Joanna Jureczko: Some remarks on Kuratowski partitions, new results

Tuesday, May 9, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Joanna Jureczko (Cardinal Stefan Wyszynski University in Warsaw)

Title: Some remarks on Kuratowski partitions, new results

Abstract:

K. Kuratowski in 1935 posed the problem whether a function $f \colon X \to Y$ from a completely metrizable space $X$ to a metrizable space $Y$ is continuous apart from a meager set.
This question is equivalent to the question about the existence of so called a Kuratowski partition, i. e. a partition $\mathcal{F}$ of a space $X$ into meager sets such that $\bigcup \mathcal{F}’$ for any $\mathcal{F}’ \subset \mathcal{F}$.
With any Kuratowski partition we may associate a $K$-ideal, i.e. an ideal of the form
$$I_{\mathcal{F}} = \{A \subset \kappa \colon \bigcup_{\alpha \in A}F_\alpha \textrm{ is meager }, F_\alpha \in \mathcal{F}\}.$$
It would seem that the information about $I_{\mathcal{F}}$ would give us full information about the ideal and the world in which it lives.
My talk is going to show that it is big simplification and localization technique from a Kuratowski partition cannot be omitted but the proof can be much simplier.
During the talk I will show among others a new proof of non-existence of a Kuratowski partition in Ellentuck topology and a new combinatorial proof of Frankiewicz – Kunen Theorem (1987) on the existence of measurable cardinals.

Chris Lambie-Hanson: Constructions from square and diamond, with an application to super-Souslin trees

Monday, May 8, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Chris Lambie-Hanson (Bar-Ilan)

Title: Constructions from square and diamond, with an application to super-Souslin trees

Abstract. In 1982, Shelah and Stanley proved that, if $\kappa$ is a regular, infinite cardinal, $2^\kappa = \kappa^+$, and there is a $(\kappa^+, 1)$-morass, then there is a $\kappa^{++}$-super-Souslin tree, which is a type of normal $\kappa^{++}$-tree that necessarily has a $\kappa^{++}$-Souslin subtree and continues to do so in any outer model in which $\kappa^{++}$ is preserved and no new subsets of $\kappa$ are present. This result establishes a lower bound of an inaccessible cardinal for the consistency strength of the conjunction of $2^\kappa = \kappa^+$ and Souslin’s Hypothesis at $\kappa^{++}$. In this talk, we will present a method for constructing objects of size $\lambda^+$ from $\square_\lambda + \diamondsuit_\lambda$, where $\lambda$ is a regular, uncountable cardinal. As an application, we will use $\square_{\kappa^+} + \diamondsuit_{\kappa^+}$ to construct a $\kappa^{++}$-super-Souslin tree. For uncountable $\kappa$, this increases Shelah and Stanley’s lower bound from an inaccessible cardinal to a Mahlo cardinal. This is joint work with Assaf Rinot.

Sabrina Ouazzani: A brief story of gaps in the infinite time Turing machines

Tuesday, May 9, 2017, 15.00
Howard House 4th Floor Seminar Room

Speaker: Dr Sabrina Ouazzani (Laboratoire d’Algorithmique, Complexité et Logique, Paris-Est Creteil University)

Title: A brief story of gaps in the infinite time Turing machines

Abstract:

I will present a part of my PhD thesis entitled “From algorithmics
to logics through infinite time computation” in which I have studied
the infinite time Turing machines model of computation from a computer
scientist’s point of view. In particular I focused on the structure of
gaps in the clockable ordinals, that is to say, ordinal times at which
no infinite time program halts.

So in this talk I will present infinite time Turing machines (ITTM),
from the original definition of the model to some new infinite time
algorithms. These algorithmic techniques will allow to highlight some
properties of the ITTM-computable ordinals and we will see that they
bring some information about the structure of gaps.

Fulgencio Lopez: A result of Dzamonja with construction schemes

Place: Fields Institute (Room 210)

Date: May 5, 2017 (14:20-15:00)

Speaker: Fulgencio Lopez, University of Toronto

Title: A result of Dzamonja with construction schemes

Abstract: We present a proof with construction schemes of the following result of Mirna Dzamonja: Forcing a Cohen real adds a Banach space of density $\omega_1$ which does not isomorphically embed in any ground model Banach space of density $\omega_1$.

Sergio Garcia-Balan: The star-Menger property on $\Psi$-spaces

Place: Fields Institute (Room 210)

Date: May 5, 2017 (13:30-14: 10)

Speaker: Sergio Garcia- Balan, York University

Title: The star-Menger property on $\Psi$-spaces

Abstract: Bonanzinga and Matveev showed that the space $\Psi(\mathcal{A})$ is strongly star-Menger if and only if $|\mathcal{A}|<\mathfrak{d}$. We will discuss what can be said about the star-Menger property on $\Psi(\mathcal{A})$.

Sam Roberts: The iterative conception of properties and comprehension

KGRC Friday Seminar – 2017‑05‑05 at 12pm

Speaker: Sam Roberts (University of Oslo, Norway)

Abstract: Mathematicians appeal to proper classes: that is, collections too large to form sets. But what are classes if not sets? One response is that classes are properties. Properties are sharply distinguished from sets: they are intensional whereas sets are extensional. Fine and Linnebo have proposed theories on which properties are “built up” in a series of stages. Unfortunately, neither of these theories imply that there are very many properties. In this talk, I will propose an improvement of these theories. More precisely, by ensuring that the stages extend far enough, I will show that the they can be modified to interpret Morse-Kelly class theory, which implies the existence of a plethora of classes.

Víctor Torres-Pérez: Rado’s Conjecture, an alternative to forcing axioms?

KGRC Research Seminar – 2017‑05‑04 at 4pm

Speaker: Víctor Torres-Pérez (TU Wien)

Abstract: Rado’s Conjecture (RC) in the formulation of Todorcevic is the statement that every tree $T$ that is not decomposable into countably many antichains contains a subtree of cardinality $\aleph_1$ with the same property.
Todorcevic has shown the consistency of this statement relative to
the consistency of the existence of a strongly compact cardinal.

Todorcevic also showed that RC implies the Singular Cardinal Hypothesis,
a strong form of Chang’s Conjecture, the continuum is at most $\aleph_2$,
the negation of $\Box(\theta)$ for every regular $\theta\geq\omega_2$,
etc. These implications are very similar to the ones obtained from traditional
forcing axioms such as MM or PFA. However, RC is incompatible even with
$MA(\aleph_1)$.

In this talk we will take the opportunity to give an overview of our
results with different coauthors obtained in the last few years together
with recent ones, involving RC, certain weak square principles and
instances of tree properties. These new implications seem to continue
suggesting that RC is a good alternative to forcing axioms. We will discuss
to which extent this may hold true and where we can find some limitations.
We will end the talk with some open problems and possible new directions.

Asaf Karagila: The Bristol model: A few steps into a Cohen real

HUJI Logic Seminar

The next meeting will be at 08/05 (in one week), Ross 63, 12:00-14:00.

Title: The Bristol model: A few steps into a Cohen real
Speaker:
Asaf Karagila
Abstract: We will take a close look at the first few steps of the construction of the Bristol model, which is a model intermediate to L[c], for a Cohen real c, satisfying V\neq L(x) for all x.

Ashutosh Kumar: On possible restrictions of null and meager ideal

BIU seminar in Set Theory

On 04/05/2017, 10-12, Building 604, Room 103

Speaker:  Ashutosh Kumar

Title: On possible restrictions of null and meager ideal

Abstract. Fremlin asked if the null ideal restricted to a non null set of reals could be isomorphic to the non stationary ideal on omega_1. Eskew asked if the null and the meager ideal could both be somewhere countably saturated. We’ll show that the answer to both questions is yes. Joint work with Shelah.

Alessandro Vignati: Set theoretical dichotomies in the theory of continuous quotients

Place: Fields Institute (Room 210)

Date: April 28, 2017 (13:30-15:00)

Speaker: Alessandro Vignati, York University

Title: Set theoretical dichotomies in the theory of continuous quotients

Abstract: We state and (depending on time) prove some dichotomies of set theoretical nature arising in the theory of continuous quotients. In particular we show that the assumption of CH on one side, and of Forcing Axioms on the other, affects the nature of possible embeddings of certain corona algebras, as well as the behavior of their automorphisms group. This is partly joint work with P. McKenney.