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



2015
Fridays

Seminars

June 26, 2015

Franklin Tall, PFA(S)[S] II
This is a continuation of last week's lecture. Last week's lecture
was largely motivation; this lecture will be mainly technical, developing
the method. If you really want to attend and missed last week, contact
me and I will give you something to read.

June 19, 2015

Franklin Tall, PFA(S)[S] and locally countable subspaces
of compact countably tight spaces.
I have lectured many times in the seminar on Stevo’s method
of forcing with a coherent Souslin tree S over a model of PFA restricted
to posets that preserve S, since it has many interesting applications
in settheoretic topology. However I believe the current cohort of
graduate students has not seen an actual proof of this sort. Since
the seminar is suffering from a lack of speakers, I plan to give a
sporadic series of lectures featuring such proofs. In particular,
as soon as I understand it sufficiently well, I want to give Alan
Dow’s proof that in such models, first countable perfect preimages
of omega_1 include copies of omega_1. This is the capstone of the
proof of the consistency of every hereditarily normal manifold of
dimension > 1 being metrizable. First of all, however, I want to
prove a technical theorem that is necessary for the manifold result,
and for many other results concerning under what conditions locally
compact normal spaces are paracompact. This particular theorem –
getting locally countable collections to be sigmadiscrete  is perhaps
not of wide interest, but the method of getting an uncountable set
in such a model to be the union of countably many “nice”
subsets (rather than just including an uncountable nice subset) should
have more applications. The “proof” I gave of this result
in the seminar five years ago turned out to have a gap. The gap is
bridged by a clever idea of Stevo. The proof will appear in a joint
paper.

June 12, 2015

Asger Törnquist, Definable maximal orthogonal families
in forcing extensions
Two Borel probability measures nu and mu on Cantor space are orthogonal
if there is a Borel set which has measure 1 for nu, but measure 0
for mu. An orthogonal family of measures is a family of pairwise orthogonal
measures; it is maximal if it is maximal under inclusion.
Maximal orthogonal families of measures can't be analytic; this
is a theorem of Preiss and Rataj (1985). A few years ago, Vera Fischer
and I showed that in L there is a Pi11 (lightface) maximal orthogonal
family (a "mof") of measures in L, but that adding a Cohen
real to L destroys all Pi11 mofs. Subsequently, it was shown that
the same holds if we add a random real (FriedmanFischerT.).
This motivated the question: Can a Pi11 mof coexist with a nonconstructible
real? In this talk we answer this by showing there is a Pi11 mof
in the Sacks and Miller extensions of L. By contrast, we will see
that in the Mathias extension of L there are no Pi11 mofs, and in
the process of doing so we will obtain a new proof of the PreissRataj
theorem.
This is joint work with David Schrittesser.

May 22, 2015

Francisco Kibedi, Maximal Saturated Linear Orders
In his 1907 paper about pantachies (maximal linearly ordered
subsets of the space of realvalued sequences partially ordered by eventual
domination), Felix Hausdorff poses several questions that he was unable
to answer, including a question he labels $(\alpha)$: Is there a pantachie
with no $(\omega_1, \omega_1)$gaps?
Hausdorff knew that CH implies the answer is no; in other words,
under CH, a pantachie must have $(\omega_1, \omega_1)$gaps. However,
Hausdorff's question turns out to be independent of ZFC. We answer
question $(\alpha)$ by proving something a bit stronger, namely, Con(ZFC
+ $\lnot$CH + $\exists$ a maximal saturated linear order of size continuum
in the space of realvalued sequences partially ordered by eventual
domination). We then extend this result to include Martin's Axiom
 i.e., we prove Con(ZFC + MA + $\lnot$CH + $\exists$ a maximal
saturated linear order of size continuum in the space of realvalued
sequences partially ordered by eventual domination).
Note: This seminar will be held in the Bahen Center, room BA 1220.

May 1, 2015

Alessandro Vignati
Forcing axioms and Operator algebras: a lifting theorem for reduced
products of matrix algebras
Inspired by the work of Farah and others in the application of forcing
axioms to operator algebras, we prove a correspondent of a lifting
theorem in a continuous setting. Analyzing different kinds of maps
from the reduced product of matrix algebra into a corona of a nuclear
C*algebra, we provide different notions of wellbehaved lifting,
and we show how forcing axioms imply their existence, in contrast
to the results obtained under the Continuum Hypothesis. Secondly,
we show some consequences of such a behavior. All required definitions
will be given. This is joint work with Paul McKenney.

April 17, 2015

Robert Raphael
On the countable lifting property for C(X)
Suppose that Y is a subspace of a Tychonoff space X so that the induced
ring homomorphism $C(X) \rightarrow C(Y)$ is onto. We show that a
countable set of pairwise orthogonal functions in C(Y) can be lifted
to a pairwise orthogonal preimage in C(X). The question originally
arose in vector lattices. Topping published the result for vector
lattices using an erroneous induction, but two years later Conrad
gave a counterexample. This is joint work with A.W.Hager.

April 10, 2015

Christopher Eagle
Model Theory of Compacta
Topological spaces do not fit well into the framework of firstorder
model theory; nevertheless, tools from model theory have had some
success in applications to compacta. Modeltheoretic ideas have been
used in topology in two ways: First, by finding suitable firstorder
structures to use as standins for topological spaces, and second,
by directly "dualizing" notions from model theory. We will
describe both of these methods, and compare them to a newer approach
which applies realvalued logic to the rings of continuous complexvalued
functions of compact spaces.
Using continuous logic we show that the pseudoarc is a coexistentially
closed continuum, answering a question of P. Bankston. We also show
that the only compact metrizable spaces $X$ where $C(X)$ has quantifier
elimination in continuous logic are the onepoint space, the twopoint
space, and the Cantor set. This is joint work with Isaac Goldbring
and Alessandro Vignati.

March 27, 2015

Dana Bartosova
About the conjecture that oligomorphic groups have metrizableuniversal
minimal flows
We will discuss a conjecture of Lionel Nguyen van Th\'e as in the
title. It was shown by Andy Zucker to be equivalent to whether every
class of finitary approximations of a countable ultrahomogeneous structure
with oligomorphic automorphism group has a finite Ramsey degree. We
look at the problem from the Boolean algebra point of view. An interesting
example in this context is the automorphism group of a topological
structure whose natural quotient is the pseudoarc, which is a work
in progress with Aleksandra Kwiatkowksa (UCLA).

March 20, 2015
Room 230

Mike Pawliuk
Amenability and Directed Graphs Part 2 : Cherlin's List
Last week Miodrag spoke in general about Amenability, Fraisse classes
and consistent random expansions. This talk will be more specific
and focus on checking the amenability and unique ergodicity of the
automorphism groups of the directed graphs on Cherlin's list. In addition,
we will present a type of product of Fraisse classes that behaves
nicely with respect to amenability and unique ergodicity.

March 13, 2015 
Miodrag Sokic
Amenability and directed graphs
Amenability for locally compact and countable groups has been extensively
studied. In this talk we will give some results in the case of nonarchimedean
groups. In particular, we consider groups of anthropomorphism of structures
from the Cherlin list of ultrahomogeneous directed graphs.

February 27, 2015 
Frank Tall
Some observations on the Baireness of C_k(X) for a locally compact
space X
The area inbetween Empty not having a winning strategy and Nonempty
having a winning strategy in the BanachMazur game has attracted interest
for many decades. We answer some questions Marion Scheepers asked
when he was here last year, and also prove results related to his
recent paper with Galvin and to a paper of Gruenhage and Ma. Our tools
include PFA(S)[S] and nonreflecting stationary sets.

February 27, 2015
12:00pm

Saeed Ghasemi
Rigidity of corona algebras
In my thesis I use techniques from set theory and model theory to
study the isomorphisms between certain classes of C*algebras. In
particular we look at the isomorphisms between corona algebras of
direct sums of sequences of full matrix algebras. We will see that
the question "whether any isomorphism between these C*algebras
is trivial" is independent from the usual axioms of set theory
(ZFC). I also extend the classical FefermanVaught theorem to reduced
products of metric structures. This theorem has a number of interesting
consequences. In particular it implies that the reduced powers of
elementarily equivalent structures are elementarily equivalent. We
also use this to find examples of corona algebras of direct sums of
sequences of full matrix algebras which are nontrivially isomorphic
under the Continuum Hypothesis. This gives the first example of genuinely
noncommutative structures with this property.
In the last chapter of my thesis I have shown that SAW*algebras
are not isomorphic to tensor products of two infinite dimensional
C*algebras, for any C*tensor product. This answers a question of
S. Wassermann who asked whether the Calkin algebra has this property.

February 11, 2015
3:30pm
Stewart Library

Alessandro Vignati
Set theory and amenable operator algebras
I will present my past and present work on logic and operator algebras.
First I will show the construction of a nonseparable amenable operator
algebra A with the property that every nonseparable subalgebra of
A is not isomorphic to a C*algebra, yet A is an inductive limit of
algebras isomorphic to C*algebras. Secondly, I will sketch possible
techniques, associated to Model Theory in a continuous setting, that
can be applied to operator algebras.

February 6, 2015
Room 230

Logan Hoehn
A complete classification of homogeneous plane compacta
In this topology talk, we will discuss homogeneous spaces in the
plane $R^2$. A space X is homogeneous if for every pair of points
in X, there is a homeomorphism of X to itself taking one point to
the other. Kuratowski and Knaster asked in 1920 whether the circle
is the only connected homogeneous compact space in the plane. Explorations
of this problem fueled a significant amount of research in continuum
theory, and among other things, led to the discovery of two new homogeneous
spaces in the plane: the pseudoarc and the circle of pseudoarcs.
I will describe our recent result which implies that there are no
more undiscovered homogeneous compact spaces in the plane. This is
joint work with Lex Oversteegen of the University of Alabama at Birmingham.

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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