
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES 
Set
Theory Seminar Series 201415
Fields Institute, Room 210
Friday 1:30 pm
Organizing
Committee:
Miguel Angel Mota, Ilijas Farah, Juris Steprans, Paul
Szeptycki



2015
Fridays

Upcoming Seminars
at 1:30 p.m. in the Fields Institute, Room 210

January 30 
TBA

201415

Past Seminars
Speaker and Talk Title

January 23, 2015 
David Fernandez
A model of ZFC with strongly summable ultrafilters, small
covering of meagre and large dominating number
Strongly summable ultrafilters are a variety of ultrafilters
that relate with Hindman's finite sums theorem in a way
that is somewhat analogous to that in which Ramsey ultrafilters
relate to Ramsey's theorem. It is known that the existence
of these ultrafilters cannot be proved in ZFC, however such
an existencial statement follows from having the covering
of meagre to equal the continuum. Furthermore, using ultraLaver
forcing in a short finite support iteration, it is possible
to get models with strongly summable ultrafilters and a
small covering of meagre, and these models will also have
small dominating number. Using this ultraLaver forcing in
a countable support iteration to get a model with small
covering meagre and strongly summable ultrafilters is considerably
harder, but it can be done and in this talk I will explain
how (it involves a characterisation of a certain kind of
strongly summable ultrafilter in terms of games). Interesingly,
this way we also get the dominating number equal to the
continuum, unlike the previously described model.

January 16, 2015 
Marcin Sabok
Automatic continuity for isometry groups
We present a general framework for automatic continuity
results for groups of isometries of metric spaces. In particular,
we prove automatic continuity property for the groups of
isometries of the Urysohn space and the Urysohn sphere,
i.e. that any homomorphism from either of these groups into
a separable group is continuous. This answers a question
of Melleray. As a consequence, we get that the group of
isometries of the Urysohn space has unique Polish group
topology and the group of isometries of the Urysohn sphere
has unique separable group topology. Moreover, as an application
of our framework we obtain new proofs of the automatic continuity
property for the group $\mathrm{Aut}([0,1],\lambda)$, due
to Ben Yaacov, Berenstein and Melleray and for the unitary
group of the infinitedimensional separable Hilbert space,
due to Tsankov. The results and proofs are stated in the
language of model theory for metric structures.

December 16,
Tuesday 
*CANCELLED*
BAHEN CENTRE, Room BA6183,
13:3015:00, talk by Neil Hindman. 
December 12, 2015
13:3015:00 
Dilip Raghavan.
Embedding $P(\omega)/FIN$ into the $P$points
We show under $\mathfrak{p}=\mathfrak{c}$ that $P(\omega)/FIN$
can be embedded into the $P$points under RK and Tukey reducibility.

December 5, 2015 
no seminar scheduled due to Dow Conference.

November 28, 2014
12:303:00
Stewart Library

Antonio Avilés
A combinatorial lemma about cardinals $\aleph_n$ and its
applications on Banach spa
The lemma mentioned in the title was used by Enflo and
Rosenthal to show that the Banach space $L_p[0,1]^\Gamma$
does not have an unconditional basis when $\Gamma\geq
\aleph_\omega$. In a joint work with Witold Marciszewski,
we used some variation of it to show that there are no extension
operators between balls of different radii in nonseparable
Hilbert spaces.
Istvan Juhász
Lindelof spaces of small extent are $\omega$resolvable
I intend to present the proof of the following result,
joint with L. Soukup and Z. Szentmiklossy: Every regular
space $X$ that satisfies $\Delta(X) > e(X)$ is $\omega$resolvable,
i.e. contains infinitely many pairwise disjoint dense subsets.
Here $\Delta(X)$, the dispersion character of $X$, is the
smallest size of a nonempty open set in $X$ and $e(X)$,
the extent of $X$, is the supremum of the sizes of all closedanddiscrete
subsets of $X$. In particular, regular Lindelof spaces of
uncountable dispersion character are $\omega$resolvable.

November 21, 2014 
Miodrag Sokic
Functional classes
We consider the class of finite structures with functional
symbols with respect to the Ramsey property.

November 14, 2014 
Martino Lupini
Fraisse limits of operator spaces and the noncommutative
Gurarij space
We realize the noncommutative Gurarij space introduced
by Oikhberg as the Fraisse limit of the class of finitedimensional
1exact operator spaces. As a consequence we deduce that
such a space is unique, homogeneous, universal among separable
1exact operator spaces, and linearly isometric to the Gurarij
Banach space.

November 7, 2014
Stewart Library 
Juris Steprans
The descriptive set theoretic complexity of the weakly
almost periodic functions in the dual of the group algebra
The almost periodic functions on a group G are those functions
F from G to the complex number such that the uniform norm
closure of all shifts of F is compact in the uniform norm.
The weakly almost periodic functions are those for which
the analogous statement holds for the weak topology. The
family of sets whose characteristic functions are weakly
almost periodic forms a Boolean algebra. The question of
when this family is a complete $\Pi^1_1$ set will be examined.

October 31, 2014

Speaker 1 (from 12:30
to 13:30):
Vera Fischer
Definable Maximal Cofinitary Groups and Large Continuum
A cofinitary group is a subgroup of the group of
all permutations of the natural numbers, all nonidentity
elements of which have only finitely many fixed points. A
cofinitary group is maximal if it is not properly contained
in any other cofinitary group. We will discuss the existence
of nicely definable maximal cofinitary groups in the presence
of large continuum and in particular, we will see the generic
construction of a maximal cofinitary group with a $\Pi^1_2$
definable set of generators in the presence of $2^\omega=\aleph_2$.
Speaker 2 (from 13:30 to 15:00):
Menachem Magidor
On compactness for being $\lambda$ collectionwise hausdorff
A compactness property is the statement for a structure
in a given class, if every smaller cardinality substructure
has a certain property then the whole structure has this
property. In this talk we shall deal with the compactness
for the property of a topological space being collection
wise Hausdorff. The space is X is said to be $\lambda$collection
wise Hausdorff ($\lambda$cwH) if every closed discrete
subset of X of cardinality less than $\lambda$ can be separated
by a family of open sets. X is cwH if it is $\lambda$cwH
for every cardinal $\lambda$.
We shall deal with the problem of when $\lambda$cwH implies
cwH, or just when does $\lambda$cwH implies $\lambda^+$cwH.
A classical example of Bing provides for every cardinal
$\lambda$ a space $X_\lambda$ which is $\lambda$cwH but
not $\lambda^+$cwH. So if we hope to get any level of
compactness for the the property of being cwH, we have to
restrict the class of spaces we consider. A fruitful case
is the case where we restrict the local cardinality of the
space. A motivating result is the construction by Shelah
(using supercompact cardinal) of a model of Set Theory in
which a space which is locally countable and which is $\omega_2$cwH
is cwH.
Can the Shelah result be generalized to larger cardinals
, e.g. can you get a model in which for spaces which are
locally of cardinality $\leq \omega_1$ and which are $\omega_3$cwH
are cwH? In general for which pair of cardinals $(\lambda,
\mu)$ we can have models in which a space which is locally
of cardinality $< \mu$ and which is $\lambda$cwH are
$\lambda^+$cwH? In this lecture we shall give few examples
where we get some ZFC theorems showing that for some pairs
$(\lambda, \mu)$ compactness necessarily fails, and cases
of pairs for which one can consistently have compactness
for the property of being cwH.

October 24, 2014

Jordi Lopez Abad
Ramsey properties of embeddings between finite dimensional
normed spaces
Given d=m , let E m,n be the set of all m×d matrices
(a i,j ) such that
(a) ? d j=1 a i,j =1 for every 1=i=m .
(b) max m i=1 a i,j =1 for every 1=j=d .
These matrices correspond to the linear isometric embeddings
from the normed space l d 8 :=(R d ,?·? 8 ) into
l d 8 , in their unit bases.
We will discuss and give (hints of) a proof of the following
new approximate Ramsey result:
For every integers d , m and r and every e>0 there exists
n such that for every coloring of E d,n into r many colors
there is A?E m,n and a color i<r such that A·E
d,m ?(c 1 (i)) e . Its proof uses the GrahamRothschild
Theorem on partitions of finite sets. We extend this result,
first for embeddings between \emph{polyhedral} normed spaces,
and finally for arbitrary finite dimensional normed spaces
to get the following:
For every finite dimensional normed spaces E and F , every
?>1 and e>0 , and every integer r , there is some
n such that for every coloring of Emb ? 2 (F,l n 8 ) into
r many colors there is T?Emb ? (G,l n 8 ) and some color
i<r such that T°Emb ? (F,G)?(c 1 (i)) ? 2 1+e .
As a consequence, we obtain that the group of linear isometries
of the Gurarij space is extremely amenable. A similar result
for positive isometric embeddings gives that the universal
minimal flow of the group of affine homeomorphisms of the
Poulsen simplex is the Poulsen simplex itself.
This a joint work (in progress) with Dana Bartosova (University
of Sao Paulo) and Brice Mbombo (University of Sao Paulo)

October 17, 2014

Speaker 1 (from 12:30 to 13:30):
Dana Bartosova
Finite Gowers' Theorem and the Lelek fan
The Lelek fan is a unique nondegenrate subcontinuum of
the Cantor fan with a dense set of endpoints. We denote
by $G$ the group of homeomorphisms of the Lelek fan with
the compactopen topology. Studying the dynamics of $G$,
we generalize finite Gowers' Theorem to a variety of operations
and show how it applies to our original problem. This is
joint work with Aleksandra Kwiatkowska.
Speaker 2 (from 13:30 to 15:00):
Assaf Rinot
Productivity of higher chain condition
We shall survey the history of the study of the productivity
of the kcc in partial orders, topological spaces, and Boolean
algebras. We shall address a conjecture that tries to characterize
such a productivity in Ramseytype language. For this, a
new oscillation function for successor cardinals, and a
new characteristic function for walks on ordinals will be
proposed and investigated.

October 10

Sheila Miller
Critical sequences of ranktorank embeddings and a tower
of finite left distributive algebras
In the early 1990's Richard Laver discovered a deep and
striking correspondence between critical sequences of ranktorank
embeddings and finite left distributive algebras on integers.
Each $A_n$ in the tower of finite algebras can be defined
purely algebraically, with no reference to the elementary
embeddings, and yet there are facts about the Laver tables
that have only been proven from a large cardinal assumption.
We present here some of Laver's foundational work on the
algebra of critical sequences of ranktorank embeddings
and some work of the author's, describe how the finite algebras
arise from the large cardinal embeddings, and mention several
related open problems.

October 3

Ilijas Farah
Omitting types in logic of metric structures is hard
One of the important tools for building models with prescribed
secondorder properties is the omitting types theorem. In
logic of metric structures omitting types is much harder
than in classical firstorder logic (it is Pi11 hard).
Although the motivation for this work comes from C*algebras,
the talk will mostly be on descriptive set theory. The intended
takeaway from the talk is "logic of metric structures
blends with descriptive set theory beautifully." This
is joint work with Menachem Magidor.

September 26

No seminar

September 19

Daniel Soukup
Trees, ladders and graphs
The chromatic number of a graph $G$ is the least (cardinal)
number $\kappa$ such that the vertices of $G$ can be covered
by $\kappa$ many independent sets. A fundamental problem
of graph theory asks how large chromatic number affects
structural properties of a graph and in particular, is it
true that a graph with large chromatic number has certain
obligatory subgraphs? The aim of this talk is to introduce
a new and rather flexible method to construct uncountably
chromatic graphs from non special trees and ladder systems.
Answering a question of P. Erdos and A. Hajnal, we construct
graphs of chromatic number $\omega_1$ without uncountable
infinitely connected subgraphs.

September 12

Konstantinos Tyros
A disjoint union theorem for trees
In this talk we will present an infinitary disjoint union
theorem for level products of trees. An easy consequence
of the dual Ramsey theorem due to T.J. Carlson and S.G.
Simpson is that for every Suslin measurable finite coloring
of the power set of the natural numbers, there exists a
sequence $(X_n)_{n\in\mathbb{N}}$ of disjoint nonempty
subsets of $\mathbb{N}$ such that the set
\[\Big\{\bigcup_{n\in Y}X_n:\; Y\;\text{nonempty subset
of }\mathbb{N}\Big\}\]
is monochromatic. The result that we will present is of
this sort, where the underline structure is the level product
of a finite sequence of uniquely rooted and finitely branching
trees with no maximal nodes of height $\omega$ instead of
the natural numbers.
As it is required by the proof of the above result, we develop
an analogue of the infinite dimensional version of the HalesJewett
Theorem for maps defined on a level product of trees, which
we will also present, if time permits.

September 5 
Diana Ojeda
Finite forms of Gowers' Theorem on the oscillation stability
of $c_0$
We give a constructive proof of the finite version of Gowers'
$FIN_k$ Theorem and analyze the corresponding upper bounds.
The $FIN_k$ Theorem is closely related to the oscillation
stability of $c_0$. The stabilization of Lipschitz functions
on arbitrary finite dimensional Banach spaces was proved
well before by V. Milman. We compare the finite $FIN_k$
Theorem with the Finite Stabilization Principle found by
Milman in the case of spaces of the form $\ell_{\infty}^n$,
$n\in N$, and establish a much slower growing upper bound
for the finite stabilization principle in this particular
case.

August 29

Seminar cancelled

August 22

Mike Pawliuk
Various types of products of Fraisse Classes, various types
of amenability and various types of preservation results.
This is joint work with Miodrag Sokic.
In a recent paper of Jakub Jasinski, Claude Laflamme, Lionel
Nguyen Van Thé and Robert Woodrow (Arxiv: 1310.6466)
it was shown that certain Fraisse Classes are actually Ramsey
classes. For many of those cases we have determined whether
their automorphism groups are extremely amenable or not.
Some of these spaces turn out to actually be a special type
of product of Fraisse classes. We were able to prove that
unique ergodicity (a type of amenability) is preserved under
this type of product.

August 1 
Martino Lupini
Functorial complexity of Polish and analytic groupoids
I will explain how one can generalize the theory of Borel
complexity from analytic equivalence relations to groupoids
by means of the notion of Borel classifying functor. This
framework allows one to capture the complexity of classifying
the objects of a category in a functorial way. I will then
present the first results relating the functorial complexity
of a groupoid and the complexity of its associated orbit
equivalence relation, focusing on the case of Polish groupoids:
For Polish groupoids with essentially treeable equivalence
relations any Borel reduction between the orbit equivalence
relations extends to a Borel classifying functor. On the
other hand for any countable nontreeable equivalence relation
E there are Polish groupoids of different functorial complexity
both having E as associated orbit equivalence relation.
The proof of these results involves a generalization of
some fundamental results on the descriptive set theory of
actions of Polish groups such as the BeckerKechris theorem
on Polishability of Borel Gspaces to actions of Polish
groupoids.

July 25 
Saeed Ghasemi
An analogue of FefermanVaught theorem for reduced
products of metric structures
I will give a metric version of the FefermanVaught theorem
for reduced products of discrete spaces. We will use this
to show that, under the continuum hypothesis, the reduced
powers of any metric structure over atomless layered ideals
are isomorphic. As another application, I will give an example
of two reduced products of sequences of matrix algebras
over Fin, which are elementarily equivalent, therefore isomorphic
under the CH, with no trivial isomorphisms between them.

June 27 
Christopher Eagle
Model theory of abelian real rank zero C*algebras
We consider algebras of the form $C(X)$, where $X$ is a
$0$dimensional compact Hausdorff space, from the point
of view of continuous model theory. We characterize these
algebras up to elementary equivalence in terms of invariants
of the Boolean algebra $CL(X)$ of clopen subsets of $X$.
We also describe several saturation properties that $C(X)$
may have, and relate these to topological properties of
$X$ and saturation of $CL(X)$. We will discuss some consequences
of saturation when we view $C(X)$ as a $C^*$algebra. All
the necessary background on continuous logic will be provided.
This is joint work with Alessandro Vignati.

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