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\begin{center}{\huge Simple \ Theories}\end{center}
\begin{center}{\large Byunghan Kim}\end{center}
\begin{center}{\large The Fields Institute}\end{center}

\bigskip

We recall  that  a first order theory
$T$ is said to be {\it simple} if , for any complete 
type $p\in S(B)$,  $p$ does not fork
over some set $A(\subseteq B)$ with $|A|\leq |T|$.  
It is well known that every
stable theory is simple.

The study of simple theories began with Shelah's paper [1]. 
In [2], remarkable progress was made. There,
it was shown that in simple theories, forking coincides with 
dividing  and nonforking
satisfies all the independence axioms of nonforking in 
stable theories, except the uniqueness of 
nonforking extension over a model ( uniqueness axiom).

Furthermore, in [3], it was shown that for simple 
theories, the right substitute for uniqueness axiom is 
``{\it the Independence Theorem over a model}'' which 
is a weaker form of the uniqueness axiom.
 Also it was shown conversely that
any theory equipped with a notion of independence 
satisfying all the basic axioms together with the Independence Theorem
over a model, should be simple, and that the 
independence notion is exactly nonforking. 
Moreover, the correct notion of
{\it strong type} in simple theories appears to be 
{\it Lascar strong type}, which is originally 
introduced by Lascar.
So the Independence Theorem for Lascar strong types is 
proved. Also in [4], it is proved that for a simple small $T$,
the Independence Theorem holds over an arbitrary 
algebraically closed set in 
$\cal{C}$$^{eq}$.

The first result, in relation to computing  the 
number of countable models in simple theories context, 
is proved in [5] recently. There, Lachlan's
famous theorem on  superstable theories is extended 
to a full class of  supersimple theories.

Recent model theoretic research on specific 
unstable structures such as
smoothly approximable structures, 
pseudo-finite fields and fields with an 
automorphism gives rise to another justification 
for the work on simple theories. These structures 
turn out to be simple and nonforking supplies a
good concept of independence as I mentioned previously. 
The model theoretic analysis of these
structures is of interest both for its own sake, as 
well as for applications such as to diophantine
geometry in the case of fields with an automorphism, 
or to arithmetic groups in the case of
pseudo-finite fields.

In conclusion, it can be said that  [2], [3] open up 
new research areas of simple theories which give one 
step higher hierarchy to the stability
classification. Accordingly many questions can be 
asked about simple theories trying to generalize
results from stable theories. Currently, the notion 
of canonical base for simple theories is found,
and the groups definable in simple theoreis 
are being studied.
\bigskip

\section*{References}

\noindent\hang[1] S. Shelah `Simple unstable theories', 
{\it Ann. Math. Logic} 19 , pp.177-203.
\smallskip

\noindent\hang[2] B. Kim, `Forking in simple 
unstable theories', to appear in
{\it J. of London Math. Soc}.
\smallskip

\noindent\hang[3] B. Kim and A. Pillay, `Simple theories', 
to appear in {\it Ann. of Pure
and Applied Logic.} 
\smallskip

\noindent\hang[4] B. Kim, `A note on Lascar strong types 
in simple theoreis', to appear in {\it J. of Symbolic Logic}.
\smallskip

\noindent\hang[5] B. Kim, `On the number of countable 
models of a countable supersimple theory', submitted.

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