Aurélien ALVAREZ
Orbit equivalence and free product decompositions
If a countable group acts freely, measure preservingly and ergodically on the standard non-atomic probability space, it gives rise to a type II_1 measure equivalence relation. There is a well defined notion of free product decomposition for such an equivalence relation and we are interested in rigidity results in that context. Namely if two groups admiting free product decompositions give rise to the same measure equivalence relation, does there exist a one-to-one correspondance between the measure equivalence relations induced by the factors ? We shall discuss and give results to this question.
Riccardo CAMERLO
Quotients of projective Fraïssé
limits
In this talk I will characterise all topological spaces that can be obtained
as quotients of projective Fraïssé limits of a projective Fraïssé
family of finite structures with a binary relation.
Márton ELEKES
Solvability cardinals
The following research was originally motivated by the theory of liftings.
This is a joint work with M. Laczkovich.
Let $\mathbb{R}^\mathbb{R}$ denote the set of real valued functions defined
on the real line.
A map $D: \mathbb{R}^\mathbb{R} \to \mathbb{R}^\mathbb{R}$ is said to
be a {\it difference operator}, if there are real numbers $a_i , b_i
\ (i=1,\ldots , n)$ such that $(Df)(x)=\sum_{i=1}^n a_i f(x+b_i)$ for
every $f\in \mathbb{R}^\mathbb{R} $ and $x\in \mathbb{R}$.
By a {\it system of difference equations} we mean a set of equations
$S=\{ D_i f=g_i : i\in I\}$, where $I$ is an arbitrary set of indices,
$D_i$ is a difference operator and $g_i$ is a given function for
every $i\in I$, and $f$ is the unknown function.
One can prove that a system $S$ is solvable if and only if every finite
subsystem of $S$ is solvable.
However, if we look for solutions belonging to a given class of functions
then the analogous statement is no longer true. For example, there exists
a system $S$ such that every finite subsystem of $S$ has a solution which
is a trigonometric polynomial, but $S$ has no such solutions; in fact, $S$
has no measurable solutions.
This phenomenon motivates the following definition. Let ${\cal F}$ be
a class of functions. The {\it solvability cardinal} ${\rm sc} (\cal F)$
of ${\cal F}$ is the smallest cardinal number $\kappa$ such that whenever
$S$ is a system of difference equations and each subsystem of $S$ of cardinality
ess than $\kappa$ has a solution in ${\cal F}$, then $S$ itself
has a solution in ${\cal F}$.
In this talk we will present the solvability cardinals of most function
classes that occur in analysis. As it turns out, the behaviour of ${\rm
sc} (\cal F)$ is rather erratic. For example, ${\rm sc} (\textrm{polynomials})=3$
but ${\rm sc} (\textrm{trigonometric polynomials})=\omega _1$,
${\rm sc} (\{ f: f\ \textrm{is continuous}\}) = \omega_1$ but ${\rm sc}
(\{ f: f\ \textrm{is Darboux}\}) =(2^\omega )^+$, and ${\rm sc} (\mathbb{R}^\mathbb{R}
)=\omega$.
We will also consistently determine solvability cardinals of the classes
of Borel, Lebesgue and Baire measurable
functions, and give some partial answers for the Baire class 1 and Baire
class $\alpha$ functions. We will also pose the intriguing open problem concerning
the possibility of defining natural ranks for the Baire class $\alpha$ functions.
Vassilis KANELLOPOULOS
Hausdorff measures associated
to functions of square bounded variation.
For each function of square bounded variation we define a Hausdorff-type
measure on the unit interval. We will present some properties and applications
of this measure to the structure of subspaces of James Function space.
Witold MARCISZEWSKI
On
Banach spaces whose norm-open sets are weakly $F_\sigma$-sets
We will discuss the results concerning non-separable Banach spaces whose
norm-open sets are countable unions of sets closed in the weak topology and
a narrower class of Banach spaces with a network for the norm topology which
is $\sigma$-discrete in the weak topology. In particular, we answer a question
of Arhangel'skii exhibiting various examples of non-separable function spaces
$C(K)$ with a $\sigma$-discrete network for the pointwise topology and (consistently)
we answer some questions of Edgar and Oncina.
concerning Borel structures and Kadec renormings in Banach spaces.
Julien MELLERAY
How to recognize an isometry?
I'll discuss what's known about the following problem: given a homeomorphism
f of a Polish metric space X, when is there a compatible distance d on X
such that f is a d-isometry?
Note: the question mark in the title is important - I definitely don't
have an answer to that question!
Luca MOTTO ROS
Analytic equivalence relations and
bi-embeddability.
The analysis of the structure of the analytic equivalence relations (ER
for short) under Borel-reducibility has been one of the most relevant subject
in the recent history of Descriptive Set Theory. Among ERs, a special place
is occupied by the isomorphism relation on countable models of a certain
$\mathcal{L}_{\omega_1 \omega}$-sentence $\varphi$. However, isomorphism relations
are just a special case of ER, as there are many example of ERs which are
not Borel-reducible to an isomorphism relation. On the contrary, we will
show in this talk that the bi-embeddability (resp. bi-homomorphism, bi-weak-homomorphism)
relation is able to capture the whole complexity of the ER structure: for
every ER $E$ there is an $\mathcal{L}_{\omega_1 \omega}$-sentence $\varphi$
such that $E$ is Borel-equivalent to bi-embeddability (resp. bi-homomorphism,
bi-weak-homomorphism) on the collection of countable models of $\varphi$.
This is joint work with Sy D. Friedman.
Christian ROSENDAL
A simple proof
of Gowers' dichotomy theorem
We prove an exact, i.e., formulated without Delta-expansions, Ramsey principle
for infinite block sequences in vector spaces over countable fields, where
the two sides of the dichotomic principle are represented by respectively
winning strategies in Gowers’ block sequence game and winning strategies
in the finite asymptotic game. This allows us to recover Gowers’ dichotomy
theorem for block sequences in normed vector spaces by a simple application
of the basic determinacy theorem for infinite asymptotic games.
Brian SEMMES
A game for the Borel functions
In this talk, I summarize my thesis work.
I introduce a Wadge-style game G(f) which characterizes the Borel functions
on the Baire space, in the sense that Player II has a winning strategy in
G(f) iff f is Borel. By tinkering with the rules of this game, it is
possible to characterize subclasses of Borel functions, including Baire classes
1 and 2. Using game-theoretic methods, I prove decomposition theorems
for two subclasses of Baire class 2, extending a result of Jayne and Rogers
(1982).
Miroslav ZELENY
Additive
families of low Borel classes
An important conjecture in the theory of Borel sets in non-separable metric
spaces is whether any point-countable Borel-additive family in a complete
metric spaces has a $\sigma$-discrete refinement. We confirm the conjecture
for point-countable $F_{\sigma\delta}$-additive families, generalizing thus
the result of R.W. Hansell.
