Archives of: Prague Set Theory Seminar

Kaethe Minden: Split Principles and Splitting Families

Dear all,

The seminar meets on Wednesday January 23rd at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Kaethe Minden — Split Principles and Splitting Families

The original split principle is an equivalent formulation of a cardinal
failing to satisfy the combinatorial essence of weak compactness. The
notion was then expanded by Gunter Fuchs and me to characterize the
negation of other large cardinal properties. Split principles give rise
to seemingly new large cardinals, and some new ideals, for example a
normal ideal on $\mathcal P_\kappa \lambda$ in the case of
$\lambda$-Shelahness. In this talk I introduce split principles and
connect them to certain new notions of splitting numbers being large.

Best,
David

Miha Habic: The ultrapower capturing property (part II)

Dear all,

The seminar meets on Wednesday January 16th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Miha Habic — The ultrapower capturing property (part II)

In 1993 Cummings showed that it is consistent (relative to large
cardinals) that there is a measurable cardinal kappa carrying a normal
measure whose ultrapower contains the whole powerset of kappa^+. He
showed that nontrivial large cardinal strength was necessary for this,
but it was not clear whether this capturing property had any direct
consequences. Recently Radek Honzík and I showed that it is relatively
consistent that the least measurable cardinal has this capturing
property. We also considered a local version of capturing. In this talk
I will introduce a forcing notion due to Apter and Shelah and the
modifications necessary to obtain our result.

Best,
David

Miha Habic: The ultrapower capturing property (part I)

The seminar meets on Wednesday January 9th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Miha Habic — The ultrapower capturing property (part I)

In 1993 Cummings showed that it is consistent (relative to large
cardinals) that there is a measurable cardinal kappa carrying a normal
measure whose ultrapower contains the whole powerset of kappa^+. He
showed that nontrivial large cardinal strength was necessary for this,
but it was not clear whether this capturing property had any direct
consequences. Recently Radek Honzík and I showed that it is relatively
consistent that the least measurable cardinal has this capturing
property. We also considered a local version of capturing. In this talk
I will overview the necessary large cardinal machinery and Cummings’
original argument.

The second part of the talk will take place on Wednesday January 16th.
In the second talk Miha will introduce a forcing notion due to Apter and
Shelah and the modifications necessary to obtain the result.

Michael Hrusak: Ramsey theorem with highly connected homogeneous sets

Dear all,

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

Program: Michael Hrusak — Ramsey theorem with highly connected
homogeneous sets
(joint work with J. Bergfalk and S. Shelah)

A graph $(\kappa, E)$ is highly homogeneous if all its restrictions on
complements of sets of cardinality less than $\kappa$ are homogeneous.
We investigate for which cardinals $\theta < \lambda \leq \kappa$ does
hold that for every colouring $c:[\kappa]^2\to \theta$ there exists $A$
subset of $\kappa$ of cardinality $\lambda$ and a colour $i\in\theta$
such that the graph $(A, c^{-1}(i)\cap [A]^2)$ is highly homogeneous.

Best,
David

Diana Carolina Montoya: The equality p=t and the generalized characteristics

Dear all,

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

Program: Diana Carolina Montoya — The equality p=t and the generalized
characteristics

Marriallis and Shelah solved in the positive the longstanding problem of
whether the two cardinal invariants $\mathfrak{p}$ (the
pseudointersection number) and $\mathfrak{t}$ (the tower number) are
equal. In this talk, I will review some essential points in their proof
in order to motivate the study of the analogous question for the
generalized characteristics $\mathfrak{p}(\kappa)$ and
$\mathfrak{t}(\kappa)$. I will present some results of Garti regarding
this generalization and finally some recent progress (joint work with
Fischer, Schilhan and, Soukup) in the direction of answering this question.

Best,
David

Viera Sottova: Modification of N via ideals on omega

Dear all,

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

Program: Viera Sottova — Modification of N via ideals on omega

We consider a combinatorial characterization of the null ideal N which
we modify using ideals on omega. Resulting ideal N_J is also a sigma
ideal and additionally it is subideal of N. We focus on common cardinal
invariants of this ideal and their relation to the original ones.
Joint work with D. A. Mejia.

Best,
David

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