Speaker: Tim Large (Cambridge undergrad) Abstract: We all know what it means to reflect about a line. But what about reflection in more general objects - in particular, what about a circle? The idea of ’reflecting in a circle’ sounds nonsensical, but it gives rise to the notion of geometric inversion. And unlike normal reflection, this does really strange things to the plane - lines can become circles and circles can become lines! But it turns out this can actually be an incredibly powerful tool in normal Euclidean geometry, even providing a neat proof of Pythagoras’ theorem. But what will seem at first a rather bizarre and unusual notion turns out to be not so unusual at all when we look at the plane in a slightly different way - with the aid of the stereographic projection - thus entering the world of non-Euclidean geometry. It turns out this even provides a crucial link between simple and elementary geometry, and the deep world of the complex plane - something hitherto almost entirely algebraic.