ABSTRACT: I will talk about my research that was originally
motivated by the question of when does a simple theory have the
property of $n$-dimensional amalgamation (where 2-dimensional
amalgamation corresponds to the Independence theorem for simple
theories). To answer that question, I develop the notion of
$n$-simplicity for $1\le n\le \omega$, where ``1-simple'' is the
same as ``simple.'' I will present the definitions and give
examples of simple unstable theories in each subclass. I sketch
the proofs of 3-dimensional amalgamation property for 2-simple
theories, and, under an additional assumption, $(n+1)$-dimensional
amalgamation property for $n$-simple theories. I will also talk
about some open problems.