Independence groups and measures in model theory

This project is in model theory, a branch of mathematical logic. Model theory studies structures and their definable sets. A structure is roughly speaking an (underlying) set, together with some collection of distinguished sets of n-tuples from the set, for varying n. For example one could take the underlying set to be the set of real numbers (the real line) and the distinguished subsets of the plane to be segments of straight lines and regions enclosed by these. Many parts of mathematics can be fruitfully studied as structures.An important part of the project is to study when the definable or distinguished sets in a structure can be assigned measures, in a reasonably coherent way. In the example above one could take the measure of a region in the plane to be its area. This example belongs to the first class of stuctures we are studying where we want to prove that there is exactly one way of assigning measures to distinguished sets. We also study a second broad class of structures, important examples of which are limits in some natural sense, of finite structures, and where counting in the finite structures can be used to assign measures to distinguished sets in the limit. We hope to use the measures in these cases to obtain an understanding of certain algebraic objects such as groups, which are definable in the structures concerned.