## Sakae Fuchino: Reflection Principles formulated as Löwenheim-Skolem Theorems for stationary logics and the Continuum Problem

Tuesday, January 22, 2019, 15.00
Howard House 4th Floor Seminar Room, University of Bristol

Speaker: Sakae Fuchino (Kobe University )

Title: Reflection Principles formulated as Löwenheim-Skolem Theorems for stationary logics and the Continuum Problem

Abstract:

We give characterizations of variations of Löwenheim-Skolem Theorem for stationary logic (i.e. the logic with monadic second order variables which run over countable subsets of a structure and with the quantifier “there exist stationarily many countable sets such that …”). Löwenheim-Skolem Theorems with reflection cardinal $<\aleph_2$ for this logic and some variants of it are shown to be equivalent either to the Diagonal Reflection Principle (DRP) down to internally club sets introduced by Sean Cox or this type of DRP plus CH.

The Löwenheim-Skolem Theorem for stationary logics with reflection cardinal “$<2^{\aleph_0}$” is not consistent with very large continuum. However, Löwenheim-Skolem Theorem for this logics with reflection cardinal “$<2^{\aleph_0}$” in terms of an internal interpretation of the stationary logic is consistent with the continuum being very large. The Löwenheim-Skolem Theorem for stationary logics with reflection cardinal “$<2^{\aleph_0}$” in terms of stationary logic with an internal ${\mathcal P}_{\kappa}{\lambda}$-interpretation of second order variables even implies that the continuum is (at least) weakly Mahlo.

The results presented in this talk are going to be included in a joint paper with André Ottembreit and Hiroshi Sakai.

## Keita Yokoyama: Ekeland’s variational principle in reverse mathematics

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 23 January 2018, 17:00 hrs

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

Speaker: Keita Yokoyama

Title: Ekeland’s variational principle in reverse mathematics

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

Abstract: Ekeland’s variational principle is a key theorem used in various
areas of analysis such as continuous optimization, fixed point theory and
functional analysis. It guarantees the existence of pseudo minimal
solutions of optimization problems on complete metric spaces. Let f be
a positive real valued continuous (or lower semi-continuous) function
on a complete metric space (X,d). Then, a point x in X is said to be a
pseudo minimum if f(x)=f(y)+d(x,y) implies x=y. Now, Ekeland’s
variational principle says that for any point a in X, there exists a
pseudo minimum x such that f(x)<=f(a)-d(a,x). In reverse
mathematics, it is observed that many theorems for continuous
optimization problems are provable within the system of arithmetical
comprehension (ACA-0), and thus most such problems have arithmetical
solutions. However, this is not the case for pseudo minima. We will
see that Ekeland’s variational principle or its restriction to the
space of continuous functions C([0,1]) are both equivalent to
Pi-1-1-comprehenstion. This is a joint work with Paul Shafer and David
Fernandez-Duque.

## Kaethe Minden: Split Principles and Splitting Families

Dear all,

The seminar meets on Wednesday January 23rd at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Kaethe Minden — Split Principles and Splitting Families

The original split principle is an equivalent formulation of a cardinal
failing to satisfy the combinatorial essence of weak compactness. The
notion was then expanded by Gunter Fuchs and me to characterize the
negation of other large cardinal properties. Split principles give rise
to seemingly new large cardinals, and some new ideals, for example a
normal ideal on $\mathcal P_\kappa \lambda$ in the case of
$\lambda$-Shelahness. In this talk I introduce split principles and
connect them to certain new notions of splitting numbers being large.

Best,
David

## Marcin Sabok: Measurable Hall’s theorem for actions of Z^n

Place: Fields Institute (Room 210)
Date: January 18, 2018 (13:30-15:00)
Speaker: Marcin Sabok
Title: Measurable Hall’s theorem for actions of Z^n
Abstract: In the 1920’s Tarski asked if it is possible to divide the unit square into finitely many pieces, rearrange them by translations and get a disc of area 1. It turns out that this is possible and proved by Laczkovich in the 1990’s. His decomposition, however, used non-measurable pieces and seemed paradoxical. Recently, Grabowski, Mathe and Pikhurko and Marks and Unger showed that such decompositions can be obtained using nice measurable pieces. During the talk, I will discuss a measurable version of the Hall marriage theorem for actions of finitely generated abelian groups. This result implies that for measurable actions of such groups, if two equidistributed measurable sets are equidecomposable, then they are equidecomposalble using measurable pieces. The latter generalizes the measurable circle squaring result by Grabowski, Mathe and Pikhurko. This is joint work with Tomasz Ciesla.

## Moritz Müller: Forcing against bounded arithmetic

Talk held by Moritz Müller (Universitat Politècnica de Catalunya, Barcelona, Spain) at the KGRC seminar on 2019-01-17.

Abstract: We study the following problem. Given a nonstandard model of arithmetic we
want to expand it by a binary relation that does something prohibitive,
e.g. violates the pigeonhole principle in the sense that it is the graph
of a bijection from $n+1$ onto $n$ for some (nonstandard) $n$ in the
model. The goal is to do so while preserving as much as possible of true
arithmetic. More precisely, we want the expansion to model the least
number principle for a class of formulas as large as possible. The problem
is of central importance in bounded arithmetic and propositional proof
complexity. It is not well understood. The talk describes a general method
of forcing to produce such expansions.

## Frank Stephan: Lampligher groups and automata

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 16 January 2018, 17:00 hrs

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

Speaker: Frank Stephan

Title: Lampligher groups and automata

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

Joint work with:
Sanjay Jain, Birzhan Moldagaliyev and Tien Dat Tran.

Abstract:
This talk is about representing lamplighter groups using
computational models from automata theory. It will be shown that
if G can be presented such that the full group operation
is recognised by a transducer, then the same is true for the lampgligher
group of G created by taking the restricted wreath product of
G with the group of integers Z. Furthermore, Cayley presentations,
where only multiplications with constants are recognised by transducers,
are used to study generalised lampglighter groups where one
takes the restricted wreath product of G over a d-dimensional
copy of the integers or the free group with d generators.
Additionally, if G is a finite group then the restricted wreath
product of G over the two-dimensional group of integers is Cayley
tree automatic.

The paper is available at
http://www.comp.nus.edu.sg/~fstephan/transducergroup.ps

## Tomasz Weiss: Accessible points, harmonic measure and the Riemann mapping

Seminar: Working group in applications of set theory, IMPAN

Thursday, 17.01.2019, 10:15, room 105, IMPAN

Speaker: Tomasz Weiss (UKSW)

Title: “Accessible points, harmonic measure and the Riemann mapping”

Abstact: “Let D be a bounded domain in R_n, n larger than 1. We provide an elementary proof that the set of all boundary accessible points of D is an analytic set. We investigate the nature of the set of accessible points of D when n=2 using only set theoretical methods. We provide also a view of the relation between harmonic measure in D, if n=2, D simply connected, and the Riemann mapping of D. In this talk we prove new results and give easier proofs of known results”.

Visit our seminar page which may include information on some future talks at https://www.impan.pl/~set_theory/Seminar/

## Menachem Magidor: Omitting types in the logic of metric structures

HUJI Logic Seminar

16/Jan/2019, 11-13, Ross 63.

Speaker: Menachem Magidor

Title: Omitting types in the logic of metric structures

Abstract.

(joint work with I. Farah)

The logic of metric structures was introduced by Ben Yaacov, Berenstein , Henson and Usvyatsov. It is a version of continuous logic which allows fruitful model theory for many kinds of metric structures. There are many aspects of this logic which make it similar to first order logic, like compactness, a complete proof system, an omitting types theorem for complete types etc. But when one tries to generalize the omitting type criteria to general (non-complete) types the problem turns out to be essentially more difficult than the first order situation. For instance one can have two types (in a complete theory) that each one can be omitted, but they can not be omitted simultaneously.

In the beginning of the talk we shall give a brief survey of the logic of metric structures, so the talk should be accessible also the listeners who are not familiar with the logic of metric structures.

## Miha Habic: The ultrapower capturing property (part II)

Dear all,

The seminar meets on Wednesday January 16th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Miha Habic — The ultrapower capturing property (part II)

In 1993 Cummings showed that it is consistent (relative to large
cardinals) that there is a measurable cardinal kappa carrying a normal
measure whose ultrapower contains the whole powerset of kappa^+. He
showed that nontrivial large cardinal strength was necessary for this,
but it was not clear whether this capturing property had any direct
consequences. Recently Radek Honzík and I showed that it is relatively
consistent that the least measurable cardinal has this capturing
property. We also considered a local version of capturing. In this talk
I will introduce a forcing notion due to Apter and Shelah and the
modifications necessary to obtain our result.

Best,
David

## Arno Pauly: Uniformity aspects of determinacy

Tuesday, January 8, 2019, 15.00
Howard House 4th Floor Seminar Room, University of Bristol

Speaker: Arno Pauly (Swansea University)

Title: Uniformity aspects of determinacy

Abstract:

We consider uniformity aspects of determinacy for some low-level point-classes. The formal framework for this is Weihrauch reducibility, which will be introduced. We distinguish two cases: For games on Cantor space with winning sets from the Hausdorff difference hierarchy, we find that there is a player such that the knowledge that she will win does not help the task of a constructing a winning strategy. This does not hold for open winning sets on Baire space — here knowing who wins the game makes it easier to construct a winning strategy. Open determinacy on Baire space shares all known properties with the perfect tree theorem (a closed subset of Baire space is either countable or contains a perfect subset), but it is an open question whether they are actually equivalent.

The results presented are from joint work with Takayuki Kihara and Alberto Marcone (https://arxiv.org/abs/1812.01549) and with Stephane Le Roux (https://arxiv.org/abs/1407.5587).