Model theory is a branch of mathematical logic which studies classes of mathematical structures. During the late
twentieth century the research extended from elementary classes to non-elementary classes, i.e. classes of
structures which are not axiomatizable in elementary logic. The main theme has been the attempt to generalize
tools from classification theory to cover more applications arising from other branches of mathematics. Several
different frameworks have been suggested and they are often incomparable in terms of generality.
In my Ph.D. work at the University of Helsinki I introduced the framework Finitary Abstract Elementary Classes
refining AECs introduced by Saharon Shelah. I am confident that this framework will turn out to be a crucial
innovation in generalizing classification theory to non-elementary classes. My work has been recognized
internationally and for the year 2007 I was invited to work as a visiting scholar at the University of Illinois at
Chicago with their distinguished group in model theory.
One goal of this project is to generalize tools from classification theory and especially geometric stability theory to
finitary AECs. Another goal is to compare different frameworks and find examples distinguishing and motivating
them.