Asaf Karagila: Models of Bristol

Tuesday, October 2, 2018, 15.00
Howard House 4th Floor Seminar Room, University of Bristol

Speaker: Asaf Karagila (University of East Anglia)

Title: Models of Bristol

Abstract:

The Bristol model lies in between L and a Cohen extension given by a single real. It is an unusual model of set theory without the axiom of choice in that it is not of the form L(x) for any set x. We will outline the construction, and discuss some of its consequences.

The full construction can be found in the following paper “The Bristol model: an abyss called a Cohen real”. https://arxiv.org/abs/1704.06939

Ashutosh Kumar: Saturated null and meager ideal

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 19 September 2018, 17:00 hrs

Room: S17#04-05, Department of Mathematics, NUS

Speaker: Ashutosh Kumar

Title: Saturated null and meager ideal

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

Abstract: We will sketch some ideas behind the proof of the following:
The null and the meager ideal could both be somewhere countably
saturated.

Michał Korch: The class of perfectly null sets and its transitive version

Dear all,

The seminar meets on Wednesday September 19th at 11:00 in the Institute
of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Michał Korch — The class of perfectly null sets and its
transitive version
(joint work with Tomasz Weiss)

The ideals of universally null sets (UN, sets which are null with
respect to any Borel diffused measure) and perfectly meager
sets (PM, sets which are meager when restricted to any perfect set) are
best known among the classes of special subsets of the real
line. Those two ideals were long considered to be somehow dual, though
some differences were also known. P. Zakrzewski proved that two other
earlier defined classes of sets smaller then PM coincide and are dual to
UN. Therefore he proposed to call this class universally meager sets.
The PM class was left without a counterpart, and we try to define a
class of sets which may play the role of a dual class to PM and we also
consider its transitive version. I am going to present some properties
of these classes and give few important problems which are still open
along with some new attempts and simplifications to get an answer.

Best,
David

Damjan Kalajdzievski: Bounding, splitting, almost disjointness and covering of the meager ideal

Place: Fields Institute (Room 210)
Date: , 2018 (13:30-15:00)
Speaker: Damjan Kalajdzievski
Title: Bounding, splitting, almost disjointness and covering of the meager ideal
Abstract:This talk is on joint work with Osvaldo Guzman. This talk will go over forcing $\omega_1=\mathfrak{b}=cov(\mathcal{M})<\mathfrak{s}=\omega_2$ with a countable support iteration of proper forcings. In doing so we will introduce the forcings $\mathbb{PT}(\mathcal{F})$, which are Miller trees that satisfy a restriction on splitting nodes relative to the filter $\mathcal(F)$, and discuss their properties when $\mathcal{F}$ is Canjar. The result is achieved by iterating the forcing $\mathbb{F}_\sigma*\mathbb{PT}(\mathcal{F})$, where $\mathbb{F}_\sigma$ is the forcing of $F_\sigma$ filters on $\omega$ ordered by reverse inclusion.

Jing Zhang: How to get the brightest rainbows from the darkest colorings

Mathematical logic seminar – Sep 18 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Jing Zhang
Department of Mathematical Sciences
CMU

Title:     How to get the brightest rainbows from the darkest colorings

Abstract:

We will discuss the rainbow (sometimes called polychromatic) variation of the Ramsey theorem on uncountable cardinals. It roughly says if a given coloring on n-tuples satisfies that each color is not used too many times, we can always find a rainbow subset, that is a set in which no two n-tuples from the set get the same color. We will use problems in different settings (inaccessible cardinals, successors of singular cardinals, small uncountable cardinals etc) to demonstrate that the rainbow variation is a “strict combinatorial weakening” of Ramsey theory.

Borisa Kuzeljevic: P-ideal dichotomy and some versions of the Souslin’s Hypothesis

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 12 September 2018, 17:00 hrs

Room: S17#04-05, Department of Mathematics, NUS

Speaker: Borisa Kuzeljevic

Title: P-ideal dichotomy and some versions of the Souslin’s Hypothesis

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

P-ideal dichotomy is a combinatorial set theoretic principle which
has many consequences on the universe of set theory. For example,
it implies the Souslin’s Hypothesis, Singular Cardinals Hypothesis,
and Projective Determinacy. In this talk we will analyze a relationship
between the P-ideal dichotomy and the statement that all Aronszajn
trees can be embedded into the rational line.

This is joint work with Stevo Todorcevic.

Set Theory Today – conference livestream

We are happy to announce that the talks at the Set Theory Today conference (10-14 September 2018) will be recorded and live streamed on the conference website https://sites.google.com/view/set-theory-today/startseite.

The schedule is also accessible on our website. Please feel free to follow and comment!

Place: Fields Institute (Library)
Date: September 7, 2018 (13:30-15:00)
Speaker: Osvaldo Guzman
Abstract: Let $\mathcal{I}$ be an ideal on $\omega.$ We define \textsf{cov}$^{\ast}\left( \mathcal{I}\right)$ as the least size of a family
$\mathcal{B\subseteq I}$ such that for every infinite $X\in\mathcal{I}$ there is $B\in\mathcal{B}$ for which $B\cap X$ is infinite. We say an \textsf{AD} family $\mathcal{A\subseteq I}$ is a \emph{\textsf{MAD} family restricted to }$\mathcal{I}$ if for every infinite $X\in\mathcal{I}$ there is $A\in \mathcal{A}$ such that $\left\vert X\cap A\right\vert =\omega.$ The cardinal invariant $\mathfrak{a}\left( \mathcal{I}\right)$ is defined as the least size of an infinite \textsf{MAD} family restricted to $\mathcal{I}.$ The cardinal invariants $\mathfrak{o}$ and $\mathfrak{a}_{s}$ may be seen as
particular cases of this class of invariants. In this talk, we will prove that
if the maximum of $\mathfrak{a}$ and  textsf{cov}$^{\ast}\left(\mathcal{I}\right)$ is $\omega_{1}$ then $\mathfrak{a}\left( \mathcal{I}% \right) =\omega_{1}.$ We will obtain some corollaries of this result. This is part of a joint work with Michael Hru\v{s}\'{a}k and Osvaldo Tellez.

Anton Bernshteyn: From finite combinatorics to descriptive set theory and back

Mathematical logic seminar – Sep 4 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Anton Bernshteyn
Department of Mathematical Sciences
CMU

Title:     From finite combinatorics to descriptive set theory and back

Abstract:

Many results in finite combinatorics can be extended to infinite structures via compactness – but this transfer is powered by the Axiom of Choice and leads, in general, to highly “pathological” objects. It is natural to ask, which combinatorial constructions can be performed in a “well-behaved” fashion, say, in a Borel or measurable way? This question is addressed in a young branch of descriptive set theory called descriptive combinatorics. We will discuss a class of coloring problems with the requirement that the desired coloring be Baire measurable (i.e., “topologically well-behaved”). The central result of this talk is that the existence of a Baire measurable coloring is equivalent to a purely combinatorial statement, analogs of which have for a long time been studied in finite graph theory with no relation to descriptive set theory.

Wislaw Kubis: Uniformly homogeneous structures

Dear all,

The seminar meets on Wednesday September 5th at 11:00 in the Institute
of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Wislaw Kubis — Uniformly homogeneous structures
A structure is homogeneous if every isomorphism between its finitely
generated substructures extends to an automorphism. We shall discuss a
stronger property. Namely, a structure U is uniformly homogeneous if it
is homogeneous and moreover for every finitely generated substructure A
of U there exists a group embedding e : Aut(A) –> Aut(U) such that e(f)
extends f for every f in Aut(A).
Most of the well known homogeneous structures are uniformly homogeneous.
We shall present examples showing that uniform homogeneity is strictly
stronger than homogeneity.
Some of the results are joint with S. Shelah, some other with B.
Kuzeljevic.

Best,
David