# Category Archives: Seminars

## 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

## Yinhe Peng: Basis of Countryman lines

Place: Fields Institute (Room 210)
Date: July 13 , 2018 (13:30-15:00)
Speaker:
Title: Basis of Countryman lines
Abstract: U. Abraham and S. Shelah proved that it is consistent to have a 2 element basis for Countryman lines. J. T.  Moore proved that, under PFA, these 2 Countryman lines serve as a basis for Aronszajn lines. We will show that for any positive integer $n$, it is consistent that the basis of Countryman lines has size $2^n$. We will also list some conditions that imply the basis to have maximal size – $2^{\omega_1}$.

## Assaf Shani: Borel equivalence relations and symmetric models

HUJI Logic Seminar

We will have a meeting of the Logic Seminar this Wednesday 11/7, 11-13 Ross 70A.

Title:  Borel equivalence relations and symmetric models
Speaker: Assaf Shani
Abstract. We develop a correspondence between the study of Borel equivalence relations induced by closed subgroups of $S_\infty$, and the study of symmetric models of set theory without choice, and apply it to prove a conjecture of Hjorth-Kechris-Louveau (1998).
For example, we show that the equivalence relation $\cong^\ast_{\omega+1,0}$ is strictly below $\cong^\ast_{\omega+1}$ in Borel reducibility. By results of Hjorth-Kechris-Louveau, $\cong^\ast_{\omega+1}$ corresponds to $\Sigma^0_{\omega+1}$ actions of $S_\infty$, while $\cong^\ast_{\omega+1,0}$ corresponds to $\Sigma^0_{\omega+1}$ actions of “well behaved” closed subgroups of $S_\infty$, e.g., abelian groups.
We further apply these techniques to study the Friedman-Stanley jumps. For example, we find a topology on the domain of $=^{++}$ so that $=^{++}\restriction C$ is Borel bireducible with $=^{++}$, for any comeager set $C$. This answers a question of Zapletal, based on the results of Kanovei-Sabok-Zapletal (2013).
For these proofs we analyze the models $M_n$, $n<\omega$, developed by Monro (1973), and extend his construction past $\omega$, through all countable ordinals. This answers a question of Karagila (2016), e.g., establishing separation between the $\omega$ and $\omega+1$’th Kinna-Wagner principle.

## 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

## Hossein Lamei Ramandi: A minimal Kurepa tree with respect to club embeddings

Place: Fields Institute (Room 210)
Date: July 6, 2018 (13:30-15:00)
Speaker: Hossein Lamei Ramandi
Title:  A minimal Kurepa tree with respect to club embeddings
Abstract:  We will prove that it is consistent with CH that there is a Kurepa tree which club embeds into all of its Kurepa sub trees. Moreover, the Kurepa tree we introduce has no Aronszajn sub tree.

## 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

## Ilya Shapirovsky: Locally finite varieties of modal algebras

Place: Fields Institute (Room 210)
Date: June 15, 2018 (13:30-15:00)
Speaker: Ilya Shapirovsky
Title: Locally finite varieties of modal algebras
Abstract: A modal algebra is a Boolean algebra enriched with an additive operation. Equational theories of modal algebras are called modal logics. A logic L is said to be n-tabular if, up to the equivalence in L, there exist only finitely many n-variable formulas.  L is locally tabular if it is n-tabular for all finite n. Algebraically, n-tabularity of a logic means that its n-generated free algebra is finite (thus,  local tabularity of a logic is equivalent to local finiteness of its variety).

It is known that a variety of closure algebras  is locally finite iff its one-generated free algebra is finite (Larisa Maksimova, 1975). The following question has been open since 1970s: does this equivalence hold for every variety of modal algebras? (The analogous problem is open for varieties of Heyting algebras: does 2-tabularity of an intermediate logic imply local tabularity?)

Recently, in our joint work with Valentin Shehtman, it was shown how local tabularity of modal logics can be characterized in terms of partitions of relational structures. I will discuss this criterion and then use it to construct the first example  of a 1-tabular but not locally tabular modal logic.

## 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

## Joshua Brot, Mengyang Cao, David J. Fernández-Bretón: Finiteness classes arising from Ramsey-theoretic statements in set theory without choice

Thursday, June 14, 2018, from 4 to 5:30pm
East Hall, room 4096

Speaker: Joshua Brot, Mengyang Cao, David J. Fernández-Bretón (University of Michigan)

Title: Finiteness classes arising from Ramsey-theoretic statements in set theory without choice

Abstract:

We investigate infinite sets that witness the failure of certain Ramsey-theoretic statements, such as Ramsey’s or (appropriately phrased) Hindman’s theorem; such sets may exist if one does not assume the Axiom of Choice. We will show very precise information as to where such sets are located within the hierarchy of infinite Dedekind-finite sets. The proofs involve both very pleasant combinatorial arguments (to establish certain implications) and the Fränkel-Mostowski technique to obtain permutation models of ZFA (to prove that certain other implications are not provable).