When one wants to describe the symmetries of any object or system, in mathematics or everyday life, the right language to use is group theory. How might one go about understanding the universe of all groups and what kinds of novel geometry might emerge as we explore this universe?
The understanding of the possible geometries in dimension 3 is one of the triumphs of 20th century mathematics. In his Chairman's Inaugural Public Lecture, Professor Martin Bridson explains this triumph and why such an understanding is impossible in higher dimensions.