# Category Archives: Seminars

## 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.

## Shimon Garti: The club principle and the splitting number

HUJI Logic Seminar
The next meeting of the Logic Seminar will take place Wednesday, 28/06, 16:00 – 18:00, Ross 70.  This is the last meeting of the Logic Seminar for this semester.
Speaker: Shimon Garti
Title: Tiltan
Abstract: We shall try to prove some surprising (and hopefully, correct) theorems about the relationship between the club principle (Hebrew: tiltan) and the splitting number, with respect to the classical s at omega and the generalized s at supercompact cardinals.

## Bill Chen: Variations of the stick principle

BGU seminar in Logic, Set Theory and Topology

Time: Tuesday, June 27th, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: Bill Chen (BGU)

Title: Variations of the stick principle

Abstract:
The stick principle asserts that there is a family of infinite subsets of $\omega_1$ of size $\aleph_1$ so that any uncountable subset of $\omega_1$ has some member of the family as a subset. We will give a forcing construction to separate versions of the stick principle which put a bound on the order-type of the subsets in the family. Time permitting, we will say a little about the relation of the stick principle with the existence of Suslin trees.

## David Schrittesser: News on mad families

KGRC Research Seminar  – 2017-06-22 at 4pm.

Speaker: David Schrittesser (University of Copenhagen, Denmark)

Abstract: This talk is about two results on mad families (dating from this year): Firstly, in joint work with Karen Bakke Haga and Asger Törnquist and, we link madness of certain definable sets to forcing and use this to show that under the Axiom of Projective Determinacy there are no projective mad families. Moreover, the results generalize: we may replace “being almost disjoint” by “being $J$-disjoint”, for certain ideals $J$ on the natural numbers including, e.g., Fin $\times$ Fin. The other result is an improvement of Horowitz and Shelah’s construction of a Borel maximal eventually different family of functions. We obtain a closed such family, and the result even generalizes to certain compact spaces.

## Diana Carolina Montoya Amaya: Some cardinal invariants of the generalized Baire spaces

The successful PhD defense of Diana Carolina Montoya Amaya took place Wednesday, June 14 at the KGRC. Congratulations!

Abstract: The central topic of this talk is the well-known Cardinal invariants of the continuum and it is divided in two parts: In the first one we focus on the generalization of some of these cardinals to the generalized Baire spaces $\kappa^\kappa$, when $\kappa$ is a regular uncountable cardinal. First, we present a generalization of some of the cardinals in Cichon’s diagram to this context and some of the provable ZFC relationships between them. Further, we study their values in some generic extensions corresponding to $<\!\!\kappa$-support and $\kappa$-support iterations of generalized classical forcing notions. We point out the similarities and differences with the classical case and explain the limitations of the classical methods when aiming for such generalizations. Second, we study a specific model where the ultrafilter number at $\kappa$ is small, $2^\kappa$ is large and in which a larger family of cardinal invariants can be decided and proven to be $<\!2^\kappa$.

The second part deals exclusively with the countable case: We present a generalization of the method of matrix iterations to find models where various constellations in Cichon’s diagram can be obtained and the value of the almost disjointness number can be decided. The method allows us also to find a generic extension where seven cardinals in Cichon’s diagram can be separated.

Board of examiners:

Professor Mirna Džamonja (University of East Anglia)
o.Univ.-Prof. Sy-David Friedman (Universität Wien)
ao.Univ.Prof. Martin Goldstern (TU Wien)

## Dima Sinapova: Iterating Prikry Forcing

Monday, June 26th, 2017, 10.30-12.00

Aula Lagrange, Palazzo Campana, Università di Torino

Speaker: Dima Sinapova (University of Illinois at Chicago)

Title: Iterating Prikry Forcing

Abstract:

We will present an abstract approach of iterating Prikry type forcing. Then we will use it to show that it is consistent to have finite simultaneous stationary reflection at $\kappa^+$ with not SCH at $\kappa$. This extends a result of Assaf Sharon. Finally we will discuss how we can bring the construction down to $\aleph_{\omega}$. This is joint work with Assaf Rinot.

## Stefan Hoffelner: NS saturated and Delta_1-definable

Monday, June 19, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Stefan Hoffelner (University of Vienna)

Title: NS saturated and Delta_1-definable

Abstract:

Questions which investigate the interplay of the saturation of the nonstationary ideal on $omega_1$, NS, and definability properties of the surrounding universe can yield surprising and deep results. Woodins theorem that in a model with a measurable cardinal where NS is saturated, CH must definably fail is the paradigmatic example. It is another remarkable theorem of H. Woodin that given $omega$-many Woodin cardinals there is a model in which NS is saturated and $omega_1$-dense, which in particular implies that NS is (boldface) $Delta_1$-definable. S.D. Friedman and L. Wu asked whether the large cardinal assumption can be lowered while keeping NS $Delta_1$-definable and saturated. In this talk I will outline a proof that this is indeed the case: given the existence of $M_1^{#}$, there is a model of ZFC in which the nonstationary ideal on $omega_1$ is saturated and $Delta_1$-definable with parameter $K_{omega_2^K}$ (note that $omega_2^K$ is of size $aleph_1$ in that model). In the course of the proof I will present a new coding technique which seems to be quite suitable to obtain definability results in the presence of iterated forcing constructions over inner models for large cardinals.

## Ari Brodsky: ​Constructing free Souslin trees from a proxy principle

BGU seminar in Logic, Set Theory and Topology

Time: Tuesday, June 13th, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: ​Ari Brodsky (BIU)

Title: ​Constructing free Souslin trees from a proxy principle

Abstract. ​More than 40 years ago, Jensen constructed a free Souslin tree of height $\omega_1$ from $\diamondsuit(\omega_1)$.
We show how to construct a free $\kappa$-Souslin tree, where $\kappa$ is an arbitrary regular uncountable cardinal.
This is joint work with Assaf Rinot.​

## Aleksander Cieślak: Cohen-stable families of subsets of integers

Tuesday, June 13, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Aleksander Cieślak (Wroclaw University of Science and Technology)

Title: Cohen-stable families of subsets of integers

Abstract:

A mad family is Cohen-stable if it remains maximal in any Cohen generic extension. Otherwise it is Cohen-stable. We will find condition necessary and sufficient for mad family to be Cohen-unstabe and investigate when such family exist.