Stuart Burrell

Mathematics, University of St Andrews

Tenure since 2016

Dimension and Structure in Fractal FormsStuart Burrell

Fractals, objects that have fine structure at arbitrarily small scales, appear throughout nature, mathematics and science. Beautiful patterns that are at once simple and complex, they have captured the attention of artists and scientists alike.

In his seminal 1982 work, The Fractal Geometry of Nature, Mandelbrot captures the relevance of fractals for the study of nature in the passage: "Clouds are not spheres. mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line" ([1] pp. 1).  He argues that, as opposed to the straight lines of Euclid, the intricate forms of fractal geometry are at the heart of appreciating the "altogether different level of complexity" ([1] pp. 1) found in nature.

Traditional concepts in geometry break down for even the simplest fractal structures, objects such as the famous Koch snowflake having infinite length. To make a meaningful analysis of their structure a more subtle notion of size is required. For this reason, dimension theory plays a central role in the study of fractals. Dimension is a subtle concept, and one area my project may be to see what can be learnt from the interplay of different ways of defining dimension, perhaps in the context of dynamical systems or other objects with fractal properties.

[1] Mandelbrot, B. B. The Fractal Geometry of Nature. San Francisco, Henry Holt and Company, 1982. 


Biography

In 2016 Stuart graduated with a master's degree in Mathematics with a focus on pure mathematics. For his studies he was the recipient of a number of awards, including the Arthur Hinton Read Memorial, Sanderson and Duncan Prizes, in addition to the Principal's Scholarship for Academic Excellence, the University Scholarship for Research and Leadership, the Fourth/Fifth level Pure Mathematics Medal, the Deans' List 2012-2016, and was a 2016 Institute of Mathematics Graduate prize winner.