David J. Fernández Bretón: mathfrak p=mathfrak t

Thursday, March 30, 2017, from 4 to 5:30pm
East Hall, room 3088

Speaker: David J. Fernández Bretón (University of Michigan)

Title: mathfrak p=mathfrak t

Abstract:

In a series of (hopefully at most) two talks, I will present the proof, due to Maryanthe Malliaris and Saharon Shelah in 2012, that the cardinal invariants p and t are equal, which constitutes an extremely important result in the theory of Cardinal Characteristics of the Continuum.

Zoltán Vidnyánszky: Anti-basis results for graphs of infinite Borel chromatic number

Place: Fields Institute (Room 210)

Date: March 24, 2017 (13:30-15:00)

Speaker: Zoltán Vidnyánszky, York University

Title: Anti-basis results for graphs of infinite Borel chromatic number

Abstract: One of the most interesting results of Borel graph combinatorics is the $G_0$ dichotomy, i. e., the fact that a Borel graph has uncountable Borel chromatic number if and only if it contains a Borel homomorphic image of a graph called $G_0$. It was conjectured that an analogous statement could be true for graphs with infinite Borel chromatic number. Using descriptive set theoretic methods we answer this question and a couple of similar questions negatively, showing that one cannot hope for the existence of a Borel graph whose embeddability would characterize Borel (or even closed) graphs with infinite Borel chromatic number.

Judyta Bąk: Domain theory and topological games

Tuesday, March 28, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Judyta Bąk (University of Silesia)

Title: Domain theory and topological games

Abstract:

Domain is a partially ordered set, in which there was introduced some specific relation. We say that a space is domain representable if it is homeomorphic to a space of maximal elements of some domain. In 2015 W. Fleissner and L. Yengulalp introduced a notion of $\pi$–domain representable space, which is analogous of domain representable. We prove that a player $\alpha$ has a winning strategy in the Banach–Mazur game on a space $X$ if and only if $X$ is countably $\pi$–domain representable. We give an example of countably $\pi$–domain representable space, which is not $\pi$–domain representable.

Michał Tomasz Godziszewski – Computable quotient presentations of models of arithmetic and set theory

KGRC Research Seminar – 2017-03-23 at 4pm

Speaker: Michał Tomasz Godziszewski (University of Warsaw, Poland)

Abstract:

A computable quotient presentation of a mathematical structure
$\mathbb{A}$ consists of a computable structure on the natural numbers
$(\mathbb{N}, \star, \ast, \ldots)$ (meaning that the operations and
relations of the structure are computable) and an equivalence relation
$E$ on $\mathbb{N}$, not necessarily computable but which is a
congruence with respect to this structure, such that the quotient
$(\mathbb{N}, \star, \ast, \ldots)_{/E}$ is isomorphic to the given
structure $\mathbb{A}$. Thus, one may consider computable quotient presentations
of graphs, groups, orders, rings and so on, for any kind of mathematical
structure.

In 2016 Bakhadyr Khoussainov discussed some questions and results in this area.
Part of this concerns the investigation, for a fixed equivalence relation $E$ or type
of equivalence relation, which kind of computable quotient presentations
are possible with respect to quotients modulo $E$.

We address two conjectures that Khoussainov had made and prove
various extensions of the Tennenbaum phenomenon to the case of
computable quotient presentations of models of arithmetic and set
theory. Specifically, no nonstandard model of arithmetic has a
computable quotient presentation by a c.e. equivalence relation. No
$\Sigma_1$-sound nonstandard model of arithmetic has a computable
quotient presentation by a co-c.e. equivalence relation. No nonstandard
model of arithmetic in the language $\{+, \cdot, \leq\}$ has a
computably enumerable quotient presentation by any equivalence relation
of any complexity. No model of ZFC or even much weaker set theories has
a computable quotient presentation by any equivalence relation of any
complexity. And similarly no nonstandard model of finite set theory has
a computable quotient presentation. Concluding from that, we indicate
how the program of computable quotient presentations has difficulties
with purely relational structures (as opposed to algebras).
This is joint work with Joel David Hamkins, GC CUNY.

 

Frank Stephan: Finitely generated semiautomatic groups

Update on the Logic Seminar at the National University of Singapore
(the originally announced speaker is ill and will speak on 12 April 2017)

Date: Wednesday, 22 March 2016, 17:00 hrs

Room: S17#04-05, Department of Mathematics, NUS

Speaker: Frank Stephan

Title: Finitely generated semiautomatic groups

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

The present work shows that Cayley automatic groups are semiautomatic
and exhibits some further constructions of semiautomatic groups.
Furthermore, the present work establishes that every finitely
generated group of nilpotency class $3$ is semiautomatic.

Joan Bagaria: Berkeley Cardinals

BCNSETS: BARCELONA RESEARCH GROUP IN SET THEORY
Thursday, 23 March 2017, 15:30

Speaker: Joan Bagaria

Title: Berkeley Cardinals

Abstract: Berkeley cardinals are  large cardinals whose existence
contradicts the Axiom of Choice. We will  present some recent
results (joint with P. Koellner and W. H. Woodin) about the relative
position of Berkeley cardinals in the large cardinal hierarchy, and
also about the possible cofinalities of the first Berkeley cardinal.

Dana Bartošová: Freedom of action in combinatorial terms

Mathematical logic seminar – Mar 21 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Dana Bartošová
Department of Mathematical Sciences
CMU

Title:     Freedom of action in combinatorial terms

Abstract:

A group acts freely on a compact Hausdorff space if all of its non-identity elements act without fixed points. By Veech’s theorem, every locally compact topological group admits a free action and the question arises to which other groups this property can be extended. On the other hand, elements of extremely amenable groups act with fixed points under any action but the opposite implication does not hold. We show a combinatorial reformulation of this property and ask how far it is from extreme amenability.

Andres Caicedo: MRP and squares, II

Thursday, March 23, 2017, from 4 to 5:30pm
East Hall, room 3088

Speaker: Andres Caicedo (Math Reviews)

Title: MRP and squares, II

Abstract:

Justin Moore’s mapping reflection principle (MRP) seems to capture the consistency strength of PFA, since it implies the failure of square. I continue the presentation of some refinements and extensions of this result. They are due to a variety of authors, and some remain unpublished.

David Aspero: Generic absoluteness for Chang models

Tuesday, March 21, 2017, 15.00
Howard House 4th Floor Seminar Room

Speaker: David Aspero (University of East Anglia)

Title: Generic absoluteness for Chang models

Abstract:

The main focus of the talk will be on extensions of Woodin’s classical result that, in the presence of a proper class of Woodin cardinals, C_omega^V and C_omega^{V^P} are elementarily equivalent for every set—forcing P (where C_kappa denotes the kappa—Chang model).

1. In the first part of the talk I will present joint work with Asaf Karagila in which we derive generic absoluteness for C_omega over the base theory ZF+DC.

2. Matteo Viale has defined a strengthening MM^{+++} of Martin’s Maximum which, in the presence of a proper class of sufficiently strong large cardinals, completely decides the theory of C_{omega_1} modulo forcing in the class Gamma of set—forcing notions preserving stationary subsets of omega_1, i.e., if MM^{+++} holds, P is in Gamma, and P forces MM^{+++}, then C_{omega_1}^V and C_{omega_1}^{V^P} are elementarily equivalent. MM^{+++} is the first example of a “category forcing axiom.”

In the second part of the talk I will present some recent joint work with Viale in which we extend his machinery to deal with other classes Gamma of forcing notions, thereby proving the existence of several mutually incompatible category forcing axioms, each one of which is complete for the theory of C_{omega_1}, in the appropriate sense, modulo forcing in Gamma.

Andres Caicedo: MRP and squares

Thursday, March 16, 2017, from 4 to 5:30pm
East Hall, room 2866

Speaker: Andres Caicedo (Math Reviews)

Title: MRP and squares

Abstract:

Justin Moore’s mapping reflection principle (MRP) seems to capture the consistency strength of PFA, since it implies the failure of square. I present some refinements and extensions of this result. They are due to a variety of authors, and some remain unpublished.