2014
Fridays

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



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.

201314

Past Seminars
Speaker and Talk Title

June 13 
no seminar 
June 6 
no seminar

May 30 
Asger Törnquist
Statements that are equivalent to CH and their Sigma12
counterparts.
There is a large number of "peculiar"
statements that have been shown over time to be equivalent
to the Continuum Hypothesis, CH. For instance, a wellknown
theorem of Sierpinski says that CH is equivalent to the statement
that the plane can be covered by countably many graphs of
functions (countably many of which are functions of x, and
countably many of which are functions of y.) What happens
if we consider the natural Sigma12 analogues of these statements
(in the sense of descriptive set theory)? It turns out that
then these statements are, in a surprising number of cases,
equivalent to that all reals are constructible. In this talk
I will give many examples of this phenomenon, and attempt
to provide an explanation of why this occurs. This is joint
work with William Weiss.

May 23 
Martino Lupini.
The LopezEscobar theorem for metric structures and the
topological Vaught conjecture.
I will present a generalization of the classical LopezEscobar
theorem to the logic for metric structures. As an application
I will provide a modeltheoretic reformulation of the topological
Vaught conjecture. This is joint work with Samuel Coskey.

May 16 
no seminar 
May 9 
Frank Tall
What I am working on
It might be useful, especially to grad students just starting
their research, if we had talks on the theme of "what
I am working on", rather than waiting for all of the
theorems to be proved and the presentation polished. In
view of the fact that no one else wants to talk tomorrow,
I am willing to give such a talk. My first topic grew outof
Marion Scheepers' talk a few weeks ago, and concerns the
question, due to Gruenhage and Ma, of whether, in the compactopen
topology, the space of continuous realvalued functions
on a locally compact normal space satisfies the Baire Category
Theorem. I have several consistency results using PFA(S)[S],
but am trying to settle the question in ZFC. The second
topic also concerns PFA(S)[S] (which you do not have to
know to understand my talk). I had characterizations under
PFA(S)[S] of paracompactness in locally compact normal spaces
that required the absence of perfect preimages of omega_1;
Together with Alan Dow, I have shown some of those characterizations
can be improved to just require the absence of copies of
omega_1, but others cannot. Some of this work requires an
interesting but difficult PFA(S)[S] proof of Dow that I
shall eventually present in the seminar.

May 2 
Micheal Pawliuk
Packing Hedgehogs densely into l_2 to give a trivial Gcompactification
In the early 80s Smirnov asked if every regular Gspace
admits an equivariant Gcompactification. In 1988 Megrelishvili
exhibited a Gspace that does not (essentially the metrizable
hedgehog with a nice group action). His example still leaves
open the larger question of if a regular Gspace can have
a *trivial* Gcompactification. In joint work with Pestov
and Bartosova, we will give such an example by finding many
copies of the metrizable hedgehog inside l_2.

April 25

Alessandro Vignati
An algebra whose subalgebras are characterized by density
A longstanding open problem is whether or not every amenable
operator algebra is isomorphic to a C*algebra. In a recent
paper, Y. Choi, I. Farah and N. Ozawa provided a non separable
counterexample. After an introduction, building on their
work and using the full power of a Luzin gap, we provide
an example of an amenable operator algebra A such that every
amenable nonseparable subalgebra of A is not isomorphic
to a C*algebra, while some "reasonable" separable
subalgebras are. In the end we describe some interesting
property of the constructed object related to the KadisonKastler
metric.

April 11 
Marion Scheepers
Box powers of Baire spaces.
A topological space is a Baire space if any countable sequence
of dense open subsets has a non empty intersection. In this
talk we discuss an elegant (consistent module large cardinals)
characterization of spaces that have the Baire property
in all powers, considered in the box topology.

April 4 
David Fernandez
Two microcontributions to the theory of Strongly Summable
Ultrafilters
Strongly Summable Ultrafilters are those generated by FSsets
(where FS(X) is the set of all possible sums of finitely
many elements from X (you can only add each element once)).
I will show two little results (with nice little neat proofs!)
about these: first, that every strongly summable ultrafilter
on the countable Boolean group is rapid. Second, that there
is a model where strongly summable ultrafilters (on any
abelian group really, but without loss of generality on
the countable Boolean group) exist yet Martin's axiom for
countable forcing notions fails (up until now, these ultrafilters
were only known to exist under this hypothesis).

March 28 
Konstantinos Tyros
Primitive recursive bounds for the finite version of Gowers'
$c_0$ theorem, Talk 2
In this talk we will present proofs for the finite version
of Gowers' $c_0$ theorem for both the positive and the general
case providing primitive recursive bounds. Multidimensional
versions of these result will be presented too.

March 21 
Konstantinos Tyros
Primitive recursive bounds for the finite version of Gowers'
$c_0$ theorem, Talk 1
In this talk we will present proofs for the finite version
of Gowers' $c_0$ theorem for both the positive and the general
case providing primitive recursive bounds. Multidimensional
versions of these result will be presented too.

March 14 
Tomasz Kania.
A chain condition for operators from C(K)spaces
Building upon work of Pelczynski, we introduce a chain
condition, defined for operators acting on C(K)spaces,
which is weaker than weak compactness. We prove that if
K is extremely disconnected and X is a Banach space then
an operator T : C(K) > X is weakly compact if and only
if it satisfies our condition and this is if and only if
the representing vector measure of T satisfies an analogous
chain condition on Borel sets of K. As a tool for proving
the abovementioned result, we derive a topological counterpart
of Rosenthal's lemma. We exhibit classes of compact Hausdorff
spaces K for which the identity operator on C(K) satisfies
our condition, for instance every class of compact spaces
that is preserved when taking closed subspaces and Hausdorff
quotients, and which contains no nonmetrisable linearly
ordered space (like the classes of Eberlein spaces, Corson
compact spaces etc.) serves as an example. Using a Ramseytype
theorem, due to Dushnik and Miller, we prove that the collection
of operators on a C(K)space satisfying our condition forms
a closed left ideal of B(C(K)), however in general, it does
not form a right ideal. This work is based on two papers
(one joint with K. P. Hart and T. Kochanek and the second
one joint with. R. Smith).

March 7 
Juris Steprans
Nontrivial automorphisms of $P(\omega_1)/fin$
Just as in the case of automorphisms of $P(\omega)/fin$,
an automorphism of $P(\omega_1)/fin$ will be called trivial
if it is induced by a bijection between cofinite subsets
of $\omega_1$. Since a nontrivial automorphism of $P(\omega)/fin$
can easily be extended to a nontrivial automorphism of
$P(\omega_1)/fin$ there is little interest examining the
existence of nontrivial automorphisms of $P(\omega_1)/fin$
without further restrictions. So, an automorphism of $P(\omega_1)/fin$
will be called really nontrivial if it is nontrivial,
yet its restriction to any subalgebra of the form $P(X)/fin$
is trivial when $X$ is countable. It will be shown to be
consistent with set theory that there is a really nontrivial
automorphism of $P(\omega_1)/fin$.
This is joint work with Assaf Rinot.

February 28 
Daniel Soukup
Daviestrees in infinite combinatorics
The aim of this talk is to introduce Daviestrees and present
new applications to combinatorics. Daviestrees are special
sequences of countable elementary submodels which played
important roles in generalizing arguments using CH to pure
ZFC proofs. My goal is to present two unrelated but fascinating
results due to P. Komjáth: we prove that the plane
is the union of n+2 "clouds" provided that the
continuum is at most $\aleph_n$ and that every uncountably
chromatic graph contains kconnected uncountably chromatic
subgraphs for each finite k. We hopefully have time to review
the most important open problems around the second theorem.

February 21
Room 230
*Please note room change 
Mohammed Bekkali
An overview of Boolean Algebras over partially ordered sets
Being at crossroads between Algebra, Topology, Logic, Set
Theory and the Theory of Order; the class of Boolean Algebras
over partially ordered sets offers more flexibility in representing
no zero elements and describing Stone spaces. Some constructions
and their interconnections will be discussed, motivating
along the way a list of open problems.

February 14 
Stevo Todorcevic
A new partition theorem for tress and is applications (Part
II)
In a recent joint work with Antonio Aviles, in order to
classify ktuples of analytic hereditary families of subsequences
of a fixed sequence of objects ( vectors, points in a topological
space,etc), we needded to come up with a new Ramsey theorem
for trees. The lecture will concentrate on stating the result
and, if time permits, on giving some ideas from the proof.

February 7 
Stevo Todorcevic
A new partition theorem for tress and is applications (Part
I)
In a recent joint work with Antonio Aviles, in order to
classify ktuples of analytic hereditary families of subsequences
of a fixed sequence of objects ( vectors, points in a topological
space,etc), we needded to come up with a new Ramsey theorem
for trees. The lecture will concentrate on stating the result
and, if time permits, on giving some ideas from the proof.

January 31 
Dana Bartosova
Lelek fan from a projective Fraïssé limit
The Lelek fan is the unique subcontinuum of the Cantor
fan whose set of endpoints is dense. The Cantor fan is the
cone over the Cantor set, that is $C\times I/\sim,$ where
$C$ is the Cantor set, $I$ is the closed unit interval and
$(a,b)\sim (c,d)$ if and only if either $(a=c$ and $b=d)$
or $(b=d=0)$. We construct the Lelek fan as a
natural quotient of a projective Fra\"iss\'e limit
and derive some properties of the Lelek fan and its homeomorphism
group. This is joint with Aleksandra Kwiatkowska.

January 17 
Miguel Angel Mota
Baumgartner's Conjecture and Bounded Forcing Axioms (Part
I)
Using some variants of weak club guessing we separate some
fragments of the proper forcing axiom: we show that for
every two indecomposable ordinals $\alpha < \beta$, the
forcing axiom for the class of all the $\beta$proper posets
does not imply the bounded forcing axiom for the class of
all the $\alpha$proper posets.

January 10 
Rodrigo Hernandez
Wijsman hyperspaces of nonseparable metric spaces
The hyperspace CL(X) of a topological space X (at least
T1) is the set of all nonempty closed subsets of X. The
usual choice for a topology in CL(X) is the Vietoris topology,
which has been widely studied. However, in this talk we
will consider the Wijsman topology on CL(X), which is defined
when (X,d) is a metric space. The Wijsman topology is coarser
than the Vietoris topology and in fact it depends on the
metric d, not just on the topology. The problem we will
address is that of normality of the Wijsman hyperspace.
It is known since the 70s that the Vietoris hyperspace is
normal if and only if X is compact. But a characterization
of normality of the Wijsman hyperspace is still not known.
It is conjectured that the Wijsman hyperspace if normal
if and only if the space X is separable. Jointly with Paul
Szeptycki, we have proved that if X is locally separable
and of uncountable weight, then the Wijsman hyperspace is
not normal.

December 13 
Martino Lupini.
The descriptive set theory of Polish groupoids
I will present an overview of functorial classification
within the framework of invariant descriptive set theory,
based on the notion of Polish groupoid and Borel classifying
functor. I will then explain how several results about Polish
group actions admit natural generalizations to Polish groupoids,
extending works of BeckerKechris, Effros, Hjorth, and Ramsay.

Dec. 06 
Konstantinos Tyros
An infinitary version of the FurstenbergWeiss Theorem.
In 2003 H. Furstenberg and B. Weiss obtained a
far reaching extension of the famous Szemer\'edi's theorem
on arithmetic progressions. They establish the existence of
finite strong subtrees of arbitrary height, having an arithmetic
progression as a level set, inside subsets of positive measure
of a homogeneous tree. In this talk an infinitary version
of their result will be presented.

Nov. 29 
Jan Pachl
Onepoint DTC sets for convolution semigroups
Every topological group G naturally embeds in the Banach
algebra LUC(G)*. The topological centre of LUC(G)* is defined
to be the set of its elements for which the left multiplication
is w*w*continuous. Although the definition demands continuity
on the whole algebra, for a large class of topological groups
it is sufficient to test the continuity of the left multiplication
at just one suitably chosen point; in other words, the algebra
has a onepoint DTC (Determining Topological Centre) set.
More generally, the same result holds for many subsemigroups
of LUC(G)*. In particular, for G in the same large class,
the uniform compactification (the greatest ambit) of G has
a onepoint DTC set. These results, which generalize those
previously known for locally compact groups, are from joint
work with Stefano Ferri and Matthias Neufang.

Nov. 15
**Note
Revised Location:
Stewart Library

Piotr Koszmider (Talk 1 from 14:00 to 15:00)
Independent families in Boolean algebras with some separation
properties
We prove that any Boolean algebra with the subsequential
completeness property contains an independent family of
size continuum. This improves a result of Argyros from the
80ties which asserted the existence of an uncountable independent
family. In fact we prove it for a bigger class of Boolean
algebras satisfying much weaker properties. It follows that
the Stone spaces of all such Boolean algebras contains a
copy of the CechStone compactification of the integers
and the Banach space of continuous functions on them has
linfinity as a quotient. Connections with the Grothendieck
property in Banach spaces are discussed. The talk is based
on the paper: Piotr Koszmider, Saharon Shelah; Independent
families in Boolean algebras with some separation properties;
Algebra Universalis 69 (2013), no. 4, 305  312.
Jordi Lopez Abad (Talk 2 from 15:30 to 16:30)
Unconditional and subsymmetric sequences in Banach spaces
of high density
We will discuss bounds and possible values for the minimal
cardinal number $\kappa$ such that every Banach space of
density $\kappa$ has an unconditional basic sequence, or
the corresponding cardinal number for subsymmetric basic
sequences.

Nov. 8 
Ilijas Farah.
The other KadisonSinger problem.
In their famous 1959 paper Kadison and Singer posed two
problems. The famous one was recently solved by Marcus,
Spielman and Srivastava, using work of Weaver. The other
(much more settheoretic) KadisonSinger
problem was resolved using the Continuum Hypothesis by Akemann
and Weaver in 2008. This assumption was weakened to Martin's
Axiom by myself and Weaver, but the question remains whether
the answer is independent from ZFC.

Nov. 1 
no seminar

Oct. 18 
Lionel Nguyen Van
Structural Ramsey theory and topological dynamics for automorphism
groups of homogeneous structures
In 2005, Kechris, Pestov, and Todorcevic established a
striking connection between structural Ramsey theory and
the topological dynamics certain automorphism groups. The
purpose of this talk will be to present this connection,
together with recent related results.

Oct. 11 
Eduardo Calderon
Asymptotic models and plegma families
We will discuss one of the usual ways in which Ramsey's
theorem is applied to the study of Banach space geometry
and then, by means of techniques closely following ones
first developed by S. Argyros, V. Kanellopoulos, K. Tyros,
we will introduce the concept of an asymptotic model of
higher order of a Banach space and establish a relationship
between these and higher order spreading models that extends
their result of the impossibility of always finding a finite
chain of spreading models reaching an $l_p$ space to the
context of weakly generated asymptotic models.

Oct. 4 
David Fernandez
Strongly Productive Ultrafilters
The concept of a Strongly Productive Ultrafilter on a semigroup
(known as a "strongly summable ultrafilter" when
the semigroup is additively denoted) constitute an important
concept ever since Hindman defined it, while trying to prove
the theorem that now bears his name. In a 1998 paper of
Hindman, Protasov and Strauss, it shown that strongly productive
ultrafilters on abelian groups are always idempotent, but
no further generalization of this fact had been made afterwards.
In this talk I will show (at least the main ideas, anyway)
the proof that this result holds on a large class of semigroups,
which includes all solvable groups and the free semigroup,
among others. After that, I'll discuss a special class of
strongly productive ultrafilters on the free semigroup (dubbed
"very strongly productive ultrafilters" by N.
Hindman and L. Jones), and show that they have the "trivial
products property". This means that (thinking of the
free semigroup S as a subset of the free group G) if p is
a very strongly productive ultrafilter on S, and q,r are
nonprincipal ultrafilters on G such that $qr=p$, then there
must be an element x of G such that $q=px$ and $r=x^{1}p$.
This answers a question of Hindman and Jones. Joint work
with Martino Lupini

Sept. 27 
Stevo Todorcevic
A construction scheme on $\omega_{1}$
We describe a simple and general construction scheme for
describing mathematical structures on domain $\omega_{1}$.
Natural requirements on this scheme will reduce the nonseparable
structural properties of the resulting mathematical object
to some finitedimensional problems that are easy to state
and frequently also easy to solve. The construction scheme
is in fact quite easy to use and we illustrate this by some
application mainly towards compact convex spaces and normed
spaces.

Sept. 20 
Rodrigo Hernandez
Countable dense homogeneous spaces
A separable space X is countable dense homogeneous (CDH)
if every time D and E are countable dense subsets of X,
there exists a homeomorphism $h:X\to X$ such that $h[D]=E$.
The first examples of CDH spaces were Polish spaces. So
the natural open question was whether there exists a CDH
metrizable space that is not Polish. By a characterization
result by Hrusak and ZamoraAviles, such a space must be
non Borel. In this talk, we will focus on recent progress
in this direction. In fact, we only know about two types
of CDH nonBorel spaces: nonmeager Pfilters (with the
Cantor set topology) and $\lambda$sets. Moreover, by arguments
similar to those used for the CDH $\lambda$set, it has
also been possible to construct a compact CDH space of uncountable
weight.

Sept. 13

Daniel Soukup
Monochromatic partitions of edgecolored infinite graphs
Our goal is to find well behaved partitions of edgecolored
infinite graphs following a long standing trend in finite
combinatorics started by several authors including P. Erdos
and R. Rado; in particular, we are interested in partitioning
the vertices of complete or nearly complete graphs into
monochromatic paths and powers of paths. One of our main
results is that for every 2edgecoloring of the complete
graph on $\omega_1$ one can partition the vertices into
two monochromatic paths of different colors. Our plan for
the talk is to review some results from the literature (both
on finite and infinite), sketch some of our results and
the ideas involved and finally present the great deal of
open problems we facing at the moment. This is a joint work
with M. Elekes, L. Soukup and Z. Szentmiklóssy.

Aug 30 
no seminar 
Wed.
Aug 28
3:00 p.m.
BA6180

Connor Meehan
Infinite Games and Analytic Sets
In the context of set theory, infinite games have been
studied since the mid20th century and have created an interesting
web of connections, such as with measurable cardinals. Upon
specifying a subset A of sequences of natural numbers, an
infinite game G(A) involves two players alternately choosing
natural numbers, with player 1 winning in the event that
the resulting sequence x is in A. We will give proofs of
Gale and Stewart's classic results that any open subset
A of Baire space leads to the game G(A) being determined
(i.e. one of the players has a winning strategy) and that
the Axiom of Determinacy (stating that all games are determined)
contradicts the Axiom of Choice. With the former we recreate
Blackwell's groundbreaking proof of a classical result about
coanalytic sets. A family U of subsets of Baire space is
said to have the reduction property if for any B and C in
U, there are respective disjoint subsets B* of B and C*
of C in U with the same union as B and C; Blackwell proves
that the coanalytic sets have the reduction property. Blackwell's
new proof technique with this old result revitalized this
area of descriptive set theory and began the development
for a slew of new results.

Aug 23 
Jack Wright
Nonstandard Analysis and an Application to Combinatorial
Number Theory
Since nonstandard analysis was first formalized in the
60's it has given mathematicians a framework in which to
do rigorous analysis with infinitesimals rather than epsilons
and deltas. More importantly, it has also allowed for the
application of powerful techniques from logic and model
theory to analysis (and other areas of mathematics). This
brief presentation will outline some of those tools and
discuss one particular application of them.
I will briefly state the key techniques: the transfer principle,
the internal definition principle, and the overflow principle.
I will then give an indication of the usefulness of these
techniques by showing how they have been used to garner
some technical results that might be able to help solve
the Erd\H{o}s' famous Conjecture on Arithmetic progressions.

Aug 9 
Miguel Angel Mota
Instantiations of Club Guessing. Part I
We build a model where Weak Club Guessing fails, mho holds
and the continuum is larger than the second uncountable
cardinal. The dual of this result will be discussed in a
future talks.

Aug 2 
Carlos Uzcategui
Uniform Ramsey theoretic properties
The classical Ramsey theorem holds uniformly in the following
sense. There is a Borel map that for a given coloring of
pairs and an infinite set A, it selects an infinite homogeneous
subset of A.
This fact sugests that the notions of a selective, Frechet,
p+ and q+ ideal could also holds uniformly. We will discuss
about some of those uniform Ramsey theoretic properties.

Jul 19 
Todor Tsankov
On some generalizations of de Finetti's theorem
A permutation group G acting on a countable set M is called
oligomorphic if the action of G on M^n has only finitely
many orbits for each n. Those groups are well known to modeltheorists
as automorphism groups of omegacategorical structures.
In this talk, I will consider the question of classifying
all probability measures on [0, 1]^M invariant under the
natural action of the group G. A number of classical results
in probability theory due to de Finetti, RyllNardzewski,
Aldous, Hoover, Kallenberg, and others fit nicely into this
framework. I will describe a couple of new results in the
same spirit and a possible approach to carry out the classification
in general.

Jul 12 
Ari Brodsky
A theory of nonspecial trees, and a generalization of
the Balanced BaumgartnerHajnalTodorcevic Theorem (slide
presentation)
Building on early work by Stevo Todorcevic, we describe
a theory of nonspecial trees of successorcardinal height.
We define the diagonal union of subsets of a tree, as well
as normal ideals on a tree, and we characterize arbitrary
subsets of a tree as being either stationary or nonstationary.
We then use this theory to prove a partition relation for
trees:
THEOREM:
Let $\nu$ and $\kappa$ be cardinals such that $\nu ^ {<\kappa}
= \nu$, and let $T$ be a nonspecial tree of height $\nu^+$.
Then for any ordinal $\xi$ such that $2^{\left\xi\right}
< \kappa$, and finite $k$, we have $T \to (\kappa + \xi
)^2_k$.
This is a generalization of the Balanced BaumgartnerHajnalTodorcevic
Theorem, which is the special case of the above where the
tree $T$ is replaced by the cardinal $\nu^+$.

Jul 5

Jose Iovino
Definability and Banach space geometry
A well known problem in Banach space theory, posed by Tim
Gowers, is whether every Banach space that has an explicitly
definable norm must contain one of the classical sequence
spaces. I will discuss recent progress obtained jointly
with Chris Eagle.
