# Category Archives: Seminars

## Juris Steprans: Groups with sub exponential growth and actions on the integers with unique invariant means in the Cohen model

Place: BA6183, Bahen Centre

Date: August 11, 2017 (13:30-15:00)

Speaker: Juris Steprans

Title: Groups with sub exponential growth and actions on the integers with unique invariant means in the Cohen model

Abstract: It will be shown that the objects mentioned in the title do not exist.

## Jose Iovino: Metastability and model theory

Place: BA6180, Bahen Centre

Date: August 4, 2017 (13:30-15:00)

Speaker: Jose Iovino

Title: Metastability and model theory

Abstract: The concept of metastability was introduced by Tao; it played crucial role in his ergodic convergence theorem (2008) and in Walsh’s generalization (Ann. of Math., 2012). I will discuss the fact that metastability is intimately connected with notions from model theory of metric structures. This is joint work with Xavier Caicedo and Eduardo Duenez.

## Christopher Eagle: Baire Category and the Omitting Types Theorem

Place: BA6183

Date: July 21 , 2017 (13:30-15:00)

Speaker: Christopher Eagle

Title: Baire Category and the Omitting Types Theorem

Abstract: It is well-known that the Omitting Types Theorem from model theory can be proved by topological means, and the central ingredient of that proof is the Baire Category Theorem.  The goal of this talk is to consider the extent to which the Omitting Types Theorem is equivalent to the Baire Category Theorem.  To do so, we will describe a topological framework (based on work of Robin Knight) that generalizes the classical type spaces from model theory.   Many classical logics (including first-order, infinitary, and continuous logics) fit into this general setting, and conversely we will show that each instance of the general framework yields a model-theoretic logic.  We then distinguish several version of the Omitting Types Theorem these logics may have, based on Baire Category properties of the underlying topological spaces.  All of these properties are equivalent for first-order logic, but are distinct in the general setting.  This is joint work with Frank Tall.

## Dorottya Sziráki: Open colorings on generalized Baire spaces

Thursday, July 20, 2017,  10:30–12.00

Main Lecture Hall , Alfréd Rényi Institute of Mathematics

Abstract: We study the uncountable version of a natural variant of the Open Coloring Axiom. More concretely, suppose that $\kappa$ is an uncountable cardinal such that $\kappa^{<\kappa}=\kappa$ and X is a subset of the generalized Baire space $\kappa^\kappa$ (the space of functions from $\kappa$ to $\kappa$ equipped with the bounded topology). Let OCA*(X) denote the following statement: for every partition of $[X]^2$ as the union of an open set R and a closed set S, either X is a union of $\kappa$ many S-homogeneous sets, or there exists a $\kappa$-perfect R-homogeneous set. We show that after Lévy-collapsing an inaccessible $\lambda>\kappa$ to $\kappa^+$, OCA*(X) holds for all $\kappa$-analytic subsets X of $\kappa^\kappa$. Furthermore, the Silver dichotomy for ${\Sigma}^0_2(\kappa)$ equivalence relations on $\kappa$-analytic subsets also holds in this model. Thus, both of the above statements are equiconsistent with the existence of an inaccessible $\lambda>\kappa$. We also examine games related to the above partition properties.

## Yann Pequignot: Sigma^1_2 sets and countable Borel chromatic numbers

Friday, July 21th, 2017, 10.00-12.00

Aula S, Palazzo Campana, Università di Torino

Speaker: Yann Pequignot (University of California, Los Angeles)

Title: Sigma^1_2 sets and countable Borel chromatic numbers

Abstract:

Analytic sets enjoy a classical representation theorem based on wellfounded relations. I will explain a similar representation theorem for Sigma^1_2 sets due to Marcone. This can be used to answer negatively the primary outstanding question from (Kechris, Solecki and Todorcevic; 1999): the shift graph is not minimal among the graphs of Borel functions which have infinite Borel chromatic number.

## Dániel T. Soukup: Uncountable strongly surjective linear orders

Thursday, July 13, 2017,  10:30

Seminar Room, Alfréd Rényi Institute of Mathematics

Abstract: A linear order $L$ is strongly surjective if $L$ can be mapped onto any of its suborders in an order preserving way. We review various results on the existence and non-existence of uncountable strongly surjective linear orders answering questions of Camerlo, Carroy and Marcone. In particular, $\diamondsuit^+$ implies the existence of a lexicographically ordered Suslin-tree which is strongly surjective and minimal; every strongly surjective linear order must be an Aronszajn type under $2^{\aleph_0} <2^{\aleph_1}$ or in the Cohen and other canonical models (where $2^{\aleph_0}=2^{\aleph_1}$); finally, we show that it is consistent with CH that there are no uncountable strongly surjective linear orders at all. Further details and open problems can be found in https://arxiv.org/abs/1706.10171

## Jindra Zapletal: Quotient forcings defined from group actions

Dear all,

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

Program:
Jindra Zapletal will talk about quotient forcings defined from group
actions.

Best,
David

## Saeed Ghasemi: Scattered C*-algebras

Place: Bahen Centre (Room 6183)

Date: July 7th, 2017 (13:30-15:00)

Speaker: Saeed Ghasemi, Polish Academy of Sciences

Title: Scattered C*-algebras

Abstract: By the Gelfand duality, the theory of C*-algebras can be
regarded as “non-commutative topology”. In a joint work with Piotr
Koszmider at IMPAN, we investigated the non-commutative analogues of
the scattered spaces, parallel to the classical research in
set-theoretic topology. The so called scattered C*-algebras, despite
being around in the literature, have not been subject to the tools
from set-theoretic topology. The techniques and constructions of
compact, Hausdorff scattered spaces, or equivalently (by the Stone
duality) superatomic Boolean algebras, have already led to many
fundamental results in the theory of Banach spaces of the form C(K),
or more generally Asplund spaces. In fact scattered C*-algebras were
introduced as C*-algebras which are Asplund as Banach spaces. I will
introduce the notion of the Cantor-Bendixson derivatives for these
C*-algebras, and present some of the basic properties of such
algebras. I will also show how it can be used to construct C*-algebras
with exotic properties, which are non-commutative versions of
well-known scattered spaces. In particular, the constructions of
non-commutative Psi-spaces and thin tall spaces lead to new
discoveries about the preservation of the “stability” for
non-separable C*-algebras.

## Witold Marciszewski: On factorization properties of function spaces

KGRC Research seminar on 2017-06-29 at 4pm

Speaker: Witold Marciszewski (University of Warsaw, Poland)

Abstract: For a Tychonoff space $X$, by $C_p(X)$ we denote the space of all continuous real-valued functions on $X$, equipped with the topology of pointwise convergence. One of the important questions (due to A.V. Arhangel’skii), stimulating the theory of $C_p$-spaces for almost 30 years and leading to interesting results in this theory, is the problem whether the space $C_p(X)$ is (linearly, uniformly) homeomorphic to its own square $C_p(X)\times C_p(X)$, provided $X$ is an infinite compact or metrizable space.

In my talk I will present some recent developments concerning these type of questions. In particular, I will show a metrizable counterexample to this problem for homeomorphisms. I will also show that, for every infinite zero-dimensional Polish space $X$, spaces $C_p(X)$ and $C_p(X)\times C_p(X)$ are uniformly homeomorphic.

This is a joint research with Rafal Gorak and Mikolaj Krupski.

## Peter Nyikos: Cardinality restrictions on some kinds of locally compact spaces

KGRC Research seminar on 2017-06-29 at 2pm

Speaker: Peter Nyikos (University of South Carolina, Columbia, USA)

Abstract: In what follows, “space” means “Hausdorff ($T_2$) topological space.” Some of the theorems and problems to be discussed include:

Theorem 1. It is ZFC-independent whether every locally compact, $\omega_1$-compact space of
cardinality $\aleph_1$ is the union of countably many countably compact spaces.

[$\omega_1$-compact means that every closed discrete subspace is countable. This is obviously implied by being the union of countably many countably compact spaces, but the converse is not true.]

Problem 1. Is it consistent that every locally compact, $\omega_1$-compact space of cardinality $\aleph_2$ is the union of countably many countably compact spaces?

Problem 2. Is ZFC enough to imply that there is a normal, locally countable, countably compact space of cardinality greater than $\aleph_1$?

Problem 3. Is it consistent that there exists a normal, locally countable, countably compact space of cardinality greater than $\aleph_2$?

The spaces involved in Problem 2 and Problem 3 are automatically locally compact, because regularity is already enough to give every point a countable countably compact (hence compact) neighborhood.

Problem 4 [Problem 5]. Is there an upper bound on the cardinalities of regular [resp. normal],
locally countable, countably compact spaces?

Theorem 2. The axiom $\square_{\aleph_1}$ implies that there is a normal, locally countable, countably compact space of cardinality $\aleph_2$.

The statement in Theorem 1 was shown consistent by Lyubomyr Zdomskyy, assuming $\mathfrak p > \aleph_1$ plus P-Ideal Dichotomy (PID). Counterexamples have long been known to exist under
$\mathfrak b = \aleph_1$, under $\clubsuit$, and under the existence of a Souslin tree.

Theorem 2 may be the first application of $\square_{\aleph_1}$ to construct a topological space whose existence in ZFC is unknown.