I work in analysis but with most of my research concerning fractal geometry. I am also interested in connections between fractal geometry and wider mathematics, especially
ergodic theory, dynamical systems, geometric measure theory, hyperbolic geometry, Fourier analysis and probability theory. Roughly speaking, a
fractal
is an object or process which exhibits complexity over a large range of scales, see some examples below. An important way to characterise fractals is via their
dimension. In fact, it is already a fascinating problem to
define the dimension of a fractal.
Here are some of the more specific research topics I am interested in. Follow the links to find out more!
Left: a self-affine carpet of the type introduced by Bedford and McMullen. Its lower, Hausdorff, box, and Assouad dimension are all distinct.
Centre: the Apollonian circle packing. It can be realised as the limit set of a Kleinian group.
Right: the graph of the popcorn function. Together with Haipeng Chen and Han Yu I proved that it has box dimension 4/3.