Model theory

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.

In recent years the pure and applied sides of the subject have become intertwined, and when you begin your course here you will be given a grounding in both aspects (many universities offer a basic course in predicate calculus (up to the Completeness and Compactness Theorems) but we cater for those who have not done such a course). You will then be in a position to choose your own favourite topic for further study-either as a dissertation for the MSc,or else as a research topic for a PhD.