**M.Sc./Ph.D. projects on Geometric Analysis**

Professor Stephen Buckley

Professor. Buckley's research is centred on various areas of classical and geometric analysis. The following are some suggested topics for research. In each case there are suitable questions for either a Master's or PhD thesis. To do a Master's degree in any of these topics, you need a good honours degree in Mathematics. Doing a Master's on one of these topics will bring you to a level where you can contemplate doing a PhD in a related topic.

*Poincaré-type inequalities*

In the broadest sense, these inequalities are concerned with controlling the variation of a function in a set by the size of its derivative on that set or a larger related set. Both the "variation" and the "size" mentioned here are measured in some average sense (usually different in the two cases). Typical questions involve proving such inequalities for new types of weights, sets, or geometries, or proving that there is a link between the existence of such inequalities and certain geometric properties of the set.

*Quasiconformal mappings and mappings of finite distortion*

Recent work by Heinonen and Koskela has led to a satisfactory theory of quasiconformal mappings on metric spaces of Loewner type. These spaces are essentially characterised by a certain type of Poincaré inequality, and are thus related to Sobolev spaces. One large class of problems involves the investigation of such Sobolev spaces in a general metric space setting, and the implications for quasiconformal theory that arise as a consequence. A very different type of problem involves the study in a Euclidean setting of mappings of finite distortion. Such mappings generalise quasiconformal and quasiregular mappings and have become a hot topic in the last few years.

*Gromov hyperbolicity*

A Gromov hyperbolic space is a space with a type of negatively curved non-Euclidean geometry, which can be defined in a rather general metric space setting. It originated as a unifying concept that allowed the study of various classes of finitely generated groups (such as fundamental groups of many compact manifolds, and so-called small cancellation groups) that are not abelian or in any of the "close-to-abelian" classes such as nilpotent or amenable. In the last few years though, it has found applications within geometric analysis.

One type of problem involves studying the relationship between the Gromov space itself and its Gromov boundary in various settings where these geometries occur. Another involves the attempt to understand when certain geometries associated with classical metrics on domains in Euclidean space or
**C**^{n} are Gromov; the metrics I have in mind include the quasihyperbolic, Bergman, Caratheodory, and Kobayashi metrics. Very little is known about this latter type of question, so the field is wide open. For instance, I recently discovered a link in the quasihyperbolic setting between Gromov hyperbolicity, slice conditions (along the lines of conditions used in problems involving Poincaré inequalities), and separation and Gehring-Hayman conditions, and I think there is much more that can be done in this direction.