Model theory is the study of the relationship between the language of mathematics and the actual world of mathematical objects and structures. It is one of the most active areas of logic and has many connections with other areas of mathematics. For example, the last fifteen years have seen remarkable applications to diophantine geometry and analysis, as well as strong interactions with many topics in algebra.
Current research focusses on definability theory-the study of objects that can be precisely defined using the logical language known as the first-order predicate calculus. The classical machinery of definability theory was developed by about 1970. A divergence then developed.
On the one hand, in a monumental piece of pure model theory for which Saharon Shelah was the driving force, definable objects were shown to be either classifiable by certain notions of dimension (the theory of vector spaces is a very special case), or else completely wild and unclassifiable.
On the other hand, applied model theory, extending much earlier work of Alfred Tarski and Abraham Robinson, has concentrated on the detailed model-theoretic structure of tame (though not necessarily classifiable in Shelah's very abstract sense) mathematically interesting structures such as specific fields (eg the real or complex or p-adic numbers), groups or rings. Often these are endowed with extra structure such as automorphisms, derivations, valuations or exponential (or other transcendental) functions.
Our research interests can be roughly listed as follows.
- Modules and categories
- o-minimal structures
- Model theoretic methods in real algebraic geometry
- Groups of Finite Morley Rank