Geometric group theory, particularly hyperbolic and relatively hyperbolic groups.  

Analysis on metric spaces, especially questions related to Hausdorff dimension and quasisymmetric maps.

Ph.D. projects

The two main themes of my research are geometric group theory and analysis on metric spaces.

Geometric group theory involves the study of infinite, finitely generated groups by considering how they act on appropriate spaces.  I am particularly interested in Gromov’s hyperbolic groups, and how they can be studied using their “boundary at infinity”.  (For example, three dimensional hyperbolic space, in the Poincaré ball model, has a natural sphere at infinity.)  These boundaries are metric spaces, usually fractal, and may carry a rich analytic structure.  The key question is to relate the algebraic properties of such groups with the analytic properties of their boundaries.  Interesting examples of hyperbolic groups include Gromov’s “random groups” and many examples from low dimensional topology.

I am particularly interested in the conformal dimension of the boundary.  This is a variation on Hausdorff dimension due to Pansu.  There are many spaces of interest where this dimension is not known, or even well estimated.  Conformal dimension links to my other main interest, analysis on metric spaces.  This involves the study of (non-smooth) functions on metric spaces that have no given smooth structure, but satisfy some weaker conditions.  This is motivated first by applications where the spaces that arise have only weak regularity.  A second motivation arises from the desire to understand classical results better by finding out exactly what hypotheses they require.

If you are interested in discussing potential projects in these areas, please do contact me. For more information, see my webpage:

http://www.maths.bris.ac.uk/ ̃jm13806/