Category Archives: Seminars

Daria Michlik: Symmetric products as cones

Tuesday, January 8, 2019, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Daria Michlik (Cardinal Stefan Wyszynski University in Warsaw)

Title: Symmetric products as cones

Abstract:
(join work with Alejandro Illanes and Veronica Martinez-de-la-Vega)
For a continuum $X$, let $F_n(X)$ be the hyperspace of all nonempty subsets of $X$ with at most $n$-points. The space $F_n(X)$ is called the n’th-symmetric product.
In
A. Illanes, V. Martinez-de-la-Vega, Symmetric products as cones, Topology Appl. 228 (2017), 36–46
it was proved that if $X$ is a cone, then its hyperspace $F_n(X)$ is also a cone.
During my talk I will discuss the converse problem. I will prove that if $X$ is a locally connected curve, then the following conditions are equivalent:
1. $X$ is a cone,
2. $F_n(X)$ is a cone for some $n\ge 2$,
3. $F_n(X)$ is a cone for each $n\ge 2$.

Tomasz Kochanek: Rosenthal’s lemma and its applications

Seminar: Working group in applications of set theory, IMPAN

Thursday, 10.01.2019, 10:15, room 105, IMPAN

Speaker: Tomasz Kochanek (IMPAN/UW)

Title: “Rosenthal’s lemma and its applications ”

Abstact: “In this instructional talk we will recall Rosenthal’s lemma on uniformly bounded sequences of measures and present its several classical applications in the Banach space and vector measures theory. First, we will prove the surprising Nikodym’s uniform boundedness principle and Phillips’ lemma where the application of Rosenthal’s result makes the proofs much easier than the original ones. A few further corollaries of Nikodym’s principle will be mentioned, such as the Dieudonné-Grothendieck theorem on bounded vector measures and the Seever theorem on the range of an operator into a B(Σ)-space. Next, we shall prove two beautiful consequences of Rosenthal’s lemma: the Diestel-Faires theorem and the Orlicz-Pettis theorem. If time allows, we will also briefly discuss their further deep consequences in the structural theory of Banach spaces”.

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

Natasha Dobrinen: Mini-course on Infinitary Ramsey theory

Time and Place: Tuesday, January 8 and Wednesday, January 9  at 10:30am in the KGRC lecture room (both parts) at the KGRC.

Part I.    Topological Ramsey spaces and applications to ultrafilters
Part II.   Ramsey theory on trees and applications to big Ramsey degrees

The Infinite Ramsey Theorem states that given $n,r\ge 1$ and a coloring of
all $n$-sized subsets of $\mathbb{N}$ into $r$ colors, there is an
infinite subset of $\mathbb{N}$ in which all $n$-sized subsets have the
same color.  There are several natural ways of extending Ramsey’s Theorem.
One extension is to color infinite sets rather than finite sets.  In this
case, the Axiom of Choice precludes a full-fledged generalization, but
upon restricting to definable colorings, much can still be said.  Another
way to extend Ramsey’s Theorem is to color finite sub-objects of an
infinite structure, requiring an infinite substructure isomorphic to the
original one.  While it is not possible in general to obtain substructures
on which the coloring is monochromatic, sometimes one can find bounds on
the number of colors, and this can have implications in topological
dynamics.

In Part I, we will trace the development of Ramsey theory on the Baire
space, from the Nash-Williams Theorem for colorings of clopen sets to the
Galvin-Prikry Theorem for Borel colorings, culminating in Ellentuck’s
Theorem correlating the Ramsey property with the property of Baire in a
topology refining the metric topology on the Baire space.  This refinement
is called the Ellentuck topology and is closely connected with Mathias
forcing.  Several classical spaces with similar properties will be
presented, including the Carlson-Simpson space and the Milliken space of
block sequences.  From these we shall derive the key properties of
topological Ramsey spaces, first abstracted by Carlson and Simpson and
more recently given a refined presentation by Todorcevic in his book {\em
Introduction to Ramsey spaces}.  As the Mathias forcing is closely
connected with Ramsey ultrafilters, via forcing mod finite initial
segments, so too any Ramsey space has a $\sigma$-closed version which
forces an ultrafilter with partition properties.  Part I will show how
Ramsey spaces can be used to find general schemata into which disparate
results on ultrafilters can be seen as special cases, as well as obtain
fine-tuned results for structures involving ultrafilters.

Part II will focus on Ramsey theory on trees and their applications to
Ramsey theory of homogeneous structures. An infinite structure is {\em
homogeneous} if each isomorphism between two finite substructures can be
extended to an automorphism of the infinite structure.  The rationals as a
linearly ordered structure and the Rado graph are prime examples of
homogeneous structures.  Given a coloring of singletons in the rationals,
one can find a subset isomorphic to the rationals in which all singletons
have the same color.  However, when one colors pairs of rationals, there
is a coloring due to Sierpinski for which any subset isomorphic to the
rationals has more than one color on its pairsets.  This is the origin of
the theory of {\em big Ramsey degrees}, a term coined by Kechris, Pestov
and Todorcevic, which investigates bounds on colorings of finite
structures inside infinite structures.  Somewhat surprisingly, a theorem
of Halpern and L\”{a}uchli involves colorings of products of trees,
discovered en route to a proof that the Boolean Prime Ideal Theorem is
strictly weaker than the Axiom of Choice, is the heart of most results on
big Ramsey degrees.  We will survey big Ramsey degree results on countable
and uncountable structures and related Ramsey theorems on trees, including
various results of Dobrinen, Devlin, D\v{z}amonja, Hathaway, Larson,
Laver, Mitchell, Shelah, and Zhang.

Natasha Dobrinen: Ramsey Theory of the Henson graphs

Abstract: A central question in the theory of ultrahomogeneous relational structures asks, How close of an analogue to the Infinite Ramsey Theorem does it carry? An infinite structure S is ultrahomogeneous if any isomorphism between two finitely generated substructures of S can be extended to an automorphism of S. We say that S has finite big Ramsey degrees if for each finite substructure A of S, there is a number n(A) such that any coloring of the copies of A in S can be reduced to no more than n(A) colors on some substructure S of S, which is isomorphic to the original S.

The two main obstacles to a fuller development of this area have been lack of representations and general Milliken-style theorems. We will present new work proving that the Henson graphs, the kk-clique free analogues of the Rado graph for k3, have finite big Ramsey degrees. We devise representations of Henson graphs via strong coding trees and prove Milliken-style theorems for these trees. Central to the proof is the method of forcing, building on Harrington’s proof of the Halpern-Läuchli Theorem.

Miha Habic: The ultrapower capturing property (part I)

The seminar meets on Wednesday January 9th 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 I)

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 overview the necessary large cardinal machinery and Cummings’
original argument.

The second part of the talk will take place on Wednesday January 16th.
In the second talk Miha will introduce a forcing notion due to Apter and
Shelah and the modifications necessary to obtain the result.

Miguel Moreno: An introduction to generalized descriptive set theory, part 2

BIU Infinite Combinatorics Seminar

Date : 31/12/2018 – 13:00 – 15:00

Speaker: Miguel Moreno (BIU)

Title : An introduction to generalized descriptive set theory, part 2

Abstract. After introducing the notions of $\kappa$-Borel class, $\kappa$-$\Delta_1^1$ class, $\kappa$-Borel^* class in the previous talk, in this talk, we will show the relation between this classes.
In descriptive set theory the Borel class, the $\Delta_1^1$ class, the Borel* class are the same class, this doesn’t hold in the generalized descriptive set theory, in particular under the assumption V=L the Borel* class is equal to the $\Sigma1^ 1$ class.

Ur Yaar: The Modal Logic of Forcing

HUJI Set Theory Seminar
On Wednesday, December 26, Ur Yaar will talk about the modal logic of forcing.

Title: The Modal Logic of Forcing

 
Abstract: Modal logic is used to study various modalities, i.e. various ways in which statements can be true, the most notable of which are the modalities of necessity and possibility. In set-theory, a natural interpretation is to consider a statement as necessary if it holds in any forcing extension of the world, and possible if it holds in some forcing extension. One can now ask what are the modal principles which captures this interpretation, or in other words – what is the “Modal Logic of Forcing”?
We can also restrict ourselves only to a certain class of forcing notions, or to forcing over a specific universe, resulting in an abundance of questions to be resolved.
We will begin with a short introduction to modal logic, and then present the tools developed by Joel Hamkins and Benedikt Loewe to answer these questions. We will present their answer to the original question, and then move to focus on the class of sigma-centered forcings, which I investigated in my Master’s thesis.

Matt Foreman: Games on weakly compact cardinals

TAU Forcing Seminar

Tuesday, 25/12/18

Speaker: Matt Foreman

Title: Games on weakly compact cardinals

Abstract: Attached.

Assaf Rinot: Hindman’s theorem and uncountable Abelian groups

Colloquium, Hebrew University of Jerusalem

Thu, 20/12/2018 – 14:30 to 15:30

Location:
Manchester Building (Hall 2), Hebrew University Jerusalem

Speaker: Assaf Rinot

Title: Hindman’s theorem and uncountable Abelian groups

Abstract. In the early 1970’s, Hindman proved a beautiful theorem in additive Ramsey theory asserting that for any partition of the set of natural numbers into finitely many cells, there exists some infinite set such that all of its finite sums belong to a single cell. In this talk, we shall address generalizations of this statement to the realm of the uncountable. Among other things, we shall present a negative partition relation for the real line which simultaneously generalizes a recent theorem of Hindman, Leader and Strauss, and a classic theorem of Galvin and Shelah. This is joint work with David Fernandez-Breton.

Damian Sobota: The Josefson–Nissenzweig theorem for Cp(X)-spaces

Seminar: Working group in applications of set theory, IMPAN

Thursday, 20.12.2018, 10:15, room 105, IMPAN

Speaker: Damian Sobota (Kurt Godel RC, Vienna)

Title: “The Josefson–Nissenzweig theorem for Cp(X)-spaces ”

Abstact: “The famous Josefson–Nissenzweig theorem asserts that for every infinite-dimensional Banach space X there exists a sequence (x_n*) in the dual space X* which is weak* convergent to 0 and each x_n* has norm 1. Despite the apparent simplicity of the theorem no constructive proof — even in the case of Banach spaces of continuous functions on compact spaces — has been known.
Recently, Banakh, Śliwa and Kąkol in their studies of separable quotients of topological vector spaces of the form Cp(X), i.e. spaces of continuous functions on Tychonoff spaces endowed with the pointwise convergence topology, have obtained several results characterizing those Cp(X)-spaces for which the Josefson–Nissenzweig theorem holds.
During my talk I will present some introductory facts concerning the theorem for Cp(X)-spaces, show that the existence of “Josefson–Nissenzweig” sequences for Cp(K)-spaces, where K is compact Hausdorff, is strongly related to a variant of the Grothendieck property of Banach spaces, as well as prove that every compact space obtained as a limit of an inverse system consisting only of minimal extensions admits such sequences (and the proof is constructive). This is a joint work with Lyubomyr Zdomskyy”.

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