2014
Fridays 
Upcoming Seminars
at 1:30 p.m. in the Fields Institute, Room 210

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.

201314

Past Seminars
Speaker and Talk Title

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.
