work with Neil Hindman, Howard University)

The famous `Finite sums theorem' proved by N. Hindman in 1970's says that for every partition of the integers into finitely many pieces there is an infinite set A of integers such that all finite sums of distinct elements of A belong to the same piece.

This result has been extended in various ways to partitions of other `Ramseyan structures'. A typical result says that a given combinatorial space S has the property that for every partition of S into finitely many pieces S has a large homogeneous substructure.

I will talk about theorems of Hales-Jewett and Gowers and their various extensions, both true and false.

KLAUS PETER HART, Delft University of Technology, Faculty ITS Postbus 5031, 2600 GA Delft, the Netherlands
Fun with model theory in topology

Recent years have seen an increasing use of model theory in Set-theoretic Topology. I will discuss some results that were obtained using model-theoretic methods; these include existence of universal spaces and uniqueness results for certain spaces. I will also discuss some questions that arose out of this work but for which it is not clear whether they require topology or model theory (OR JUST GOOD OLD FASHIONED insightful analysis and intricate combinatorics) to solve them.

MICHAEL HRUSAK, Inst. de Matem., UNAM, Morelia, Mexico
Combinatorics of MAD families

A classification of maximal almost disjoint (MAD) families of subsets of a countable set along the lines of Rudin-Keisler ordering of ultrafilters is presented. A connection with a problem of J. Roitmann as to whether an existence of a dominating family of size À₁ implies the existence of a MAD family of size À₁ is also discussed.

SALMA KUHLMANN, Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan  S7N 5E6
A maximality property of the Hardy field H(R_an^Powers)

Let R_an^Powers be the expansion of the ordered field of real numbers by the restricted analytic functions and the power functions. Then R_an^Powers is a polynomially bounded (o-minimal) reduct of (R_an,exp), the expansion by restricted analytic functions and the exponential function (cf. [M]). L. v. d. Dries conjectured that R_an^Powers is a maximal polynomially bounded reduct of (R_an,exp).

Let  be an elementary extension of (R_an,exp), and let Â_an^Powers be the corresponding reduct. Let H(Â) be the Hardy field of germs at infinity of R-definable (1-variable) functions. Then H(R_an^Powers) is a Hardy subfield of H(Â). In this talk, we show that H(Â_an^Powers) enjoys the following maximality property:

Let H be a polynomially bounded Hardy subfield of H(Â), containing H(Â_an^Powers), and closed under compositions and compositional inverses. Then necessarily H = H(R_an^Powers).

In particular this shows that H(Â_an^Powers) is maximal among those Hardy subfields of H(Â) arising from polynomially bounded reducts of Â. This gives a partial answer to the above conjecture. The proof of the maximality property is based on a structure theorem for H(Â) established in [K-K] and [KS].

References

[K-K ] F. V. Kuhlmann, and S. Kuhlmann, Valuation Theory of exponential Hardy fields. preprint (1998).
[KS ] S. Kuhlmann, Ordered Exponential Fields. The Fields Institute Monographs 12, Amer. Math. Soc. Publications, January 2000.
[M ] C. Miller, Expansions of the real field with power functions. Ann. Pure Appl. Logic 68(1994), 79-94.

JUSTIN MOORE, Toronto
Dimensions of Martin's axiom

It has been shown that MA_À1 is equivalent to a Ramsey theoretic statement about partitions of finite subsets of w₁. Replacing ``finite'' with ``n-element'' leads to an array of formally weaker axioms whose exact relationship to MA_À1 is currently unknown. This talk will examine some of the consequences of the 2 and 3 dimensional versions of Martin's Axiom.

DAVID PIERCE, McMaster University, Department of Mathematics and Statistics Hamilton, Ontario  L8S 4K1
Differential geometry in model theory

The model completion of the theory of differential fields can be axiomatized by the Frobenius Theorem of differential geometry. In other words: Let T be the theory of fields of characteristic zero with several commuting derivations. Then we can augment T by axioms saying the following, that a collection of differential one-forms have a common zero, provided that the differential ideal that they generate is the same as the ordinary ideal. The result is the model completion of T.

SERGEI STARCHENKO, Department of Mathematics, University of Notre Dame, Notre Dame, Indiana  46556, USA
On stable structures definable in o-minimal models

We will consider stable structures definable in o-minimal models. Our conjecture is that Zilber's principal holds for these structure. As the first step we will consider fields of finite Morley Rank and show that these fields must be pure.

SIMON THOMAS, Rutgers University, New Brunswick, New Jersey  08903, USA
Asymptotic cones of finitely generated groups

If an observer moves steadily away from the Cayley graph of a finitely generated group, then any finite configuration will eventually become indistinguishable from a single point; but he may observe certain finite configurations which resemble earlier configurations. The asymptotic cone is a topological space which encodes all of these recurring finite configurations. Unfortunately the construction of an asymptotic cone involves the choice of a nonprincipal ultrafilter on the set of natural numbers, and it was not clear whether the resulting asymptotic cone depended on the choice of the ultrafilter. In this talk, answering a question of Gromov, I shall present an example of a finitely generated group which has two non-homeomorphic asymptotic cones. This is joint work with Boban Velickovic.

STEVO TODORCEVIC, UFR de Mathématiques, Université Denis-Diderot Paris 7, Paris Cedex  05, France
Parametrized Ramsey theory