## Damian Sobota: The Josefson–Nissenzweig theorem for Cp(X)-spaces

Seminar: Working group in applications of set theory, IMPAN

Thursday, 20.12.2018, 10:15, room 105, IMPAN

Speaker: Damian Sobota (Kurt Godel RC, Vienna)

Title: “The Josefson–Nissenzweig theorem for Cp(X)-spaces ”

Abstact: “The famous Josefson–Nissenzweig theorem asserts that for every infinite-dimensional Banach space X there exists a sequence (x_n*) in the dual space X* which is weak* convergent to 0 and each x_n* has norm 1. Despite the apparent simplicity of the theorem no constructive proof — even in the case of Banach spaces of continuous functions on compact spaces — has been known.
Recently, Banakh, Śliwa and Kąkol in their studies of separable quotients of topological vector spaces of the form Cp(X), i.e. spaces of continuous functions on Tychonoff spaces endowed with the pointwise convergence topology, have obtained several results characterizing those Cp(X)-spaces for which the Josefson–Nissenzweig theorem holds.
During my talk I will present some introductory facts concerning the theorem for Cp(X)-spaces, show that the existence of “Josefson–Nissenzweig” sequences for Cp(K)-spaces, where K is compact Hausdorff, is strongly related to a variant of the Grothendieck property of Banach spaces, as well as prove that every compact space obtained as a limit of an inverse system consisting only of minimal extensions admits such sequences (and the proof is constructive). This is a joint work with Lyubomyr Zdomskyy”.

Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/

## Philipp Schlicht: Oligomorphic groups are essentially countable

Tuesday, December 18, 2018, 15.00
Howard House 4th Floor Seminar Room, University of Bristol

Speaker: Philipp Schlicht (University of Bristol )

Title: Oligomorphic groups are essentially countable

Abstract:

Model theoretic properties of a countable structure are closely connected with properties of its automorphism group. For instance, the automorphism groups of ω-categorical structures on N are precisely the oligomorphic closed subgroups of Sym(N) (a permutation group is oligomorphic if for each k there are only finitely many k-orbits). In this recent project with Andre Nies and Katrin Tent, we study the complexity of topological isomorphism of oligomorphic closed subgroups of Sym(N) in the setting of Borel reducibility. Previous work of Kechris, Nies and Tent, and independently Rosendal and Zielinski, showed that this equivalence relation is below graph isomorphism. We show that it is below a Borel equivalence relation with countable equivalence classes.

## Michael Hrusak: Ramsey theorem with highly connected homogeneous sets

Dear all,

The seminar meets on Wednesday December 19th at 11:00 in the Institute
of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Michael Hrusak — Ramsey theorem with highly connected
homogeneous sets
(joint work with J. Bergfalk and S. Shelah)

A graph $(\kappa, E)$ is highly homogeneous if all its restrictions on
complements of sets of cardinality less than $\kappa$ are homogeneous.
We investigate for which cardinals $\theta < \lambda \leq \kappa$ does
hold that for every colouring $c:[\kappa]^2\to \theta$ there exists $A$
subset of $\kappa$ of cardinality $\lambda$ and a colour $i\in\theta$
such that the graph $(A, c^{-1}(i)\cap [A]^2)$ is highly homogeneous.

Best,
David

## Daniel T. Soukup – New aspects of ladder system uniformization II

Talk held by Daniel Soukup (KGRC) at the KGRC seminar on 2018-12-13.

Abstract: We continue the previous lecture and present proofs for some of the new results. We show that $\diamondsuit$ implies that for any Aronszajn-tree $T$, there is a ladder system with a 2-colouring with no $T$-uniformization. However, if $\diamondsuit^+$ holds then for any ladder system $C$ there is an Aronszajn tree $T$ so that any monochromatic colouring of $C$ has a $T$-uniformization (cf. https://arxiv.org/abs/1806.03867).

## Asaf Karagila: On countable unions of countable sets

BIU seminar in Set Theory

December 17, 2018

Speaker: Asaf Karagila (UEA)

Title: On countable unions of countable sets

Abstract. How big can countable unions of countable sets be? Assuming the axiom of choice, countable. Not assuming the axiom of choice, it is not hard to arrange situation where there are many incomparable cardinals which are the countable union of countable sets. But none of them are “particularly large”. While a countable union of countable sets can at most be mapped onto $\omega_1$, its power set can be made much larger. We prove an old (and nearly forgotten) theorem of Douglass Morris, that it is consistent that for every $\alpha$ there is a set which is a countable union of countable sets, but its power set can be mapped onto $\alpha$.

## Ilijas Farah: On the model theory of C*-algebras

HUJI Logic Seminar
12/December/18, 11 am, in Ross 63.

Speaker: Ilijas Farah

Title: On the model theory of C*-algebras

Abstract. Ultrapowers and reduced products play a central role in the Elliott classification program for separable (nuclear, etc.) C*-algebras. Although an ultrapower of a separable C*-algebra A is quite different from the reduced product ℓ∞(A)/c0(A)

, these massive algebras are interchangeable in many (but not quite all) concrete applications. I will present a theorem
that attempts to give an abstract explanation of this phenomenon. This preliminary result applies to some other axiomatizable categories, and its proof does not use any of the nontrivial theory of C*-algebras.

No previous knowledge of C*-algebras is required; they appear primarily as a motivation.
This is preliminary part of a joint work with Christopher Schafhauser.

## Piotr Koszmider: Controlling linear operators on C(K)s through the rigidity of K

Seminar: Working group in applications of set theory, IMPAN

Thursday, 13.12.2018, 10:15, room 105, IMPAN

Speaker: Piotr Koszmider (IM PAN)

Title: “Controlling linear operators on C(K)s through the rigidity of K”

Abstact: ” In the second talk of the series devoted to classical phenomena in Banach spaces of the form C(K) we will see how linear operators on a C(K) can be represented by continuous maps from K into the space of the Radon measures on K with the weak* topology. Previously presented results concerning weakly compact subsets in M(K) will allow us to obtain a “geometric” understading of the rigidity conditions on the algebra of all linear operators (having few operators modulo weakly compact operators) as versions of topological or Boolean rigidity conditions (having few continuous maps or few Boolean endomorphisms) which can be combinatorially imposed on K.”.

Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/

## Aristotelis Panagiotopoulos: Higher dimensional obstructions for star-reductions

The last meeting before the break. Happy holidays.
Seminar will resume in the New Year

Mathematical logic seminar – Dec 11 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Aristotelis Panagiotopoulos
Department of Mathematics
Caltech

Title:     Higher dimensional obstructions for star-reductions

Abstract:

In this talk we will consider *-reductions between orbit equivalence relations. These are Baire measurable reductions which preserve generic notions, i.e., preimages of comeager sets are comeager. In short, *-reductions are weaker than Borel reductions in the sense of definability, but as we will see, they are much more sensitive to the dynamics of the orbit equivalence relations in question.

Based on a past joint work with M. Lupini we will introduce a notion of dimension for Polish G-spaces. This dimension is always 0 whenever the group G admits a complete and left invariant metric, but in principle, it can take any value n within 0,1,….∞ For each such n we will produce a free action of S∞ which is generically n-dimensional and we will deduce that the associated orbit equivalence relations are pairwise incomparable with respect to *-reductions.

This is in joint work with A. Kruckman.

## Sakae Fuchino: Downward Lowenheim Skolem Theorems for stationary logics and the Continuum Problem

Tuesday, December 11, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Sakae Fuchino (Kobe University)

Title: Downward Lowenheim Skolem Theorems for stationary logics and the Continuum Problem

Abstract:

Downward Lowenheim Skolem Theorems of extended logics can be considered as reflection principles. In this talk we consider Downward Lowenheim Skolem Theorems of variations of stationary logic. Some of the strongest forms of reflection principles formulated in this way imply CH while some other imply that the continuum is very large. The results presented in this talk are further development of the results presented in the talk I gave last year in Wroclaw and will be a part of a joint paper with Hiroshi Sakai and Andre Ottenbreit Maschio Rodrigues.

## Diana Carolina Montoya: The equality p=t and the generalized characteristics

Dear all,

The seminar meets on Wednesday December 12th at 11:00 in the Institute
of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Diana Carolina Montoya — The equality p=t and the generalized
characteristics

Marriallis and Shelah solved in the positive the longstanding problem of
whether the two cardinal invariants $\mathfrak{p}$ (the
pseudointersection number) and $\mathfrak{t}$ (the tower number) are
equal. In this talk, I will review some essential points in their proof
in order to motivate the study of the analogous question for the
generalized characteristics $\mathfrak{p}(\kappa)$ and
$\mathfrak{t}(\kappa)$. I will present some results of Garti regarding
this generalization and finally some recent progress (joint work with
Fischer, Schilhan and, Soukup) in the direction of answering this question.

Best,
David