A topological group is extremely amenable if it does not have a fixed-point free action on a compact Hausdorff space. I'll talk about a new example of extremely amenable groups of functions with topology of convergence with respect to a diffused submeasures. The main result unifies earlier work of Christensen-Herer and Pestov. We deduce our results from a Ramsey-type result and give an example showing there is no concentration of measure in this context.

This is a joint work with Slawek Solecki.

A class C of separable Banach spaces is said to be strongly bounded if whenever A is an analytic subset of C, there exists a space in C which contains an isomorphic copy of any X in A.

It is proved that the classes REFL of separable reflexive spaces and SD of spaces with separable dual are strongly bounded. This gives another proof of a result of E. Odell and T. Schlumprecht answering a question of J. Bourgain: there exists a separable reflexive Banach space which is universal for the class of separable uniformly convex spaces.

Joint work with Pandelis Dodos.

In 1984, S. Shelah obtained the consistency of b = w₁ < s = w₂ using countable support iteration of proper forcing notions. The method can not be further generalized since subsequent iterations would collapse the continuum. However finite support iteration of c.c.c. forcing notions does not have this disadvantage and we succeed to extend the above result obtaining a model of b = w₁ < s = m for m arbitrary regular cardinal. The c.c.c. forcing notions which we use are closely related to the partial order used originally by S. Shelah.

We present some new constructions of Boolean algebras achieved by combinatorial methods which have applications in the theory of Banach spaces.

The purpose of this talk is to present a framework for studying weakly-null sequence of Banach spaces using the Ramsey property of the families of finite sets of integers called Barriers, introduced by Nash-Williams. We focus on "partial unconditionality" properties of weakly-null sequences of Banach spaces. Inspired by a recent work of S. J. Dilworth, E. Odell, Th. Schlumprecht and A. Zsak, we give a general notion of partial unconditionality that covers most of the known cases, including the classical Elton's near unconditionality, convex unconditionality or Schreier unconditionality, and some new ones.

The method reduces the problem to the understanding of mappings j: B ® FIN×c₀, where FIN denotes the family of finite sets, B Í FIN is a barrier, and c₀ is the Banach space of sequences of real numbers converging to zero. We present several combinatorial results concerning these mappings, starting with the simpler mappings j: B ® FIN. One of the main results here is that every mapping j: B ® c₀ has a restriction which is, up to perturbation, what we call a L-mapping, i.e., j has a precise Lipschitz property and satisfies that the support suppj(s) of j(s) is included in s for every s Î B. L-mappings allow to define naturally a weakly-null sequence, called L-sequence, associated to them.

Finally our approach shows that if for some notion of unconditionality \mathfrak F there is a weakly-null sequence with no \mathfrakF-unconditional subsequence, then there must be an L-sequence with no \mathfrak F-unconditional subsequence.

This is a joint work with S. Todorcevic.

In this talk we will consider what can be said about a linear order which contains neither real nor Aronszajn suborders. It is easily seen that s-scattered orders fit this criterion. Baumgartner constructed an example which is not s-scattered and contains neither a real nor Aronszajn suborder. Baumgartner's example is necessarily not minimal with respect to not being s-scattered. We have shown that PFA implies any minimal non-s-scattered order of size À₁ must either be a real or Aronszajn type. A version of this theorem for larger linear orders will also be discussed, along with the relevance to a theorem of Laver which asserts that the s-scattered orders are well quasi-ordered. The work presented is joint work with Tetsuya Ishiu.

We study the unitary group of separable infinite dimensional complex Hilbert space as a discrete group and show that all of its actions by isometries on a metric space have orbits of finite diameter. This property is enough to ensure that the unitary group also satisfies properties FH and FA of Serre.

This is a joint work with Eric Ricard of CNRS, Université Franche-Comté.

Let I₀ be the s-ideal of subsets of a Polish group generated by Borel sets which have perfectly many pairwise disjoint translates. I will present a Fubini-type theorem that holds between I₀ and the s-ideals of Haar measure zero sets and of meager sets. I will show how to use this result to give a simple proof of a generalization of a theorem of Balcerzak-Roslanowski-Shelah stating that I₀ on 2^N strongly violates the countable chain condition.

A partial order is said to have the n-localization property if every real added by the partial order can be approximated by an n-branching tree in the ground model. Well known arguments establishing this type of property for variants of Sacks forcing do not generalize (in the obvious fashion) to the case of Silver forcing. It will be shown that, nevertheless, similar results do hold.