The Nobel Prize in Chemistry 2011 was awarded to Dan Shechtman for his 1982 discovery of a new form of matter, now known as quasicrystals. Their structure and properties challenged the foundations of crystallography because, according to the laws of classical crystallography, such materials simply could not exist. This talk will offer a glimpse at the underlying mathematical structure of aperiodic order. The mathematics of aperiodic order describes the construction and the properties of aperiodically ordered structures in space, such as point sets or tilings. Some of their intriguing properties will be explained using simple examples and lots of pictures.

Fractals are sets that have a very intricate structure and contain many copies of themselves at different scales. They are not easily described using classical geometry and so a new theory of fractal geometry has been developed. Such sets have attracted increasing attention in recent years, partly due to the development of striking computer graphics and partly due to a realisation that many natural phenomena such as cloud boundaries and coastlines are best approximated by fractals. This lecture will explore the nature of a variety of fractals, with particular interest in those arising through the repeated application of a function involving complex numbers - the theory of which is known as complex dynamics.