Category Archives: Seminars

Stefan Hoffelner: NS saturated and Δ_1-definable

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

Program:
Stefan Hoffelner — NS saturated and Δ_1-definable

Abstract:
Questions which investigate the interplay of the saturation of the
nonstationary ideal on ω_1, NS, and definability properties of the
surrounding universe can yield surprising and deep results. Woodins
theorem that in a model with a measurable cardinal where NS is
saturated, CH must definably fail is the paradigmatic example. It is
another remarkable theorem of H. Woodin that given ω-many Woodin
cardinals there is a model in which NS is saturated and ω-dense, which
in particular implies that NS is (boldface) Δ_1-definable. The latter
statement is of considerable interest in the emerging field of
generalized descriptive set theory, as the club filter is known to
violate the Baire property.
With that being said the following question, asked first by S.D.
Friedman and L. Wu seems relevant: is it possible to construct a model
in which NS is both Δ_1-definable and saturated from less than ω-many
Woodins? In this talk I will outline a proof that this is indeed the
case: given the existence of M_1^#, there is a model of ZFC in which the
nonstationary ideal on ω_1 is saturated and Δ_1-definable with parameter
ω_1. In the course of the proof I will present a new coding technique
which seems to be quite suitable to obtain definability results in the
presence of iterated forcing constructions over inner models for large
cardinals.

Andy Zucker: A direct solution to the Generic Point Problem

Mathematical logic seminar – Sep 5 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Andy Zucker
Department of Mathematical Sciences
CMU

Title:     A direct solution to the Generic Point Problem

Abstract:

We provide a new proof of a recent theorem of Ben-Yaacov, Melleray, and Tsankov. If G is a Polish group and X is a minimal, metrizable G-flow with all orbits meager, then the universal minimal flow M(G) is non-metrizable. In particular, we show that given X as above, the universal highly proximal extension of X is non-metrizable.

Juris Steprans: Groups with sub exponential growth and actions on the integers with unique invariant means in the Cohen model

Place: BA6183, Bahen Centre

Date: August 11, 2017 (13:30-15:00)

Speaker: Juris Steprans

Title: Groups with sub exponential growth and actions on the integers with unique invariant means in the Cohen model

Abstract: It will be shown that the objects mentioned in the title do not exist.

Jose Iovino: Metastability and model theory

Place: BA6180, Bahen Centre

Date: August 4, 2017 (13:30-15:00)

Speaker: Jose Iovino

Title: Metastability and model theory

Abstract: The concept of metastability was introduced by Tao; it played crucial role in his ergodic convergence theorem (2008) and in Walsh’s generalization (Ann. of Math., 2012). I will discuss the fact that metastability is intimately connected with notions from model theory of metric structures. This is joint work with Xavier Caicedo and Eduardo Duenez.

Christopher Eagle: Baire Category and the Omitting Types Theorem

Place: BA6183

Date: July 21 , 2017 (13:30-15:00)

Speaker: Christopher Eagle

Title: Baire Category and the Omitting Types Theorem

Abstract: It is well-known that the Omitting Types Theorem from model theory can be proved by topological means, and the central ingredient of that proof is the Baire Category Theorem.  The goal of this talk is to consider the extent to which the Omitting Types Theorem is equivalent to the Baire Category Theorem.  To do so, we will describe a topological framework (based on work of Robin Knight) that generalizes the classical type spaces from model theory.   Many classical logics (including first-order, infinitary, and continuous logics) fit into this general setting, and conversely we will show that each instance of the general framework yields a model-theoretic logic.  We then distinguish several version of the Omitting Types Theorem these logics may have, based on Baire Category properties of the underlying topological spaces.  All of these properties are equivalent for first-order logic, but are distinct in the general setting.  This is joint work with Frank Tall.

Dorottya Sziráki: Open colorings on generalized Baire spaces

Thursday, July 20, 2017,  10:30–12.00

Main Lecture Hall , Alfréd Rényi Institute of Mathematics

Abstract: We study the uncountable version of a natural variant of the Open Coloring Axiom. More concretely, suppose that $\kappa$ is an uncountable cardinal such that $\kappa^{<\kappa}=\kappa$ and X is a subset of the generalized Baire space $\kappa^\kappa$ (the space of functions from $\kappa$ to $\kappa$ equipped with the bounded topology). Let OCA*(X) denote the following statement: for every partition of $[X]^2$ as the union of an open set R and a closed set S, either X is a union of $\kappa$ many S-homogeneous sets, or there exists a $\kappa$-perfect R-homogeneous set. We show that after Lévy-collapsing an inaccessible $\lambda>\kappa$ to $\kappa^+$, OCA*(X) holds for all $\kappa$-analytic subsets X of $\kappa^\kappa$. Furthermore, the Silver dichotomy for ${\Sigma}^0_2(\kappa)$ equivalence relations on $\kappa$-analytic subsets also holds in this model. Thus, both of the above statements are equiconsistent with the existence of an inaccessible $\lambda>\kappa$. We also examine games related to the above partition properties.

Yann Pequignot: Sigma^1_2 sets and countable Borel chromatic numbers

Friday, July 21th, 2017, 10.00-12.00

Aula S, Palazzo Campana, Università di Torino

Speaker: Yann Pequignot (University of California, Los Angeles)

Title: Sigma^1_2 sets and countable Borel chromatic numbers

Abstract:

Analytic sets enjoy a classical representation theorem based on wellfounded relations. I will explain a similar representation theorem for Sigma^1_2 sets due to Marcone. This can be used to answer negatively the primary outstanding question from (Kechris, Solecki and Todorcevic; 1999): the shift graph is not minimal among the graphs of Borel functions which have infinite Borel chromatic number.

Dániel T. Soukup: Uncountable strongly surjective linear orders

Thursday, July 13, 2017,  10:30

Seminar Room, Alfréd Rényi Institute of Mathematics

Abstract: A linear order $L$ is strongly surjective if $L$ can be mapped onto any of its suborders in an order preserving way. We review various results on the existence and non-existence of uncountable strongly surjective linear orders answering questions of Camerlo, Carroy and Marcone. In particular, $\diamondsuit^+$ implies the existence of a lexicographically ordered Suslin-tree which is strongly surjective and minimal; every strongly surjective linear order must be an Aronszajn type under $2^{\aleph_0} <2^{\aleph_1}$ or in the Cohen and other canonical models (where $2^{\aleph_0}=2^{\aleph_1}$); finally, we show that it is consistent with CH that there are no uncountable strongly surjective linear orders at all. Further details and open problems can be found in https://arxiv.org/abs/1706.10171

Jindra Zapletal: Quotient forcings defined from group actions

Dear all,

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

Program:
Jindra Zapletal will talk about quotient forcings defined from group
actions.

Best,
David

Saeed Ghasemi: Scattered C*-algebras

Place: Bahen Centre (Room 6183)

Date: July 7th, 2017 (13:30-15:00)

Speaker: Saeed Ghasemi, Polish Academy of Sciences

Title: Scattered C*-algebras

Abstract: By the Gelfand duality, the theory of C*-algebras can be
regarded as “non-commutative topology”. In a joint work with Piotr
Koszmider at IMPAN, we investigated the non-commutative analogues of
the scattered spaces, parallel to the classical research in
set-theoretic topology. The so called scattered C*-algebras, despite
being around in the literature, have not been subject to the tools
from set-theoretic topology. The techniques and constructions of
compact, Hausdorff scattered spaces, or equivalently (by the Stone
duality) superatomic Boolean algebras, have already led to many
fundamental results in the theory of Banach spaces of the form C(K),
or more generally Asplund spaces. In fact scattered C*-algebras were
introduced as C*-algebras which are Asplund as Banach spaces. I will
introduce the notion of the Cantor-Bendixson derivatives for these
C*-algebras, and present some of the basic properties of such
algebras. I will also show how it can be used to construct C*-algebras
with exotic properties, which are non-commutative versions of
well-known scattered spaces. In particular, the constructions of
non-commutative Psi-spaces and thin tall spaces lead to new
discoveries about the preservation of the “stability” for
non-separable C*-algebras.