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CAIMS Minisymposia

Set Theory and its Applications
Org: Alain Louveau (Paris VI) and Stevo Todorcevic (Toronto; Paris Dauphine)

M. BURKE, University of Prince Edward Island
Simultaneous approximation and interpolation of increasing functions by increasing entire functions

When A and B are countable dense subsets of R, it follows from a well-known result of Cantor that f[A]=B for some order-isomorphism f of R. A theorem of K. F. Barth and W. J. Schneider states that f can be taken to be the restriction to R of an entire function. S. Shelah established a consistent analog of Cantor's result for sets of cardinality À1 by building a model where 2À0 > À1 and second category sets of cardinality À1 exist while any two sets of cardinality À1 which are nonmeager in every interval are order-isomorphic. In earlier work, we proved that the order-isomorphism in Shelah's theorem can be taken to be the restriction to R of an entire function. Using an approximation theorem of L. Hoischen, we also showed that the order-isomorphism f can be taken so that it and its first n derivatives approximate those of a given nondecreasing surjection g of class Cn. Hoischen's theorem also gives equality of the derivatives of f and g on a closed discrete set. We incorporate that improvement into our earlier result.

The following special case of the theorem is provable in ZFC. Let g: R ® R be a nondecreasing Cn surjection. Let e: R ® R be positive and continuous. Let E Í R be a closed discrete set on which g is strictly increasing. Let each of {Ai}, {Bi} be a sequence of pairwise disjoint countable dense subsets of R such that for each i Î N and x Î E we have x Î Ai if and only if g(x) Î Bi. Then there is an entire function f: C ® C such that f[R] Í R and the following properties hold.

    (a) For all x Î R\E, Df(x) > 0.
    (b) For k=0,...,n and all x Î R, |Dkf(x)-Dkg(x)| < e(x).
    (c) For k=0,...,n and all x Î E, Dkf(x)=Dkg(x).
    (d) For each i Î N, f[Ai]=Bi.

ILIJAS FARAH, York University, 4700 Keele Street, Toronto, ON M3J 1P3
The commutant of B(H) in its ultrapower

For an infinite-dimensional separable Hilbert space H let B(H) be its algebra of bounded linear operators. Fix a free ultrafilter U on N and consider the ultrapower of B(H). Eberhard Kirchberg asked whether the commutant of B(H) in its ultrapower is equal to the scalar multiples of the identity. I will answer 3/4 of his question.

This is a joint work with N. Christopher Phillips.

VERA FISCHER, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3
The consistency of b = k < s = k+

Using a proper forcing notion, in 1984 S. Shelah obtained the consistency of b = w1 < s = w2. We obtain a s-centered suborder of Shelah's poset which preserves the unboundedness of a given unbounded family and adds a real not split by the ground model reals. Thus under an appropriate finite support iteration of length k+ we obtain the consistency of b = k < s = k+ for k an arbitrary regular uncountable cardinal.

CLAUDE LAFLAMME, University of Calgary, Department of Mathematics and Statistics, 2500 University Dr. NW, Calgary, Alberta, Canada T2N 1N4
Distinguishing Number of Countable Homogeneous Relational Structures

The distinguishing number of a graph G is the smallest positive integer r such that G has a labeling of its vertices with r labels for which there is no non-trivial automorphism of G preserving these labels.

M. Albertson and K. Collins computed the distinguishing number for various finite graphs, and W. Imrich, S. Klavzar and V. Trofimov computed the distinguishing number of some infinite graphs, showing in particular that the Random Graph has distinguishing number 2.

We compute the distinguishing number of various other finite and countable homogeneous structures, including undirected and directed graphs, and posets. We show that this number is in most cases two or infinite, and besides a few exceptions conjecture that this is so for all primitive homogeneous countable structures.

Joint work with L. Nguyen Van The and N. Sauer.

D. LECOMTE, Université Paris 6, Institut de Mathématiques de Jussieu, Equipe d'Analyse Fonctionnelle, Tour 46-0, boîte 186, 4 place Jussieu, 75252 Paris cedex 05, France
Analytic sets in any dimension

We study the extension to infinite products of the following two dichotomies concerning analytic subsets of two Polish spaces: the Kechris-Solecki-Todorcevic dichotomy about graphs, and the Debs-Lecomte dichotomy about potentially P0x sets.

P. MATET, Laboratoire de Mathematiques, Université de Caen-CNRS, BP 5186, 14032 Caen Cedex, France
Weak Saturation of Ideals on Pk (l)

Suppose k is an uncountable successor cardinal, l > k is a cardinal of cofinality less than k, and J is a k-complete, fine, proper ideal on Pk(l). Then, as shown by Chris Johnson and Yo Matsubara, Pk (l) can be partitioned into l many pieces not in J. What about getting more pieces, say cof( Pk (l), Í ) many? We use pcf theory to show that this can be achieved in a number of cases.


VLADIMIR PESTOV, Université d'Ottawa, 585 av. King Edward, Ottawa, Ontario, Canada K1N 6N5
Actions isométriques affines de groupes polonais sur les espaces de Banach sans points fixes

Cet ouvrage conjoint avec Lionel Nguyen Van Thé (Université de Calgary) propose une caractérisation des groupes topologiques possédant la propriété de point fixe lorsqu'ils agissent par actions isométriques affines sur les espaces de Banach. Entre autres, il est démontré que seuls les groupes précompacts possèdent cette propriété, et que dans le cas séparable, il suffit de considérer les actions sur un unique espace de Banach áU ñ, appelé l'espace de Holmes (l'unique espace de Banach engendré par l'espace métrique universel d'Urysohn U). Enfin, nous montrons les exemples de groupes polonais qui ne peuvent pas agir de façon propre par isométries sur les espaces de Banach, voire les espaces métriques complets. Tels sont le groupe unitaire U(l2), le groupe symétrique infini S¥, le groupe des homéomorphismes Homeo+[0,1], et bien d'autres.

JURIS STEPRANS, Fields Institute