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# Dr John Mackay

## Dr John Mackay

BSc(Edin.), PhD(Mich.)

Lecturer in Pure Mathematics
### Area of research

Geometric group theory and analysis on metric spaces
## Summary

## Ph.D. projects

## Biography

#### BSc (Hons) Mathematics

#### PhD Mathematics

## Keywords

## Memberships

### Organisations

### Pure Mathematics

### Probability, Analysis and Dynamics

### Research themes

## Recent publications

- All details
- Overview
- About
- Research
- Publications

BSc(Edin.), PhD(Mich.)

Office G.17

Fry BuildingWoodland Road,

BS8 1UG

(See a map)

+44 (0) 117 42 84823

john.mackay@bristol.ac.uk

Geometric group theory, particularly hyperbolic and relatively hyperbolic groups.

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

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:

University of Edinburgh (2003)

University of Michigan (2008)

- geometric group theory
- hyperbolic groups
- analysis on metric spaces

- Hume, D, Mackay, J & Tessera, R, 2019, ‘Poincaré profiles of groups and spaces’.
*Revista Matemática Iberoamericana*. - Druţu, C & Mackay, JM, 2019, ‘Random groups, random graphs and eigenvalues of p-Laplacians’.
*Advances in Mathematics*, vol 341., pp. 188-254 - Cashen, C & Mackay, J, 2019, ‘A metrizable topology on the contracting boundary of a group’.
*Transactions of the American Mathematical Society*. - Mackay, J & Sisto, A, 2019, ‘Quasi-hyperbolic planes in relatively hyperbolic groups’.
*Annales Academiae Scientiarum Fennicae Mathematica*. - Kleiner, B & Mackay, JM, 2016, ‘Differentiable structures on metric measure spaces: A primer’.
*Annali della Scuola Normale Superiore di Pisa - Classe di Scienze*, vol XVI., pp. 41-64 - Duchin, M, Jankiewicz, K, Klimer, S, Lelièvre, S, Mackay, J & Sánchez, A, 2016, ‘A sharper threshold for random groups at density one-half’.
*Groups, Geometry and Dynamics*, vol 10., pp. 985-1005 - Mackay, JM, 2016, ‘Conformal dimension via subcomplexes for small cancellation and random groups’.
*Mathematische Annalen*, vol 364., pp. 937-982 - Mackay, JM & Przytycki, P, 2015, ‘Balanced walls for random groups’.
*Michigan Mathematical Journal*, vol 64., pp. 397-419 - Mackay, JM, 2014, ‘Quasi-circles through prescribed points’.
*Indiana University Mathematics Journal*, vol 63., pp. 403-417 - Mackay, JM, Tyson, JT & Wildrick, K, 2013, ‘Modulus and Poincaré inequalities on non-self-similar Sierpiński carpets’.
*Geometric and Functional Analysis*, vol 23., pp. 985-1034

View complete publications list in the University of Bristol publications system

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