## Damian Sobota: Josefson-Nissenzweig theorem for $C(K)$-spaces

Tuesday, March 26, 2019, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Damian Sobota (Universityof Viena)

Title: Josefson-Nissenzweig theorem for $C(K)$-spaces

Abstract:

The Josefson-Nissenzweig theorem is a powerful tool in Banach space theory. Its special version for Banach spaces of continuous functions reads as follows: for a given infinite compact space K there exists a sequence $(\mu_n)$ of normalized signed Radon measures on K such that the integrals $\mu_n(f)$ converge to $0$ for any function $f$ in $C(K)$. During my talk I will investigate when the sequence $(\mu_n)$ can be chosen in such a way that every $\mu_n$ is just a finite linear combination of Dirac point measures (in other words, $\mu_n$ has finite support). This will appear to have connections with the Grothendieck property of Banach spaces and complementability of the space $c_0$. In particular, I’ll present a very elementary proof that $c_0$ is always complemented in a space $C(K\times K)$.

## Saeed Ghasemi: AF-algebras with Cantor-set property

Dear all,

The seminar meets on Wednesday March 27th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Saeed Ghasemi — AF-algebras with Cantor-set property

A separable AF-algebra is a C*-algebra which is (isomorphic to) the
inductive limit of a direct sequence of finite dimensional C*-algebras.
We introduce a class of separable AF-algebras, called AF-algebras with
Cantor-set property, which are, in some contexts, suitable
noncommutative analogues of the Cantor set. One of the main features of
AF-algebras with Cantor-set property is that they are all Fraisse
limits of some category of finite dimensional C*-algebras and left
invertible embeddings. As a consequence of this, many properties of the
Cantor set that can be proved using the Fraisse theory, such as the
homogeneity and universality, also can also be proved for AF-algebras
with Cantor-set property. In fact, the category of all finite
dimensional C*-algebras and left invertible embeddings is a Fraisse
category and its Fraisse limit is a separable AF-algebra with Cantor-set
property which has the universality property that maps surjectively onto
any separable AF-algebra.*
This is a joint work with Wieslaw Kubis.

*- All of these results can be restated and proved in the language of
partially ordered abelian groups without mentioning any C*-algebras.

Best,
David

## Ashutosh Kumar: Order dimension of Turing degrees

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 27 March 2019, 17:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: Ashutosh Kumar

Title: Order dimension of Turing degrees

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

The order dimension of a partially ordered set (P,<)
is the smallest size of a family F of linear orders,
each extending <, such that the intersection of F is
the given ordering <.

Higuchi, Lempp, Raghavan and Stephan asked if the order dimension
of Turing degrees could be decided in ZFC. We show that the answer is no.

This is joint work with Dilip Raghavan.

## Spencer Unger: Stationary reflection and the singular cardinals hypothesis

HUJI Logic Seminar

Tomorrow, Spencer Unger will speak in our logic seminar about Stationary reflection and the singular cardinals hypothesis. As usual we meet at 11am in Ross 63. Looking forward to seeing you there,

Title: Stationary reflection and the singular cardinals hypothesis.

Abstact. We examine reflection of stationary sets at successors of singular cardinals and its connection with cardinal arithmetic. For instance it has been open whether the failure of the singular cardinal hypothesis at a singular cardinal mu of uncountable cofinality implies the existence of a nonreflecting stationary subset of mu^+. In recent joint work with Omer Ben-Neria and Yair Hayut we have shown that the answer is no modulo the consistency of some large cardinals. In this talk, we survey some instances of methods used in the proof. In particular, we show how to construct Prikry sequences over iterated ultrapowers and exploit them for combinatorial proofs.

## Ralf Schindler: Variants of the extender algebra and their applications

BIU Infinite Combinatorics Seminar

Mon, 25/03/2019 – 13:00

Speaker: Ralf Schindler (Münster)

Title: Variants of the extender algebra and their applications

Abstract. In the 1970’ies, Bukowský identified a beautiful and handy criterion for when V is a forcing extension of a given inner model, which proved very useful recently in set theoretical geology. In the 1990’ies, Woodin isolated his extender algebra which makes use of a large cardinal, a Woodin cardinal. It turns out that Bukowský’s theorem and Woodin’s extender algebra may be presented in a uniform fashion – one proof and one forcing gives both results. We will present the proof and then discuss its application in inner model theoretic geology. This is joint work with Grigor Sargsyan and Farmer Schlutzenberg.

## Antonio Aviles: Twisted sums of spaces of continuous functions

Place: Fields Institute (Room 210)
Date: March 22, 2019 (13:30-15:00)
Speaker: Antonio Aviles
Title: Twisted sums of spaces of continuous functions
Abstract: Given two Banach spaces $Z$ and $X$, can we find a Banach space $Y$ that contains $X$ as an uncomplemented subspace and $Y/X = Z$? We will mention two instances of this problem connected to set theoretic questions. When $X = c_0$ and $Z=C(K)$ is a space of continuous functions on a nonmetric compactum, the answer may be negative under $MA_{\omega_1}$ but it is always positive under CH (joint work with W. Marciszewski and G. Plebanek). When $X = \ell_\infty/c_0$ and $Z=c_0(\mathfrak{c})$, the answer is positive provided splitting chains exist in $\mathcal{P}(\omega)/fin$ (joint work with P. Borodulin-Nadzieja, F. Cabello, D. Chodounsk\'{y} and O. Guzm\'{a}n)

## Yair Hayut: Strong compactness and the filter extension property

Talk held by Yair Hayut (KGRC) at the KGRC seminar on 2019-03-21.

Abstract

The notion of strongly compact cardinal is one of the earliest large cardinal axioms, yet it is still poorly understood.

I will review some classical and semi-classical connections between partial strong compactness, the strong tree property and the filter extension property, getting a level-by-level equivalence and an elementary embedding characterization.

This analysis is especially interesting for the property “every κκ-complete filter on $\kappa$ can be extended to a $\kappa$ -complete ultrafilter” (where $\kappa$  is uncountable). This property was isolated by Mitchell and was named “$\kappa$ -compactness” by Gitik. In his recent paper, Gitik showed that some definable versions of it have a relatively low consistency strength, yet others provide an inner models with a Woodin cardinal. Applying the equivalence above to this case, I will improve the previously known lower bound for $\kappa$ -compactness.

Then, I’ll move to a more speculative area, and conjecture that $\kappa$ -compactness is equiconsistent with a certain large cardinal axiom in the realm of subcompact cardinals. I will give a few arguments in favour of this conjecture.

## European Set Theory Conference 2019 – registration reminder

Let us remind you that registration is open (still with the Early Fee) for the  European Set Theory Conference. We welcome contributed talks and encourage you to take advantage of the support from ASL.

Details: European Set Theory Conference

Please note that the registration for the Advanced Class 2019 (Young Set Theory Workshop) is separate.

## Advanced Class 2019 (Young Set Theory Workshop) – registration reminder

Let us remind you that registration is open (still with the Early Fee) for the Advanced Class 2019 (Young Set Theory Workshop). We welcome poster submissions and encourage you to take advantage of the support from ASL.

Please note that the registration for the European Set Theory Conference is separate.

## Jing Zhang: Poset dimension and singular cardinals

Mathematical logic seminar – Mar 19 2019
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Jing Zhang
CMU

Title:     Poset dimension and singular cardinals

Abstract:

The dimension of a poset (P, ≤P) is defined as the least cardinal λ such that there exists a λ-sized collection of linear extensions of P realizing P, that is to say a ≤P b if and only a ≤ b in any linear extension in the collection. We will focus on the poset Pα(κ), that is the poset of subsets of κ of size less than α partially ordered by inclusion, and determine completely the dimension of such posets under GCH. Then we will mention a few consistency results when GCH fails. In particular, we point out the connection between the dimension of the poset Pα (2κ) and the density of 2κ under the <α-box product topology, and show it is consistent that they are different.