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\begin{center}{\huge On topological stability conjecture}\end{center}
\begin{center}{\large Ludomir Newelski (Wroclaw)}\end{center}

\bigskip

The subject is related to Vaught's conjecture. 
Assume $T$ is a complete theory, $M$ is 
a countable model of $T$ and $Q$ is a $0$-definable 
subset of $M$. Then ${\rm Aut}(Q)$ acts on $S(Q)$,
the orbits are called pseudo-types.
Notice that this is a Borel action of a 
Polish group on a Polish space,
and the orbits are Borel, too.
The topological stability conjecture asserts 
that if $T$ has < continuum countable models,
then all (good) pseudo-types are 
``topologically nice'' ($t$-stable). 
I proved several special cases of 
this conjecture, mainly for stable
and superstable theories. These special 
cases have interesting consequences,
which I will discuss.

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