Category Archives: Seminars

Thilo Weinert: On Order-Types in Polarised Partition Relations

Talk held by Thilo Weinert (KGRC) at the KGRC seminar on 2018-11-08.

Abstract: The history of the polarised partition relation goes back to the original seminal paper by Erdős and Rado from 1956. For the ordinary partition relation after some time one also investigated order-types. For the polarised partition relation however, I am only aware of three papers where order-type played a role and here nothing beyond well-orders ever seems to have been considered.

We are attempting to rectify this. We will present an analogue of a theorem of Jones and a potentially vacuous generalisation of a proposition of Garti and Shelah. Furthermore we will show limits to further generalisations and analogues and will exhibit some open problems.

This is joint work with Lukas Daniel Klausner.

David Fernández-Bretón: Algebraic Ramsey-theoretic results with small monochromatic sets

BIU seminar in Set Theory

November 5, 2018

Speaker: David J. Fernández Bretón (KGRC)

Title: Algebraic Ramsey-theoretic results with small monochromatic sets

Abstract: We will explore some (recent and not so recent; some positive,
some negative) Ramsey-type results (each of which is due to some subset
of the set {Komj\’ath, Hindman, Leader, H.S. Lee, P. Russell, Shelah, D.
Soukup, Strauss, Rinot, Vidnyánszky, myself}) where abelian groups are
coloured, and one attempts to obtain monochromatic sets defined in terms
of the group structure. We will focus specifically on two families of
very recent results: the first one concerns colouring groups with
uncountably many colours, attempting to obtain finite monochromatic
FS-sets; the second one concerns colouring groups (most of the time, our
group of interest is the real line $\mathbb R$ with its usual addition)
with finitely many colours, attempting to obtain countably infinite
monochromatic sumsets.

Robert Passmann: The de Jongh Property for a Subtheory of CZF

Tuesday, November 6, 2018, 10:30
Seminar room N0.003, Mathematical Institute, University of Bonn

Speaker: Robert Passmann (Amsterdam)

Title: The de Jongh Property for a Subtheory of CZF

Abstract:

After recalling intuitionistic logic, Kripke semantic, and (Heyting) algebra-valued models for set theory, we will introduce the notions of loyalty and faithfulness: An algebra-valued model is called loyal if its propositional logic is the propositional logic of its underlying algebra; it is called faithful if all elements of the underlying algebra are truth values of sentences in the language of set theory in the model. We will then analyse the loyalty and faithfulness of a particular construction of Kripke models of set theory due to Iemhoff, and, by using classical models of ZFC set theory and the forcing technique, prove the de Jongh property for the constructive set theory CZF${}^*$ satisfied by Iemhoff’s models, i.e., for any propositional formula $\phi(p_0,…,p_{n-1})$ with propositional letters $p_0,…,p_{n-1}$, it holds that intuitionistic logic IPC proves $\phi(p_0,…,p_{n-1})$ if and only if CZF${}^*$ proves $\phi(\psi_0,…,\psi_{n-1})$ for all set theoretical sentences $\psi_0,…,\psi_{n-1}$. More precisely, we show that CZF${}^*$ has the de Jongh property with respect to every logic that can be characterised by a class of Kripke frames.

George Barmpalias: Compression of data streams down to their information content

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 7 November 2018, 17:00 hrs

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

Speaker: George Barmpalias

Title: Compression of data streams down to their information content

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

Abstract:
According to Kolmogorov complexity, every finite binary string is
compressible to a shortest code – its information content – from
which it is effectively recoverable. We investigate the extent to
which this holds for infinite binary sequences (streams). We devise a
new coding method which uniformly codes every stream X into an
algorithmically random stream Y, in such a way that the first n bits
of X are recoverable from the first I(X[1..n]) bits of Y, where I is any
partial computable information content measure which is defined on all
prefixes of X, and where X[1..n] is the initial segment of X of
length n. As a consequence, if g is any computable upper bound on the
initial segment prefix-free complexity of X, then X is computable from
an algorithmically random Y with oracle-use at most g. Alternatively
(making no use of such a computable bound g) one can achieve an
oracle-use bounded above by K(X[1..n])+log(n). This provides a strong
analogue of Shannon’s source coding theorem for algorithmic
information theory.

William Chen: ANTICHAINS, THE STICK PRINCIPLE, AND A MATCHING NUMBER

Place: Fields Institute (Library)
Date:  September 2, 2018 (13:30-15:00)
Speaker: William Chen
Title: ANTICHAINS, THE STICK PRINCIPLE, AND A MATCHING NUMBER
Abstract: This talk is about some cardinal invariants related to $\omega_1$. The
antichain number is the least cardinal for which there does not exist a
subcollection of that size with pairwise finite intersections, and the
matching number is the least cardinal for which there exists a
subcollection X of that size of order-type $\omega$ subsets of $\omega_1$
so that every uncountable subset of $\omega_1$ has infinite intersection
with a member of X. We explore how these invariants behave in various
forcing extensions. Joint work with Geoff Galgon.

Jing Zhang: A Ramsey theorem for (repeated) sums

Mathematical logic seminar – Oct 30 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Jing Zhang
Department of Mathematical Sciences
CMU

Title:     A Ramsey theorem for (repeated) sums

Abstract:

The motivation is the question: for any finite coloring f: R -> r , does there exist an infinite X such that X + X is monochromatic under f? Hindman, Leader and Strauss showed the answer is negative if CH holds. Komjáth, Leader, Russell, Shelah, Soukup and Vidnyánszky showed the positive answer is consistent relative to the existence of a certain large cardinal. I will demonstrate how to eliminate the use of large cardinals. Other variations of the statement will also be discussed, including some ZFC results.

Aleksandra Kwiatkowska: Infinite permutation groups

Tuesday, October 30, 2018, 10.00
Seminar room N0.003, Mathematical Institute, University of Bonn

Speaker: Aleksandra Kwiatkowska (Münster)

Title: Infinite permutation groups

Abstract:

We discuss several results on infinite permutation groups, that is, closed subgroups of the symmetric group on a countable set, or equivalently, automorphism groups of countable structures. We will focus on ample genericity, where a topological group G has ample generics if for every n, the diagonal conjugacy action of G on Gn has a comeager orbit, on similarity classes, and on topological generators of permutation groups. For example, we show that for a permutation group G, under mild assumptions, for every n and an n-tuple f in G, the countable group generated by f is discrete, or precompact, or the conjugacy class of f is meager. Finally, we will focus on automorphism groups of structures equipped with a definable linear order, such as the ordered random graph, the ordered rational Urysohn metric space, the ordered random poset, the ordered random boron tree, and many other extremely amenable permutation groups. In particular, we give new examples of such groups which have a comeager conjugacy class. This is joint work with Maciej Malicki.

David Uhrik: Composing discontinuous functions

Dear all,

The seminar meets on Wednesday October 31st at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: David Uhrik — Composing discontinuous functions

In this talk we’ll look at composing functions from the Young hierarchy,
which is defined similarly as the well-known Baire hierarchy of
functions but we only consider monotone sequences. It will be shown that
these compositions behave nicely, i.e. the resulting function is again
an element of the hierarchy and its rank is bounded above. On the other
hand we can decompose these functions into a composition of two with
lower rank. In the end I’ll say something about the “failed” attempt to
generalise this hierarchy to convergence according to an ideal.

Best,
David

David Schrittesser: The Ramsey property, MAD families, and their multidimensional relatives

Talk held by David Schrittesser (KGRC) at the KGRC seminar on 2018-10-25.

Abstract: Suppose every set of real numbers has the Ramsey property and “uniformization on Ellentuck-comeager sets” as well as Dependent Choice hold (as is the case under the Axiom of Determinacy, but also in Solovay’s model). Then there are no MAD families. As it turns out, there are also no (Fin x Fin)-MAD families, where Fin x Fin is the two-dimensional Fubini product of the ideal of finite sets. We also comment on higher dimensional products.

All results are joint work with Asger Törnquist.

Arturo Martínez-Celis: On the Michael Space Problem

Seminar: Working group in applications of set theory, IMPAN

Thursday, 25.10. 2018, 10:15, room 105, IMPAN

Speaker: Arturo Martínez-Celis (IM PAN)

Title: “On the Michael Space Problem”

Abstact: “A Lindelöf Topological space is Michael if it has non-Lindelöf product with the space of the irrational numbers. These kind of spaces were introduced by Ernest Michael in 1963 and it is still unknown if one can be constructed in ZFC. We will introduce the notion of Michael ultrafilter, which implies the existence of a Michael space. We will also discuss the relation between this kind of ultrafilters and some classical cardinal invariants and we will use this to study the behaviour of this notion in some models of set theory”.

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