## Viera Sottova: Modification of N via ideals on omega

Dear all,

The seminar meets on Wednesday December 5th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Viera Sottova — Modification of N via ideals on omega

We consider a combinatorial characterization of the null ideal N which
we modify using ideals on omega. Resulting ideal N_J is also a sigma
ideal and additionally it is subideal of N. We focus on common cardinal
invariants of this ideal and their relation to the original ones.
Joint work with D. A. Mejia.

Best,
David

## Miloš Kurilić: Vaught’s Conjecture for Monomorphic Theories

Place: Fields Institute (Room 210)
Date: November 30, 2018 (1:00-2:00)
Speaker: Miloš Kurilić
Title: Vaught’s Conjecture for Monomorphic Theories
Abstract: A complete first order theory of a relational signature is called monomorphic iff
all its models are monomorphic (i.e.\ have all the $n$-element substructures
isomorphic, for each positive integer $n$).
We show that a complete theory ${\mathcal T}$ having infinite models is monomorphic
iff it has a countable monomorphic model
and confirm the Vaught conjecture for monomorphic theories.
More precisely, we prove that if ${\mathcal T}$ is a complete monomorphic theory
having infinite models, then the number of its non-isomorphic countable models,
$I({\mathcal T} ,\omega)$, is either equal to $1$ or to ${\mathfrak c}$. In addition,
$I({\mathcal T},\omega )= 1$ iff some countable model of ${\mathcal T}$ is simply
definable by an $\omega$-categorical linear order on its domain.

## Jordi Lopez-Abad: Approximate Ramsey property of normed spaces

Place: Fields Institute (Room 210)
Date: November 30, 2018 (2:00-3:00)
Title: Approximate Ramsey property of normed spaces
Abstract: We will present and discuss the known examples approximate Ramsey
properties of normed spaces. In some cases, this can be seen as a multidimensional Borsuk-Ulam Theorem, and as a reformulation of the extreme amenability of the automorphism group of an appropriate space (a Fraisse limit). Concerning the proofs, we will sketch them for the 3 main families of spaces: $\{\ell_p^n\}_$ for $p=1,2,\infty$.

## David Chodounsky: Generic ultrafilters

Place: Fields Institute (Room 210)
Date: November 30, 2018 (12:00-1:00)
Speaker: David Chodounsky
Title: Generic ultrafilters
Abstract: S. Todorcevic proved that given the presence of large
cardinals, selective ultrafilters are precisely ultrafilters
P(omega)/FIN-generic over L(R). We generalize this result. We provide a
characterization of P(omega)/I-generic ultrafilters over L(R) for an
arbitrary F_sigma ideal I.
This is a joint work with Jindra Zapletal.

## Daniel Soukup: New aspects of ladder system uniformization

Talk held by Daniel Soukup (KGRC) at the KGRC seminar on 2018-11-29.

Abstract: Given a tree $T$ of height $\omega_1$, we say that a ladder system colouring $(f_\alpha)_{\alpha\in \lim\omega_1}$ has a $T$-uniformization if there is a function $\varphi$ defined on a subtree $S$ of $T$ so that for any $s\in S_\alpha$ of limit height and almost all $\xi\in dom (f_\alpha)$, $\varphi(s\upharpoonright \xi)=f_\alpha(\xi)$.

In sharp contrast to the classical theory of uniformizations on $\omega_1$, J. Moore proved that CH is consistent with the statement that any ladder system colouring has a $T$-uniformization (for any Aronszajn tree $T$). Our goal is to present a fine analysis of ladder system uniformization on trees pointing out the analogies and differences between the classical and this new theory. We show that if $S$ is a Suslin tree then CH implies that there is a ladder system colouring without $S$-uniformization, but $MA(S)$ implies that any ladder system colouring has even an $\omega_1$-uniformization.

Furthermore, it is consistent that for any Aronszajn tree $T$ and ladder system $\mathbf C$ there is a colouring of $\mathbf C$ without a $T$-uniformization; however, and quite surprisingly, $\diamondsuit^+$ implies that for any ladder system $\mathbf C$ there is an Aronszajn tree $T$ so that any monochromatic colouring of $\mathbf C$ has a $T$-uniformization. We also prove positive uniformization results in ZFC for some well-studied trees of size continuum. (cf. https://arxiv.org/abs/1806.03867 and https://arxiv.org/abs/1803.03583)

## James Cummings: Shelah’s singular compactness theorem

Mathematical logic seminar – Nov 27 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         James Cummings
Department of Mathematical Sciences
CMU

Title:     Shelah’s singular compactness theorem

Abstract:

Shelah’s singular compactness theorem is a general result showing that a singular cardinal λ has properties reminiscent of those enjoyed by large cardinals: for example

If G is an abelian group of size λ and every subgroup of G with size less than λ is free, then G is free.

If X is a family of size λ of countable sets, and every subfamily of size less than λ has a transversal, then X has a transversal.

I will prove a version of the singular compactness theorem, and discuss some complementary consistency results for λ regular.

## Piotr Koszmider: The Grothendieck property for Banach spaces of continuous functions

Seminar: Working group in applications of set theory, IMPAN

Thursday, 29.11.2018, 10:15, room 105, IMPAN

Speaker: Piotr Koszmider (IM PAN)

Title: “The Grothendieck property for Banach spaces of continuous functions”

Abstact: “In the first talk of the series devoted to classical phenomena in Banach spaces of the form C(K) we will see how weakly compact sets in the dual space to C(K) generalize finite subsets of K. The concrete goal will be to motivate the Grothendieck property (weak and weak* convergence of sequences coincide in the dual) for C(K)s as a generalization of K having no nontrivial convergent sequence and to prove that l∞≡C(βN) has the Grothendieck property. This will require the proof of the Grothendieck-Dieudonne characterization of weakly compact sets in the spaces of measures. All the results and proofs presented during the talk are in classical texbooks, but we will try to represent combinatorial and topological bias, leading in the following talks to more set-theoreic issues. The purpose of this series of talks is to introduce particpants with the set-theoretic topological background to some topics related to Banach spaces of the form C(K). The area is quite sensitive to infinitary combinatorics, e.g., Talagrand: CH implies that there is infinite K such that C(K) is Grothendieck but does not have l∞ as its quotient; Haydon, Levy, Odell: p=2^ω>ω_1 implies that every Grothendieck C(K) for K infinite has l∞ as its quotient “.

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

## Serhii Bardyla: A topologization of graph inverse semigroups

Tuesday, November 27, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Serhii Bardyla (Ivan Franko National University of Lviv)

Title: A topologization of graph inverse semigroups

Abstract:

We characterize graph inverse semigroups which admit only discrete locally compact semigroup topology. It will be proved that if a directed graph $E$ is strongly connected and contains a finite amount of vertices then a locally compact semitopological graph inverse semigroup $G(E)$ is either compact or discrete. We describe graph inverse semigroups which admit compact semigroup topology and construct a universal object in the class of graph inverse semigroups. Embeddings of graph inverse semigroups into compact-like topological semigroups will be investigated. Also, we discuss some open problems.

## Saharon Shelah: The spectrum of the existence of a universal model

HUJI Logic Seminar

21 November 2018

Speaker: Saharon Shelah

Title: The spectrum of the existence of a universal model

Abstract. The existence of a universal model (of a theory T in a cardinal lambda) is a natural question in model theory and set theory. We shall deal with new sufficient conditions for non-existence.

## 7th European Set Theory Conference, Vienna, July 1-5, 2019

The 7th European Set Theory Conference of the European Set Theory Society will take place in Vienna,  July 1 – 5, 2019.

The list of invited speakers include:

• Jörg Brendle (Kobe University)
• Mirna Džamonja (University of East Anglia)
• Moti Gitik (Tel Aviv University)
• Alexander Kechris (Caltech)
• Piotr Koszmider (Polish of Academy of Sciences)
• Maryanthe Malliaris (University of Chicago)
• Justin Moore (Cornell University)
• Dima Sinapova (University of Illinois at Chicago)
• Slawomir Solecki (Cornell University)
• Boaz Tsaban (Bar-Ilan University)
• Anush Tserunyan (University of Illinois at Urbana-Champaign)
• Matteo Viale (University of Torino)