## Dana Bartosova: When can we act freely?

Place: Fields Institute (Stewart Library)

Date: May 19, 2017 (13:30-15:00)

Speaker: Dana Bartosova, CMU

Title: When can we act freely?

Abstract:  A topological group admits a free action if there is a compact
Hausdorff space on which the group acts without fixed points. I will
discuss this notion and explain how to translate it into colourings of
graphs and Ramsey type properties.

## Jarosław Swaczyna: Haar-small sets

Tuesday, May 23, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Jarosław Swaczyna (Lodz University of Technology)

Title: Haar-small sets

Abstract:

In locally compact Polish groups there is a very natural $\sigma$-ideal of null sets with respect to Haar-measure. In non locally compact groups there is no Haar measure, however Christensen introduced a notion of Haar-null sets which is an analogue of locally compact case. In 2013 Darji introduced a similar notion of Haar-meager sets. During my talk I will present some equivalent definition of Haar-null sets which leads us to joint generalization of those notions. This is joint work with T. Banakh, Sz. Głąb and E. Jabłońska.

## Victoria Gitman: A model of second-order arithmetic with the choice scheme in which $\Pi^1_2$-dependent choice fails

KGRC research seminar – 2017‑05‑18 at 4pm

Speaker: Victoria Gitman (CUNY Graduate Center, New York, USA)

Abstract:

In second-order arithmetic, the choice scheme is the scheme of assertions, for every second-order formula $\varphi(n,X,A)$, that if for every $n$ there is a set $X$ such that $\varphi(n,X,A)$ holds, then there is a single set $Y$ whose $n$-th slice $Y_n$ witnesses $\varphi(n,Y_n,A)$. While full second-order arithmetic ${\textrm Z}_2$ implies the choice scheme for $\Sigma^1_2$-assertions, the reals of the Feferman-Lévy model form a model of ${\textrm Z}_2$ in which $\Pi^1_2$-choice fails. The dependent choice scheme is the analogue ${\textrm DC}$ for second-order arithmetic and it asserts, for every second-order formula $\varphi(X,Y,A)$, that if for every set $X$ there is another set $Y$ such that $\varphi(X,Y,A)$ holds, then there is a single set $Z$, viewed as an $\omega$-sequence of sets, such that for every $n$, $\varphi(Z\upharpoonright n,Z_n,A)$ holds. The theory ${\textrm Z}_2$ implies $\Sigma^1_2$-dependent choice, and Simpson has conjectured that there is a model of ${\textrm Z}_2$ with the choice scheme in which $\Pi^1_2$-dependent choice fails. We prove Simpson’s conjecture by constructing a symmetric submodel of a forcing extension in which ${\textrm AC}_\omega$ holds, but ${\textrm DC}$ fails for a $\Pi^1_2$-definable relation on the reals.

We force over $L$ with a tree iteration of Jensen’s forcing (a ccc subposet
of Sacks forcing adding a unique generic real) along the tree ${}^{\lt\omega}\omega_1$, adding a tree, isomorphic to ${}^{\lt\omega}\omega_1$, of finite sequences of reals ordered by extension, such that that the sequences on level $n$ are $L$-generic for the $n$-length iteration of Jensen’s forcing. We extend the uniqueness of generic reals properties of Jensen’s forcing (obtained earlier by Jensen and later by Lyubetsky and Kanovei) by showing that in the tree iteration extension, the only sequences of reals $L$-generic for the $n$-length iteration of Jensen’s forcing are those explicitly added on level $n$ of the generic tree. The uniqueness property implies that the generic tree is $\Pi^1_2$-definable.

The theorem arose out of our attempts to separate the analogues of the
choice scheme and the dependent choice scheme over Kelley-Morse set theory,
and we conjecture that an appropriate generalization of our arguments will
now achieve this result.

This is joint work with Sy-David Friedman.

## Nick Ramsey: Definability in the absolute Galois group of a PAC field

HUJI Logic Seminar

The next meeting of the Logic Seminar will be in 15/05, 14:00 – 16:00 in Sprinzak Building, Room 101.  Please note the change in time and place.
Definability in the absolute Galois group of a PAC field

Speaker: Nick Ramsey
Abstract:  A field K is called pseudo-algebraically closed (PAC), if every absolutely irreducible variety defined over K has a K-rational point.  This class of fields was introduced in 1968 by Ax on the way to his famous algebraic characterization of the pseudo-finite fields: a field is elementarily equivalent to an ultra-product of finite fields if and only if it is perfect, has free profinite absolute Galois group on a single generator, and is PAC.  In the decades that followed, the PAC fields were an intensive object of study within model-theoretic algebra.  Cherlin, van den Dries, and MacIntyre gave complete invariants for the first-order theory of a PAC field in terms of its characteristic, absolute numbers, and the ‘co-theory’ of its absolute Galois group, in a formalism they called ‘co-logic’.  This ‘co-logic’ approach was presented within first-order logic by Chatzidakis who, later on, showed how to connect model-theoretic properties of the theory of the absolute Galois group of a PAC field to the model-theoretic properties of the field itself.  We will survey this work in detail and give some recent applications to classification-theoretic questions concerning PAC fields.  Time-permitting, we will also talk about some work in progress and many open problems in this area.

## Stefan Hoffelner: $\text{NS}_{\omega_1}$ saturated and a $\Sigma^{1}_{4}$-definable wellorder on the reals

KGRC Research Seminar – 2017‑05‑11 at 4pm.

Speaker: Stefan Hoffelner (KGRC)

Abstract: The investigation of the saturation of the nonstationary ideal $\text{NS}_{\omega_1}$ has a long tradition in set theory. In the early 1970’s K. Kunen showed that, given a huge cardinal, there is a universe in which $\text{NS}_{\omega_1}$ is $\aleph_2$-saturated. The assumption of a huge cardinal has been improved in the following decades, using very different techniques, by many set theorists until S. Shelah around 1985 realized that already a Woodin cardinal is sufficient for the consistency of the statement “$\text{NS}_{\omega_1}$ is saturated”.

Due to work of H. Woodin on the one hand and G. Hjorth on the other, there is a surprising and deep connection between definable wellorders of the reals and the saturation of $\text{NS}_{\omega_1}$: In a universe with a measurable cardinal and $\text{NS}_{\omega_1}$ saturated, it is impossible to have a $\Sigma^1_3$-wellorder. This leads naturally to the question whether there is a universe in which $\text{NS}_{\omega_1}$ is saturated and its reals have a
$\Sigma^1_{4}$-wellorder. In my talk I will outline a proof that this is indeed the case; assuming the existence of $M_1^{\#}$ there is a model with a $\Sigma^1_{4}$-definable wellorder on the reals in which  $\text{NS}_{\omega_1}$ is saturated.

This is joint work with Sy-David Friedman.

## Andrey Morozov: Infinite time Blum-Shub-Smale machines for computability in analysis

Monday, May 15, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Andrey Morozov (Sobolev Institute of mathematics, Novosibirsk)

Title: Infinite time Blum-Shub-Smale machines for computability in analysis

## Grigory Mashevitzky: Action of endomorphism semigroups on definable sets

​​​​BGU seminar in Logic, Set Theory and Topology

Time: Tuesday, May 9th, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: Grigory Mashevitzky (BGU)

Title: Action of endomorphism semigroups on definable sets

Abstract:
I plan to discuss the construction, examples and some applications the Galois-type correspondence between subsemigroups of the endomorphism semigroup End(A) of an algebra A and sets of formulas. Such Galois-type correspondence forms a natural frame for studying algebras by means of actions of different subsemigroups of End(A) on definable sets over A. Between possible applications of this Galois correspondence is a uniform approach to geometries defined by various fragments of the initial language.

The next prospective application deals with effective recognition of sets and effective computations with properties that can be defined by formulas from a fragment of the original language. In this way one can get an effective syntactical expression by semantic tools.

Yet another advantage is a common approach to generalizations  of the main model theoretic concepts to the sublanguages of the first order language. It also reveals new connections between well-known concepts. One more application concerns the generalization of the unification theory or more generally Term Rewriting Theory to the logic unification theory.

## Joanna Jureczko: Some remarks on Kuratowski partitions, new results

Tuesday, May 9, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Joanna Jureczko (Cardinal Stefan Wyszynski University in Warsaw)

Title: Some remarks on Kuratowski partitions, new results

Abstract:

K. Kuratowski in 1935 posed the problem whether a function $f \colon X \to Y$ from a completely metrizable space $X$ to a metrizable space $Y$ is continuous apart from a meager set.
This question is equivalent to the question about the existence of so called a Kuratowski partition, i. e. a partition $\mathcal{F}$ of a space $X$ into meager sets such that $\bigcup \mathcal{F}’$ for any $\mathcal{F}’ \subset \mathcal{F}$.
With any Kuratowski partition we may associate a $K$-ideal, i.e. an ideal of the form
$$I_{\mathcal{F}} = \{A \subset \kappa \colon \bigcup_{\alpha \in A}F_\alpha \textrm{ is meager }, F_\alpha \in \mathcal{F}\}.$$
It would seem that the information about $I_{\mathcal{F}}$ would give us full information about the ideal and the world in which it lives.
My talk is going to show that it is big simplification and localization technique from a Kuratowski partition cannot be omitted but the proof can be much simplier.
During the talk I will show among others a new proof of non-existence of a Kuratowski partition in Ellentuck topology and a new combinatorial proof of Frankiewicz – Kunen Theorem (1987) on the existence of measurable cardinals.

## Chris Lambie-Hanson: Constructions from square and diamond, with an application to super-Souslin trees

Monday, May 8, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Chris Lambie-Hanson (Bar-Ilan)

Title: Constructions from square and diamond, with an application to super-Souslin trees

Abstract. In 1982, Shelah and Stanley proved that, if $\kappa$ is a regular, infinite cardinal, $2^\kappa = \kappa^+$, and there is a $(\kappa^+, 1)$-morass, then there is a $\kappa^{++}$-super-Souslin tree, which is a type of normal $\kappa^{++}$-tree that necessarily has a $\kappa^{++}$-Souslin subtree and continues to do so in any outer model in which $\kappa^{++}$ is preserved and no new subsets of $\kappa$ are present. This result establishes a lower bound of an inaccessible cardinal for the consistency strength of the conjunction of $2^\kappa = \kappa^+$ and Souslin’s Hypothesis at $\kappa^{++}$. In this talk, we will present a method for constructing objects of size $\lambda^+$ from $\square_\lambda + \diamondsuit_\lambda$, where $\lambda$ is a regular, uncountable cardinal. As an application, we will use $\square_{\kappa^+} + \diamondsuit_{\kappa^+}$ to construct a $\kappa^{++}$-super-Souslin tree. For uncountable $\kappa$, this increases Shelah and Stanley’s lower bound from an inaccessible cardinal to a Mahlo cardinal. This is joint work with Assaf Rinot.

## Sabrina Ouazzani: A brief story of gaps in the infinite time Turing machines

Tuesday, May 9, 2017, 15.00
Howard House 4th Floor Seminar Room

Speaker: Dr Sabrina Ouazzani (Laboratoire d’Algorithmique, Complexité et Logique, Paris-Est Creteil University)

Title: A brief story of gaps in the infinite time Turing machines

Abstract:

I will present a part of my PhD thesis entitled “From algorithmics
to logics through infinite time computation” in which I have studied
the infinite time Turing machines model of computation from a computer
scientist’s point of view. In particular I focused on the structure of
gaps in the clockable ordinals, that is to say, ordinal times at which
no infinite time program halts.

So in this talk I will present infinite time Turing machines (ITTM),
from the original definition of the model to some new infinite time
algorithms. These algorithmic techniques will allow to highlight some
properties of the ITTM-computable ordinals and we will see that they
bring some information about the structure of gaps.