
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



2014
Fridays

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

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.

201415

Past Seminars
Speaker and Talk Title

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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