## Michael Hrusak: Katetov order

This is an inaugural Set Theory Seminar in Mexico City, a (possibly temporary) branch of the Morelia Topology and Set Theory seminar.

## Alexander Kreuzer: On the Uniform Computational Content of Computatibility Theory

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 2 September 2015, 17:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: Alexander Kreuzer

Title: On the Uniform Computational Content of Computatibility Theory

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

Abstract:
We demonstrate that the Weihrauch lattice can be used to study the
uniform computational content of computability theoretic properties
and theorems in one common setting.
The properties that we study include diagonal non-computability,
hyperimmunity, complete extensions of Peano arithmetic, 1-genericity,
Martin-Loef randomness and cohesiveness. The theorems that we include
in our case study are the Low Basis Theorem of Jockusch and Soare, the
Kleene-Post Theorem and Friedman’s Jump Inversion Theorem. It turns
out that all the aforementioned properties and many theorems in
computability theory, including all theorems that claim the existence
of some Turing degree, have very little uniform computational content.
They are all located outside of the upper cone of binary choice (also
known as LLPO) and we call problems with this property
indiscriminative. Since practically all theorems from classical
analysis whose computational content has been classified are
discriminative, our observation could yield an explanation for why
theorems and results in computability theory typically have very
little direct consequences in other disciplines such as analysis.

Two notable exceptions to this are the Low Basis Theorem which is
discriminative, this is perhaps why it is considered to be one of the
most applicable theorems in computability theory, and the Baire
category theorem (or to be precise certain formulations of it) which
is an indiscriminative principle occurring in mathematical
analysis. We will see that the Baire category theorem is related to
1-genericity.

In some cases a bridge between the indiscriminative world and the
discriminative world of classical mathematics can be established via a
suitable residual operation and we demonstrate this in case of the
cohesiveness problem, which turns out to be the quotient of two
discriminative problems, namely the limit operation and the jump of
Weak Koenig’s Lemma.

This is joint work with Vasco Brattka and Matthew Hendtlass.

## Recent Developments in Axiomatic Set Theory

Dates

Wednesday, September 16, 2015, 14:00 p.m. – Friday, September 18, 2015, 12:00 p.m.
Venue
Room 420, RIMS Kyoto
Organizer
Masahiro Shioya (University of Tsukuba), E-mail: shioya _at_ math.tsukuba.ac.jp

# Program (as of August 21)

September 16 (Wed.)
14:20-14:50 Masaru Kada and Takuto Kato: Variants of AC under ZF minus union
15:10-15:40 Masaru Kada and Souji Shizuma: Some remarks on in nite hat guessing games
16:00-16:50 Makoto Takahashi: On non -shortness of Axiom A posets with frame systems
September 17  (Thu.)
10:00-10:50 Teruyuki Yorioka: Some consistency results with the existence of a non special Aronszajn tree
11:10-12:00 David Chodounsky: F-Mathias reals and generic filters
14:00-14:50 Joel David Hamkins: Upward closure in the generic multiverse of a countable model of set theory
15:10-16:00 Toshimichi Usuba: Set-theoretic geology and large large cardinals
16:20-17:00 Tadatoshi Miyamoto: Side condition methods and morasses

September 18 (Fri.)
9:00-9:50 Sakae Fuchino: TBA
10:10-11:00 Hiroshi Sakai: TBA

## Franklin Tall: PFA(S)[S] and countably compact spaces, continued

Place: Fields Institute (Room 210)

Date: August 28th, 2015 (13:30-15:00)

Speaker: Franklin Tall

Title: PFA(S)[S] and countably compact spaces, continued

Abstract:

We continue with Alan Dow’s proof that PFA(S)[S] implies every first countable perfect pre-image of $\omega_1$ includes a copy of $\omega_1$.

## Frank Stephan: Covering the Recursive Sets

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 19 August 2015, 17:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: Frank Stephan

Title: Covering the Recursive Sets

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

The talk gives an overview of recent work which solves two problems of
Brendle, Brooke-Taylor, Ng and Nies. One construction is based on a
construction of Khan and Miller from 2014, the other construction is a
direct construction utilising martingales. Furthermore, the talk
introduces the notions of infinitely often subuniform families and
shows first results for these; these concepts also permit to get some
result towards the solution of the third problem of Brendle,
Brooke-Taylor, Ng and Nies.

## Franklin Tall: PFA(S)[S] and countably compact spaces

Place: Fields Institute (Room 210)

Date: August 14th, 2015 (13:30-15:00)

Speaker: Franklin Tall

Title: PFA(S)[S] and countably compact spaces

Abstract:

We will continue with Dow’s proof and related material.

## Borisa Kuzeljevic: Isomorphic Substructures of Fraisse Limits

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 12 August 2015, 17:00 hrs

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

Speaker: Borisa Kuzeljevic

Title: Isomorphic Substructures of Fraisse Limits

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

We will present some results on embedding linear orders into the
posets of isomorphic substructures of Fraisse limits. For a relational
structure X we denote this poset by P(X).
Because each chain in a poset can be extended to a maximal one, it is
enough to characterize the class M(X) of order types of all
maximal chains in P(X). For example, if the structure
X is either the rational line, the random k-uniform
hypergraph, the random poset or some of the Henson graphs, then a linear
order belongs to the class M(X) if and only if it is isomorphic
to the order type of a compact set of reals whose minimum is not isolated.
We will also show that our construction is applicable to any
Fraisse structure whose age satisfies the strong amalgamation
property. This is a joint work with Milos Kurilic.

## Dana Bartošová: On the structure of the universal minimal flows via (near) ultrafilters

Place: Fields Institute (Room 210)

Date: August 7th, 2015 (13:30-15:00)

Speaker: Dana Bartošová

Title: On the structure of the universal minimal flows via (near) ultrafilters

Abstract:

I will recall how we can see universal minimal flows as spaces of (near) ultrafilters and which results have been obtained by this approach. Then I will focus on problems whose solution I believe to make a good use of this viewpoint, in particular, the unique amenability and unique ergodicity problems, metrizability of universal minimal flows or the Ellis problem.

## Franklin Tall: PFA(S)[S] and countably compact spaces, II

Place: Fields Institute (Room 210)

Date: July 31st, 2015 (13:30-15:00)

Speaker: Franklin Tall

Title: PFA(S)[S] and countably compact spaces, II

Abstract:

We give some more applications and then start going through the details of Alan’s proof. It is not necessary to have attended my previous talk in order to understand this one.

## Maxim Burke: Products of derived structures on topological spaces

Place: Fields Institute (Room 210)

Date: July 24th, 2015 (13:30-15:00)

Speaker: Maxim Burke

Title: Products of derived structures on topological spaces

Abstract:

We consider topological spaces X equipped with an algebra A of subsets of X and an ideal I of A. Motivated by the example of the Jordan measurable subsets of R, we consider the “derived structure” obtained by replacing A by the algebra of elements of A whose boundaries belong to I, and I by its intersection with that algebra.

In previous joint work with N. D. Macheras and W. Strauss, these derived structures were classified (under some assumptions) and densities were computed for them. In more recent work with the same co-authors, we extend that work in the context of products of derived structures. We examine when product operations preserve densities and other types of liftings. We will discuss some topological and set-theoretic issues that arise in the context of this work.