This PhD project is in geometric group theory: a young, diverse, active and fascinating branch of mathematics at the intersection of algebra, geometry, combinatorics and topology. The main goal of geometric group theory is to reveal new relationships between algebraic properties of groups and the geometric properties of spaces they act on, and to use these techniques to answer long-standing problems in adjacent areas of mathematics and computer science.
Among the most fundamental relationships between the algebraic and geometric properties of groups is that any group homomorphism with closed image and bounded kernel is a coarse embedding with respect to any choice of proper left invariant metrics on the groups. Coarse embeddings between metric spaces are interesting in their own right, and arise just as naturally in many other disciplines, such as topology, Lorentzian geometry and theoretical computer science.
Despite this, coarse embeddings between groups are incredibly poorly understood. A major reason for this is that there are few methods to obstruct coarse embeddings. A monotone coarse invariant (MCI) is an invariant which behaves monotonically with respect to coarse embeddings – growth and asymptotic dimension are two classical examples of such invariants. In the last ten years, the number and diversity of MCIs has increased dramatically, with new MCIs inspired by techniques from combinatorics, analysis, topology, electrical network theory and neural networks. The goal of the project is to further develop these MCIs as tools to tackle key problems in geometric group theory, fractal geometry and beyond.
Prospective students should have completed courses in group theory and metric spaces. Experience in any of the following is also desirable: algebraic topology, analysis, combinatorics, (non-Euclidean) geometry, Lie theory.
To apply, please submit an application via our website and state that you are applying for this project in your application: http://www.bristol.ac.uk/study/postgraduate/2022/sci/phd-mathematics/
If you have any queries please email [Email Address Removed]
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