Structural model theory, definable measures and representations

Model theory is a branch of mathematical logic which has proved to be extremely versatile, allowing new insights into many different areas of mathematics, ranging from all core topics in algebra to geometry, analysis and beyond. Model theory has also benefited from different branches of mathematics which have allowed new insights into purely model theoretic objects.
Among the different flavours of model theory, structural model theory is interested in classifying different theories thanks to purely combinatorial concepts such as omega-categoricity, stability or independence. These concepts run deep through mathematics, with examples of stable structures ranging from diverse classes of fields to free groups, for example. Stability and omega-categoricity have proved to be especially fruitful concepts for model theory as they allow useful characterisations of the fine structure of models with these properties. For example, by the classical Ryll-Nardzewski theorem and related results, the automorphism group of the countable model of an omega-categorical theory determines its model theoretic properties entirely. Another useful result is that countable omega-stable theories have saturated models in every cardinality and these yield valuable information about the whole class of models of such a theory.
My research will begin with an investigation of the feasibility of sharpening our understanding of automorphism groups of various classes of models thanks to representation theory. This question has been largely unexplored so far, with the main result currently coming from the article 'Unitary Representations of Oligomorphic Groups', Todor Tsankov, 2012. In this article, the author proves a strong theorem about the unitary representations of automorphism groups of countable models of omega-categorical theories with weak elimination of imaginaries. This promising result raises the question of whether anything can be said about the representations of different kinds of automorphism groups, such as those which are associated with homogeneous models of stable non-omega-categorical theories.
The prospect of building a representation theory of groups which occur naturally in model theory raises the dual question of whether it is possible to give a model theoretic account of representations. This will be the second strand of my research. This topic has been largely untouched but it seems that it might be possible to give an insightful treatment of unitary representations in the context of continuous logic. As a starting point, there is a rich literature on measures in model theory. These have been studied thoroughly in conjunction with stability and NIP thanks to continuous logic. Moreover, it is well-known that a group G which acts on a measure space X gives rise naturally to a unitary representation of G on L^2(X). I will investigate under what conditions on a definable measure, invariant under a definable group, one finds in a similar way a representation, and how structural properties such as NIP and stability bear on these representations.
These two strands in my research - the representation theory of automorphism groups and a model theoretic treatment of representations - are connected by a general duality which has been observed between automorphism groups of models and definable groups. Although this duality is rather imprecise, one can hope that a model theoretic understanding of representations can shed light on this duality, and this could potentially lead to a new understanding of problems in the representation theory of groups which occur across different areas of mathematics. Therefore, one can hope that a model theoretic understanding of unitary representations could pave the way for further interactions between model theory and algebra.
This project falls within the EPSRC Logic and Combinatorics, and Algebra research areas.