## Piotr Koszmider: Dimension drop phenomena and compact supports in noncommutative topology

Place: Fields Institute (Room 210)

Date: December 15, 2017 (13:30-15:00)

Speaker: Piotr Koszmider

Title: Dimension drop phenomena and compact supports in noncommutative topology

Abstract: “When X is a locally compact Hausdorff space, continuous functions on X with compact support can approximate every continuous function in C_0(X). There is a natural notion of elements with compact supports for general, not necessarily commutative, C*-algebras and a  result of Blackadar saying that  in every separable C*-algebra one can choose from such elements  an approximate unit (Blackadar calls it an almost idempotent approximate unit).

We address the issue of the existence of such an approximate unit for general, not necessarily separable C*-algebra and show that such  approximate units exist in every C*-algebra of density omega_1, that they do not exist in some C*-algebras of density min{2^k: 2^k>continuum} and that their existence in all operator algebras acting on the separable Hilbert space is  independent from ZFC. The infinitary combinatorics used involves CH, Canadian trees and Q-sets.

No knowledge of noncommutative mathematics beyond multiplication of 2×2 matrices will be assumed. These are the results of a joint research project with Tristan Bice available at arxiv.org/pdf/1707.09287.pdf

## Olena Karlova: Extension of Borel maps and Borel retracts of topological spaces

Tuesday, December 19, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Olena Karlova (Chernivtsi National University, Ukraine)

Title: Extension of Borel maps and Borel retracts of topological spaces

Abstract:

We will discuss the problem of extension of (dis)continuous maps between topological spaces. Concepts of Baire and Borel retracts of topological spaces will be introduced. Some open problems will be considered.

## Moritz Müller: On the relative strength of finitary combinatorial principles

Talk held by Moritz Müller (KGRC) at the KGRC seminar on 2017-12-14 at 4pm .

Abstract: Define a finitary combinatorial principle to be a first-order sentence which is valid in the finite but falsifiable in the infinite. We aim to compare the strength of such principles over a weak arithmetic. We distinguish “weak” and “strong” principles based on their behaviour with respect to finite structures that are only partially defined. The talk sketches a forcing proof of a theorem stating that over relativized $T^1_2$ “weak” principles do not imply “strong” ones.

## Jose Iovino: Definability in linear functional analysis

Place: Fields Institute (Room 210)

Date:  December 8 , 2017 (13:30-15:00)

Speaker: Jose Iovino

Title: Definability in linear functional analysis

Abstract: I will discuss some recent results in the theory of second-order definability and applications of these results in Banach space theory.

## 2nd Pan Pacific International Conference on Topology and Applications, November 13–17, 2017

The 2nd Pan Pacific International Conference on Topology and Applications (2nd PPICTA) had a special session in Set Theory, to which we now provide the slides.

(50min. talks)
1. David Chodounsky (Institute of Mathematics CAS)
Sacks indestructible ultrafilters and the HL property

2. David Fernández-Bretón (University of Michigan)
Algebraic Ramsey-Theoretic Statements with an Uncountable Flavour

3. Diego A. Mejía (Shizuoka University)
Ultrafilter limits in matrix iterations

4. Liuzhen Wu (Chinese Academy of Sciences)
Simultaneously reflection principle, successors of singulars and Jonsson cardinals

(30min. talks)
1. Joerg Brendle (Kobe University)
Constant prediction, strong porosity, and unsymmetric game ideals

2. Longyun Ding (Nankai University)
Equivalence relations and Borel reduction

3. Jialiang He (Sichuan University)
Tukey reduction on F_sigma ideal

4. Daisuke Ikegami (Tokyo Denki University)
On supercompactness of omega_1

5. Takayuki Kihara (Nagoya University)
Borel isomorphisms at the second level

6. Hiroshi Sakai (Kobe University)
On reflection principles of indescribable sets

7. André Ottenbreit Maschio Rodrigues (Kobe University)
Uncountable Fordor-type reflection principle and reflection of meta-Lindelöfness

## Yair Hayut: Subcompact cardinals

TAU forcing seminar

On 04/Dec/17, 9-11, Yair Hayut will be speaking on Subcompact cardinals.

Abstract. During the investigation of the existence of squares in core models, Jensen isolated the concept of subcompact cardinals. Subcompactness is weaker than supercompactness and stronger than superstrength. It is a natural candidate for the consistency strength of simultaneous failure of $\square(\kappa)$ and $\square_\kappa$. A work of Schimmerling and Zeman indicates that under some mild assumptions, subcompactness is the only possibility for failure of squares at core models of the form L[E].

In this talk I will define the relevant large cardinals notions, and talk about Zeman’s theorem for the consistency of failure $\square_{\aleph_{\omega}}$ from a measurable subcompact cardinal.

## Bill Chen: A small Dowker space

BIU seminar in Set Theory

Speaker: Bill Chen (BGU)

There will be two talks: November 27 and December 4.

First talk: A small Dowker space, part 1

Abstract. A topological space is said to be “Dowker” if it is normal but its product with the unit interval is not normal. In this lecture, we shall present a construction, due to Balogh, of a Dowker space of size continuum.

Second talk: A small Dowker space, part 2

Abstract. Last week, we presented an approach for constructing a Dowker space of size continuum, ending up with a statement of a lemma that would yield such a space. In this talk, we shall prove this lemma.

## Jonathan Verner: Coloring metric spaces

Dear all,

The seminar meets on Wednesday December 6th at 11:00 in the Institute of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program:
Jonathan Verner — Coloring metric spaces: A report from the oracle

“I will present a short and elegant argument, due to S. Shelah, showing how one can construct a metric space $X$ of size $2^\omega$ such that for any countable metric space  $K$ and any countable coloring of $X$ there is a monochromatic scaled copy of $K$ in $X$”.

## David J. Fernández Bretón: More Ramsey-theoretic statements: uncountably many colours, finite monochromatic sets

Thursday, December 7, 2017, from 4 to 5:30pm
East Hall, room 3096

Speaker: David J. Fernández Bretón (University of Michigan)

Title: More Ramsey-theoretic statements: uncountably many colours, finite monochromatic sets

Abstract:

Hindman’s theorem states that for every colouring of an infinite abelian group with finitely many colours, there will be an infinite set whose finite sums are monochromatic. Increasing the number of colours to infinitely many makes the theorem fail, as does keeping the number of colours finite but requiring an uncountable monochromatic finite-sum set. Recently Komjáth discovered, however, that by increasing the number of colours to be infinite (even uncountable), and at the same time decreasing the size of our desired monochromatic set to a finite number, it is possible to obtain some positive Ramsey-theoretic results. In this talk I will discuss some of these results, as well as some improvements and generalizations of them that I found over the summer, jointly with my REU student Sung-Hyup Lee.

## Jan Pachl: Topological centres for group actions

Place: Fields Institute (Room 210)

Date: December 1, 2017 (13:30-15:00)

Speaker: Jan Pachl

Title: Topological centres for group actions

Abstract: Based on joint work with Matthias Neufang and Juris Steprans. By a variant of Foreman’s 1994 construction, every tower ultrafilter on $\omega$ is the unique invariant mean for an amenable subgroup of $S_\infty$, the infinite symmetric group. From this we prove that in any model of ZFC with tower ultrafilters there is an element of $\ell_1(S_\infty)^{\ast\ast} \setminus \ell_1(S_\infty)$ whose action on $\ell_1(\omega)^{\ast\ast}$ is w* continuous. On the other hand, in ZFC there are always such elements whose action is not w* continuous.