Outline:

Fractals exhibit disorder on all scales. As a simple example of such, consider the exactly self-similar Sierpinski gasket or carpet. Increasing the complexity, one might introduce randomness into the construction, so that the self-similarity is only statistical. Studying the structure of fractals, both deterministic and random, and investigating stochastic processes upon them has demanded the development of a range of novel techniques. Whilst these have answered a number of interesting questions (e.g. how far does the Brownian motion on a Sierpinski gasket travel in a given time?), there are still many interesting open problems remaining (e.g. the same problem for the Sierpinski carpet is open). I am happy to discuss further details of possible research questions with interested candidates. Those wishing to study such problems with me should have a strong background in probability and/or analysis.