Archives of: Prague Set Theory Seminar

Saeed Ghasemi: AF-algebras with Cantor-set property

Dear all,

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

Program: Saeed Ghasemi — AF-algebras with Cantor-set property

A separable AF-algebra is a C*-algebra which is (isomorphic to) the
inductive limit of a direct sequence of finite dimensional C*-algebras.
We introduce a class of separable AF-algebras, called AF-algebras with
Cantor-set property, which are, in some contexts, suitable
noncommutative analogues of the Cantor set. One of the main features of
AF-algebras with Cantor-set property is that they are all Fraisse
limits of some category of finite dimensional C*-algebras and left
invertible embeddings. As a consequence of this, many properties of the
Cantor set that can be proved using the Fraisse theory, such as the
homogeneity and universality, also can also be proved for AF-algebras
with Cantor-set property. In fact, the category of all finite
dimensional C*-algebras and left invertible embeddings is a Fraisse
category and its Fraisse limit is a separable AF-algebra with Cantor-set
property which has the universality property that maps surjectively onto
any separable AF-algebra.*
This is a joint work with Wieslaw Kubis.

*- All of these results can be restated and proved in the language of
partially ordered abelian groups without mentioning any C*-algebras.

Best,
David

Jonathan Verner: Towers in filters, cardinal invariants, and Luzin type families, part II

Dear all,

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

Program: Jonathan Verner will continue his talk from the last seminar:

Towers in filters, cardinal invariants, and Luzin type families

Jonathan will present results from his recent paper (with J. Brendle, B.
Farkas);
We investigate which filters on ω can contain towers, that is, a modulo
finite descending sequence without any pseudointersection. We prove the
following results:
(1) Many classical examples of nice tall filters contain no towers.
(2) It is consistent that tall analytic P-filters contain towers of
arbitrary regular height.
(3) It is consistent that all towers generate non-meager filters.
(4) The statement “Every ultrafilter contains towers.” is independent of
ZFC.

Best,
David

Jonathan Verner: Towers in filters, cardinal invariants, and Luzin type families

Dear all,

The seminar meets on Wednesday March 13th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Jonathan Verner — Towers in filters, cardinal invariants, and
Luzin type families

Jonathan will present results from his recent paper (with J. Brendle, B.
Farkas);
We investigate which filters on ω can contain towers, that is, a modulo
finite descending sequence without any pseudointersection. We prove the
following results:
(1) Many classical examples of nice tall filters contain no towers.
(2) It is consistent that tall analytic P-filters contain towers of
arbitrary regular height.
(3) It is consistent that all towers generate non-meager filters.
(4) The statement “Every ultrafilter contains towers.” is independent of
ZFC.

Best,
David

Michal Doucha: Definable pseudometrics and Borel reductions between them

Dear all,

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

Program: Michal Doucha — Definable pseudometrics and Borel reductions
between them

I will introduce a “continuous generalizatio” of the theory of
definable equivalence relations and Borel reductions between them.
Equivalence relations will be replaced by pseudometrics and reductions
between them will be replaced by certain uniformly continuous maps. I
will explain our motivation and prove some basic results. I will present
some open problems whose solutions may require completely new ideas from
invariant descriptive set theory. It will be based on a joint paper with
Marek Cúth and Ondřej Kurka.

Jan Grebík: Vizing’s Theorem for Graphings

Dear all,

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

Program: Jan Grebík — Vizing’s Theorem for Graphings

I will show that the measurable edge chromatic index of a graphing G
(bounded degree Borel graph with invariant measure) with maximum degree
D is D+1. This is a joint work with Oleg Pikhurko.

Best,
David

Asaf Karagila: Staring into a Cohen real: the Bristol model

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

Program: Asaf Karagila — Staring into a Cohen real: the Bristol model

What kind of intermediate models can be found when adding a Cohen real,
c, to L? If we are concerned with models of ZFC, then answer is simple:
L itself, or Cohen extensions of L.
But when models of ZF are of interest, this fails spectacularly. The
Bristol model is a model intermediate to L[c] which is not even
constructible from a set. We will discuss the details of the
construction, and the consequences it has on the models which are
trapped between L and L[c].

Jan Hubička: Combinatorial proofs of the extension property for partial automorphisms

Dear all,

The seminar meets on Wednesday February 13th at 11:00 in the Institute of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Jan Hubička — Combinatorial proofs of the extension property for partial automorphisms

Class K of finite structures has extension property for partial automorphisms (EPPA) if for every A in K there exists B in K such that every partial automorphism of A (that is isomorphism of two substructures of A) extends to automorphism of B. Hrushovski, in 1992,
shown that the class of all finite graphs has EPPA.  This result was used by Hodges, Hodkinson, Lascar and Shelah to show that the random
graph has small index property. This motivated search for new classes with EPPA. I will show (and partly prove) new general theorem giving a structural condition for class having EPPA. The theorem is a strengthening of the Herwig–Lascar theorem, but the proof techniques are new, combinatorial and completely self-contained.

I will also discuss connection to structural Ramsey theory.

This is joint work with Jaroslav Nesetril and Matej Konecny.

Best,
David

Jordi Lopez Abad: Amalgamation and Ramsey properties of $\ell_p^n$’s

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

Program: Jordi Lopez Abad — Amalgamation and Ramsey properties of
$\ell_p^n$’s

We will give a proof of the fact that $\{\ell_p^n\}$ have the
approximate Ramsey property and a strong form of amalgamation (they are
Fraïssé classes, in a metric sense). The proofs are divided into 3
cases: $p=\infty$, $p=2$ and $p\neq 2,\infty$. We will also discuss the
case of the families of all finite dimensional subspaces of $L_p(0,1)$
for $p\neq 2,\infty$ and of $C[0,1]$.

 

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