## 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

## Jindřich Zapletal: Geometric forcing

Dear all,

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

Program: Jindřich Zapletal — Geometric forcing
In joint effort with Paul Larson we develop a theory which makes
possible a complete control over simple forcing extensions of the
Solovay model and detailed comparison of various consequences of the
axiom of choice.

Czech version: Jindřich Zapletal — Geometrický forcing
Ve společném úsilí s Paulem Larsonem rozvíjíme teorii, která umožnuje
úplnou kontrolu nad jednoduchými forcingovými rozšířeními Solovayova
modelu a porovnáváni různých důsledků axiomu výběru.

Best,
David

## Wednesday seminar

Dear all,

The seminar meets on Wednesday June 27th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Jindra Zapletal will present a talk.

Best,
David

## Mirna Džamonja: Higher order versions of the logic of chains close

Dear all,

The seminar meets on Wednesday June 20th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Mirna Džamonja — Higher order versions of the logic of chains
close

First order logic of chains was discovered by Carol Karp and revisited
in recent work of Dz. with Jouko Vaananen. The results have shown that
the logic, defined through a singular cardinal of countable cofinality,
behaves very much like the first order logic. In our new joint work, we
study higher order versions of the logic of chains and their fragments
to defend the thesis that in this context we can also recover
similarities with the ordinary logic. We also discuss the idea of
infinite computation.

Best,
David

## Viera Šottová: Ideal version of selection principle S1(P,R)

The seminar meets on Wednesday June 6t at 11:00 in the Institute of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Speaker: Viera Šottová

Title: Ideal version of selection principle $S_1(\mathcal P,\mathcal R)$.

Abstract: attached.

## Damian Sobota: Rosenthal families and ultrafilters

Dear all,

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

Program: Damian Sobota — Rosenthal families and ultrafilters

Rosenthal’s lemma is a simple technical result with numerous
applications to measure theory and Banach space theory. The lemma in its
simplest form reads as follows: “For every infinite real-entried matrix
(m(n,k): n,k in N) such that every entry is non-negative and the sum of
every row is <=1, and every epsilon>0, there exists an infinite subset A
of N such that for every k in A we have sum_{n in A, n\neq
k}m_n^k<epsilon.” A natural question arises whether we can choose the
set A from a previously fixed family F of infinite subsets of N. If F
has such a property, then we call it Rosenthal. Thus, Rosenthal’s lemma
states that [N]^omega is Rosenthal. During my talk I’ll present some
necessary or sufficient conditions for a family to be Rosenthal and
prove that under MA(sigma-centered) there exists a P-point which is a
Rosenthal family but not a Q-point. (No Banach space will appear during
the talk.)

Best,
David

## Jonathan Verner: Ultrafilters and nonstandard models of arithmetic

Dear all,

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

Program: Jonathan Verner — Ultrafilters and nonstandard models of
arithmetic

(This is a talk postponed from April 18th.)

Best,
David