Archives of: Prague Set Theory Seminar

Stefan Geschke: There are no universal minimal metric flows for countable discrete groups

Wednesday, November 14, 2018, 11:00
Prague – IM AS CR, Zitna 25, seminar room, front building, third floor

Speaker: Stefan Geschke

Title: There are no universal minimal metric flows for countable discrete groups

Abstract:

Let $G$ be a topological group. A $G$-flow is a compact space $X$ together with a continuous action of $G$. The morphisms between $G$-flows are continuous maps that respect the group action. A flow is minimal if it has no proper (nonempty) subflows. A flow $X$ is universal in a class $\mathcal C$ if it is in $\mathcal C$ and for every flow $Y \in \mathcal C$ there is an epimorphism from $X$ onto $Y$. Using Fürstenberg’s structure theorem for distal flows, Foreman and Beleznay showed that there are no universal minimal metric $\mathbb Z$-flows.

Every group $G$ acts in a natural way on the space $2^G$. Gao and Jackson showed that for every countable discrete group $G$, $2^G$ has a perfect set of minimal subflows. We show that this implies that there are no universal minimal metric flows for any countable discrete group $G$.

David Uhrik: Composing discontinuous functions

Dear all,

The seminar meets on Wednesday October 31st at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: David Uhrik — Composing discontinuous functions

In this talk we’ll look at composing functions from the Young hierarchy,
which is defined similarly as the well-known Baire hierarchy of
functions but we only consider monotone sequences. It will be shown that
these compositions behave nicely, i.e. the resulting function is again
an element of the hierarchy and its rank is bounded above. On the other
hand we can decompose these functions into a composition of two with
lower rank. In the end I’ll say something about the “failed” attempt to
generalise this hierarchy to convergence according to an ideal.

Best,
David

Fabiana Castiblanco: Capturing tree forcing notions and some preservation results

The seminar meets on Wednesday October 24th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Fabiana Castiblanco — Capturing tree forcing notions and some
preservation results

In this talk, we will introduce the concept of capturing forcing notions
in order to show that various tree posets such as Sacks (S), Silver (V),
Mathias (M), Laver (L) and Miller (ML) forcing preserve the existence of
sharps for reals. Furthermore, these tree forcing notions preserve
levels of Projective Determinacy. As a consequence of this fact we
obtain that Σ^1_{n+3}-P-absoluteness holds for P∈T := {S, V, M, L, ML}
under the assumption of Π^1_{n+1}-determinacy.
If time permits, as an application of our results, we will see that if
Π^1_{n+1}-determinacy holds, each P∈T does not add new orbits to
∆^1_{n+3}-thin transitive relations.

Best,
David

Miha Habič: Surgery and generic coding

Dear all,

The seminar meets on Wednesday October 10th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.
Note that there will be no seminar on Wednesday October 17th.

Program: Miha Habič — Surgery and generic coding

There has been some interest recently in nonamalgamability phenomena
between countable models of set theory, and forcing extensions of a
fixed model in particular. Nonamalgamability is typically achieved by
coding some forbidden object between a collection of models in such a
way that each model on its own remains oblivious, but some combination
of them can recover the forbidden information.

In this talk we will examine the problem of coding arbitrary information
into a generic filter, focusing on two particular examples. First, I
will present some results of joint work with Jonathan Verner where we
consider surgical modifications to Cohen reals and sets of indices where
such modifications are always possible. Later, I will discuss a recent
result of S. Friedman and Hathaway where they achieve, using different
coding, nonamalgamability between arbitrary countable models of set
theory of the same height.

Best,
David

Victor Torres Perez: Combinatorial Principles without MA

Dear all,

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

Program: Victor Torres Perez — Combinatorial Principles without MA

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 ℵ_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 ℵ_2, the
negation of Box(θ) for every regular θ ≥ ω_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(ℵ_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.
For example, we will also discuss some recent results regarding the
P-ideal dichotomy (which can be consistent with the negation of MA(ℵ_1))
and square principles.

Best,
David

Viera Šottová: The critical cardinality of spaces with respect to the ideal version of S_1(P,R)

Dear all,

The seminar meets on Wednesday September 26th at 11:00 in the Institute
of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Viera Šottová — The critical cardinality of spaces with
respect to the ideal version of S_1(P,R)

I will present the second part a joint paper with J. Šupina which
focuses on critical cardinality of the ideal version of sequence
selection principles. If I, J are arbitrary ideals on omega, then the
critical cardinality of S_1(I-Gamma, J-Gamma) can be described by the
cardinal lambda(I,J) in the case of covers. Therefore we investigate
lambda(I,J) and its relation to the other cardinals. This will be the
main point of this talk.

Best,
David

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

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

Aleksandra Kwiatkowska: Universal minimal flows of the homeomorphism groups of Ważewski dendrites

Dear all,

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

Program: Aleksandra Kwiatkowska — Universal minimal flows of the
homeomorphism groups of Ważewski dendrites

For each P ⊆ {3,4,…,ω} there is a continuum called Ważewski dendrite
W_P, we can construct it in the framework of the Fraisse theory. If P is
finite, we prove that the universal minimal flow of the homeomorphism
group H(W_P) is metrizable and we compute it explicitly. This answers a
question of Duchesne. If P is infinite, we show that the universal
minimal flow of H(W_P) is not metrizable. This provides examples of
topological groups which are Roelcke precompact and have a
non-metrizable universal minimal flow with a comeager orbit.

Best,
David

Jan Grebík: Borel chromatic numbers

Dear all,

The seminar meets on Wednesday July 18th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Jan Grebík — Borel chromatic numbers
I will present an overview of known facts about Borel chromatic numbers
of Borel graphs. I will also state a problem concerning continuous
actions of summable ideal.

Best,
David