Groups definable in tame expansions of o-minimal structures

Summary

This project lies at the nexus of model theory (mathematical logic), group theory and combinatorics. The main objects of study are groups definable in various structures, which can be of topological/geometric nature, such as o-minimal structures and tame expansions of them, or more generally of combinatorial nature, such as structures with NIP (not the independence property). The NIP property is also of interest to statistics and machine learning.

In the o-minimal setting, definable groups have been fairly understood with perhaps the most notable result being the solution of Pillay’s conjecture, which draws an explicit connection between those groups and real Lie groups. Extensions of Pillay’s conjecture in one of many possible expansions of o-minimal structures will be investigated in this project. Concrete examples include the expansion of the real field by a predicate for (a) the set of real algebraic numbers, (b) a dense independent set, (c) the set of all rational powers of 2, (d) the set of all integer powers of 2, or (e) any subgroup of the real multiplicative group with the Mann property. Those settings have recently seen the development of model-theoretic tools, which will be used in this project.

In the more general, NIP setting, fewer tools are available, and concrete questions will involve first the development of an understanding of definable sets in NIP expansions of o-minimal structures, and then its application to the study of definable groups.

The successful applicant will benefit from a large and exceptionally vibrant research group in mathematical logic, including 7 permanent academic staff with expertise in model theory, set theory, recursion theory, proof theory, categorical logic, and logic in computer science. The logic group consistently also includes several postdocs and PhD students, runs 4 regular seminar series and is a node of several regional and international research networks. The group is an active participant in the MAGIC consortium, which provides specialist lecture courses for mathematics postgraduates at a network of 20 UK Universities.

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